[{"content":"In this section we first define the connective K-theory spectrum $\\mathrm{k}(\\mathcal{C})$ of a category with finite colimits, then extend it to the non-connective K-theory spectrum $\\mathrm{K}(\\mathcal{C})$.\nConnective K-theory Construction 1 (Cospan category). Let $\\mathcal{C}$ be a category with pushouts. Applied to the saturated triple $(\\mathcal{C}^{\\mathrm{op}}, \\mathcal{C}^{\\mathrm{op}}, \\mathcal{C}^{\\mathrm{op}})$, the twisted arrow construction produces a complete Segal anima $\\mathrm{N}\\mathsf{Span}(\\mathcal{C}^{\\mathrm{op}})$; denote the corresponding category by $\\mathsf{coSpan}(\\mathcal{C})$. Concretely:\nObjects of $\\mathsf{coSpan}(\\mathcal{C})$ are the objects of $\\mathcal{C}$.\nA morphism $X \\to Y$ in $\\mathsf{coSpan}(\\mathcal{C})$ is a cospan\nA $2$-morphism between cospans $[X \\to W \\leftarrow Y]$ and $[X \\to W' \\leftarrow Y]$ is an equivalence $W \\xrightarrow{\\sim} W'$ compatible with the two legs:\nThe cospan category inherits a symmetric monoidal structure from the coproduct on $\\mathcal{C}^{\\sqcup}$ (not the coproduct monoidal structure on $\\mathsf{coSpan}(\\mathcal{C})$ itself).\nDefinition 2 (Connective algebraic K-theory). Let $\\mathcal{C}$ be a category with finite colimits. Its connective algebraic K-theory is \\[ \\mathrm{k}(\\mathcal{C}) \\coloneqq \\Omega\\left| \\mathrm{N}\\mathsf{Span}(\\mathcal{C}^{\\mathrm{op}}) \\right|. \\] For $n \\ge 0$ set $\\mathrm{k}_n(\\mathcal{C}) \\coloneqq \\pi_n(\\mathrm{k}(\\mathcal{C}))$.\nSince $\\mathsf{coSpan}(\\mathcal{C})$ is symmetric monoidal, its geometric realisation $\\left| \\mathrm{N}\\mathsf{Span}(\\mathcal{C}^{\\mathrm{op}}) \\right|$ is naturally a commutative monoid in $\\mathsf{An}$. Group completion — equivalently, passage to loops — turns it into a commutative group in $\\mathsf{An}$. The equivalence between commutative groups in anima and connective spectra $\\mathsf{Sp}_{\\ge 0}$ produces the associated spectrum \\[ \\mathbb{B}^{\\infty}\\!\\left(\\Omega\\left| \\mathrm{N}\\mathsf{Span}(\\mathcal{C}^{\\mathrm{op}}) \\right|\\right) \\in \\mathsf{Sp}_{\\ge 0}, \\] which we also denote by $\\mathrm{k}(\\mathcal{C})$ when there is no risk of confusion.\nIntuition At the abstract level, $\\mathrm{k}(\\mathcal{C})$ already has a commutative group structure. More concretely, its construction encodes the additive relations coming from pushouts, with the initial object corresponding to the zero element $[\\varnothing] = 0$.\nTo see this relation strictly inside $|\\mathsf{coSpan}(\\mathcal{C})|$, choose $\\varnothing$ as the basepoint. For each object $X$ of $\\mathcal{C}$, define two morphisms in $\\mathsf{coSpan}(\\mathcal{C})$: \\begin{align*} a_X \u0026amp;\\coloneqq (X \\to X \\leftarrow \\varnothing) \\qquad \\text{(a morphism } X \\to \\varnothing\\text{),} \\ b_X \u0026amp;\\coloneqq (\\varnothing \\to X \\leftarrow X) \\qquad \\text{(a morphism } \\varnothing \\to X\\text{).} \\end{align*} The composite $a_X \\circ b_X$ is the self-map of $\\varnothing$ given by the cospan $\\varnothing \\to X \\leftarrow \\varnothing$; this is a loop at the basepoint, whose homotopy class we denote $[X]$ (or $p_X$).\nMore generally:\nFor a morphism $g = (X \\to A \\leftarrow \\varnothing)\\colon X \\to \\varnothing$, the composite $g \\circ b_X$ has central pushout $A$, so it represents the loop $p_A$. For a morphism $f = (\\varnothing \\to B \\leftarrow X)\\colon \\varnothing \\to X$, the composite $a_X \\circ f$ has central pushout $B$, hence is $p_B$. The composite $g \\circ f$ traces out the loop $p_{A \\sqcup_X B}$ via the pushout $A \\leftarrow X \\to B$. In the abelian group $\\pi_1(|\\mathsf{coSpan}(\\mathcal{C})|)$ one has (in additive notation) \\[ [g \\circ f] = [g \\circ b_X] - [a_X \\circ b_X] + [a_X \\circ f], \\] so \\[ [A \\sqcup_X B] = [A] - [X] + [B] \\quad\\Longrightarrow\\quad [A \\sqcup_X B] + [X] = [A] + [B]. \\] Geometrically: every pushout diagram in $\\mathsf{coSpan}(\\mathcal{C})$ supplies a $2$-cell inside the geometric realisation, making the homotopy identity above hold. In particular the cofibre sequence $X \\xrightarrow{f} Y \\to Y/X$ corresponds to the pushout $\\varnothing \\sqcup_X Y \\simeq Y/X$; taking $A = \\varnothing$ and $B = Y$ gives \\[ [Y/X] + [X] = [\\varnothing] + [Y] = [Y]. \\] Thus, at the level of $\\pi_1$, the relation $[Y] = [X] + [Y/X]$ is faithfully witnessed.\nThis construction, however, is less transparent at higher homotopy levels. We now give a second, more homotopically lucid construction of algebraic K-theory — the Waldhausen $S$-construction.\nFor this second construction we need $\\mathcal{C}$ to be pointed with finite colimits. Before giving the simplicial object itself, here is the motivating picture.\nWe seek an anima $W$ whose fundamental group is $\\mathrm{k}_0(\\mathcal{C})$ and whose higher homotopy groups give the higher K-groups directly. So $W$ should carry the following data:\na basepoint $0$;\nfor each $X \\in \\mathcal{C}$ a loop in $W$, representing $[X] \\in \\pi_1(W) \\simeq \\mathrm{k}_0(\\mathcal{C})$;\nfor each cofibre sequence $X \\to Y \\to Y/X$, the relation $[Y] = [X] + [Y/X]$ in $\\pi_1(W)$. Equivalently, every map $f\\colon X \\to Y$ should satisfy $[Y] = [Y/X] + [X]$ where $Y/X \\coloneqq \\mathrm{cofib}(f)$;\nhigher coherence data: for a sequence $X \\to Y \\to Z$ we want $[Z] = [X] + [Y/X] + [Z/Y]$. Two natural routes derive this:\nvia $X \\to Z \\to Z/X$ and $Y/X \\to Z/X \\to Z/Y$, giving $[Z] = [X] + [Z/X]$ and $[Z/X] = [Y/X] + [Z/Y]$; or via $Y \\to Z \\to Z/Y$ and $X \\to Y \\to Y/X$, giving $[Z] = [Y] + [Z/Y]$ and $[Y] = [X] + [Y/X]$. Each route is the concatenation of two $2$-cells; the four $2$-cells bound a $2$-sphere inside $|W|$, and the composable pair $(f,g)$ supplies a $3$-cell filling it. All of this is packaged in the $\\mathsf{S}_3$ diagram and this pattern extends: length-$n$ sequences give objects of $\\mathsf{S}_n\\mathcal{C}$, consisting of all subquotients $X_j / X_i$ and their pushout relations; the boundary of each such object is a $(n-1)$-sphere built from lower-order $\\mathsf{S}$-data, and the new $\\mathsf{S}_n$-data provides the $n$-cell that fills it.\nConstruction 3 (Waldhausen S-construction). Let $\\mathcal{C}$ be a pointed category with finite colimits. Its Waldhausen $\\mathsf{S}$-construction is the simplicial object $\\mathsf{S}_{\\bullet}\\mathcal{C}$ whose level $n$, $\\mathsf{S}_n\\mathcal{C} \\subseteq \\mathsf{Fun}(\\mathsf{Ar}[n], \\mathcal{C})$, is the full subcategory spanned by $x\\colon \\mathsf{Ar}[n] \\to \\mathcal{C}$ with\n$x_{i,i} = 0$ for $0 \\le i \\le n$;\nevery square\nwith $0 \\le i \u003c j \u003c n$ is a pushout.\nThe simplicial-category structure comes from the functoriality in $[n]$ of $\\mathsf{Fun}(\\mathsf{Ar}[n], \\mathcal{C})$. In other words, $\\mathsf{S}_n\\mathcal{C}$ is the category of diagrams of the shape\nwith every square a pushout.\nProposition 4. When $\\mathcal{C}$ is a pointed category with finite colimits, the constructions Construction 3 and Construction 1 produce the same algebraic K-theory. We next show that $\\mathcal{C}$ and its pointed completion have the same algebraic K-theory.\nConstruction 5. Let $\\mathcal{C}$ be a category with finite colimits. Consider the pointed version $\\mathsf{Ind}(\\mathcal{C})_*$ of $\\mathsf{Ind}(\\mathcal{C})$ and the canonical composite \\[ \\mathcal{C} \\xrightarrow{\\;y\\;} \\mathsf{Ind}(\\mathcal{C}) \\xrightarrow{\\;(-)_+\\;} \\mathsf{Ind}(\\mathcal{C})_*. \\] Let $\\mathcal{C}_+$ denote the smallest full subcategory generated under finite colimits by the essential image of this composite. One checks that the functor $\\mathcal{C} \\to \\mathcal{C}_+$ is the pointed completion of $\\mathcal{C}$: it turns $\\mathcal{C}$ into a pointed category with finite colimits.\nEquipping $\\mathsf{Cat}^{\\mathrm{rex}}$ with the Lurie tensor product, $\\mathcal{C}_+$ is naturally identified with $\\mathcal{C} \\otimes \\mathsf{An}_*^{\\mathrm{fin}}$. Since the Lurie tensor product preserves colimits in each variable, the assignment $\\mathcal{C} \\mapsto \\mathcal{C}_+$ preserves filtered colimits.\nAnother useful point: let $\\mathsf{Cat}^{\\mathrm{rex}}_* \\subset \\mathsf{Cat}^{\\mathrm{rex}}$ be the subcategory of pointed categories with finite colimits and finite-colimit-preserving pointed functors. Then $\\mathsf{Cat}^{\\mathrm{rex}}_*$ is closed under colimits in $\\mathsf{Cat}^{\\mathrm{rex}}$.\nProposition 6. The finite-colimit-preserving functor $\\mathcal{C} \\to \\mathcal{C}_+$ induces an equivalence $\\mathrm{k}(\\mathcal{C}) \\simeq \\mathrm{k}(\\mathcal{C}_+)$. Proof. First suppose $\\mathcal{C}$ has a terminal object $*$. Then $\\mathcal{C}_+ = \\mathcal{C}_{*/}$, so the task reduces to showing that the functor \\[ \\mathsf{coSpan}(\\mathcal{C}) \\to \\mathsf{coSpan}(\\mathcal{C}_+) \\] induces an equivalence on geometric realisations.\nThere is a forgetful functor $\\mathcal{C}_{*/} \\to \\mathcal{C}$ preserving pushouts (but not the initial object). Consider the composites \\[ \\mathcal{C}_+ \\to \\mathcal{C} \\to \\mathcal{C}_+, \\qquad X \\mapsto X_+ = X \\sqcup *, \\] (with basepoint the newly adjoined $*$), and the reverse composite \\[ \\mathcal{C} \\to \\mathcal{C}_+ \\to \\mathcal{C}, \\qquad X \\mapsto X_+. \\] Since the $\\mathsf{coSpan}$ construction depends only on pushouts, it is functorial in pushout-preserving functors. In both composites the canonical map $X \\to X \\sqcup *$ is a natural transformation from the identity to the composite, and for each morphism $f\\colon X \\to Y$ the square\nis a pushout. By [sheaves-on-manifolds, Prop. 3.1.9] , a natural transformation whose squares are all pushouts induces a homotopy between the $\\mathsf{coSpan}$ realisations. Thus each composite is homotopic to the identity on $\\mathrm{k}$, and the forgetful and $(-)_+$ functors induce mutually inverse equivalences $\\mathrm{k}(\\mathcal{C}) \\simeq \\mathrm{k}(\\mathcal{C}_+)$ whenever $\\mathcal{C}$ has a terminal object.\nIn general, view $\\mathsf{Ar}(\\mathcal{C}) \\to \\mathcal{C}$ as the left Bousfield localisation of $\\mathcal{C}$ at $t$-pushout maps, so that $\\mathcal{C}$ is a colimit of its over-categories, \\[ \\mathcal{C} \\simeq \\operatorname*{colim}_{X \\in \\mathcal{C}} \\mathcal{C}_{/X}. \\] Each $\\mathcal{C}_{/X}$ has the terminal object $\\mathrm{id}_X$, so the first part gives $\\mathrm{k}(\\mathcal{C}_{/X}) \\xrightarrow{\\sim} \\mathrm{k}((\\mathcal{C}_{/X})_+)$. Since $\\mathcal{C}$ has finite colimits it is filtered as an index, and $\\mathcal{C}_+ \\simeq \\operatorname*{colim}_X (\\mathcal{C}_{/X})_+$. Filtered colimits are preserved by both $\\mathrm{k}$ and $(-)_+$ ( ), so the equivalence lifts to \\[ \\mathrm{k}(\\mathcal{C}) \\simeq \\operatorname*{colim}_X \\mathrm{k}(\\mathcal{C}_{/X}) \\xrightarrow{\\;\\sim\\;} \\operatorname*{colim}_X \\mathrm{k}((\\mathcal{C}_{/X})_+) \\simeq \\mathrm{k}(\\mathcal{C}_+). \\] $\\square$ Basic properties Definition 2 is well-behaved on morphisms. Writing $\\mathsf{Cat}^{\\mathrm{rex}}$ for categories with finite colimits and right-exact functors between them, Definition 2 upgrades to a functor \\[ \\mathrm{k}\\colon \\mathsf{Cat}^{\\mathrm{rex}} \\to \\mathsf{Sp}_{\\ge 0}. \\]Recall that both $\\mathsf{Cat}^{\\mathrm{rex}}$ and $\\mathsf{Sp}_{\\ge 0}$ are semi-additive, i.e. finite products coincide with finite coproducts.\nThe geometric-realisation functor is essentially a sifted colimit, so it commutes with finite products in $\\mathsf{An}$; the loop functor $\\Omega$, as a right adjoint, preserves limits. Combining these, $\\mathrm{k}$ preserves finite products, and in a semi-additive category that amounts to preserving finite coproducts. More informally, for functors $F, G\\colon \\mathcal{C} \\to \\mathcal{D}$, \\[ \\mathrm{k}(F \\sqcup G) \\simeq \\mathrm{k}(F) + \\mathrm{k}(G). \\]In fact $\\mathrm{k}$ also preserves filtered colimits:\nProposition 7. $\\mathrm{k}\\colon \\mathsf{Cat}^{\\mathrm{rex}} \\to \\mathsf{Sp}_{\\ge 0}$ preserves filtered colimits. Proof. First, filtered colimits in $\\mathsf{Cat}^{\\mathrm{rex}}$ agree with those in $\\mathsf{Cat}$. The claim then reduces to showing that, for a filtered colimit $\\mathcal{C} \\simeq \\operatorname*{colim}_i \\mathcal{C}_i$ in $\\mathsf{Cat}$ (so $\\mathcal{C}^{\\mathrm{op}} \\simeq \\operatorname*{colim}_i \\mathcal{C}_i^{\\mathrm{op}}$), one has \\[ \\operatorname*{colim}_{i} \\mathrm{N}\\mathsf{Span}(\\mathcal{C}_i^{\\mathrm{op}}) \\simeq \\mathrm{N}\\mathsf{Span}(\\mathcal{C}^{\\mathrm{op}}) \\] in $\\mathsf{CSeg}(\\mathsf{An})$. Complete Segal objects are simplicial objects, and simplicial colimits are levelwise. At level $n$, $\\mathrm{N}\\mathsf{Span}(\\mathcal{C})_n = \\mathrm{Hom}_{\\mathsf{AdTrip}}(\\mathsf{TwAr}([n]), \\mathcal{C})$, and $\\mathrm{Hom}_{\\mathsf{AdTrip}}(\\mathsf{TwAr}([n]), -)$ preserves filtered colimits, so the claim follows.\n$\\square$ Stabilisation We now extend connective K-theory to stable categories. Since $\\mathcal{C}$ has only finite colimits, we cannot stabilise by taking limits directly; instead we dualise and build the Spanier–Whitehead category.\nDefinition 8 (Spanier–Whitehead category). Let $\\mathcal{C}$ be a category with finite colimits. Its Spanier–Whitehead category is \\[ \\mathsf{SW}(\\mathcal{C}) \\coloneqq \\operatorname{colim}\\!\\left( \\mathcal{C}_+ \\xrightarrow{\\Sigma_+} \\mathcal{C}_+ \\xrightarrow{\\Sigma_+} \\cdots \\right), \\] with colimit computed in $\\mathsf{Cat}^{\\mathrm{rex}}$. By Gabriel–Ulmer duality $\\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}} \\simeq \\mathsf{Pr}_{\\aleph_0}^L$ and the fact that Ind does not distinguish a small category from its idempotent completion ($\\mathsf{Ind}(\\mathcal{A}) \\simeq \\mathsf{Ind}(\\mathcal{A}^{\\mathrm{idem}})$), we get $\\mathsf{Ind}(\\mathsf{SW}(\\mathcal{C})) \\in \\mathsf{Pr}_{\\aleph_0}^L$. Colimits in $\\mathsf{Pr}_{\\aleph_0}^L$ are limits in $\\mathsf{Pr}_{\\aleph_0}^R$, and the forgetful functor $\\mathsf{Pr}_{\\aleph_0}^R \\to \\mathsf{Cat}$ preserves limits, which gives \\[ \\mathsf{Ind}(\\mathsf{SW}(\\mathcal{C})) \\simeq \\mathsf{Sp}(\\mathsf{Ind}(\\mathcal{C})) \\simeq \\mathsf{Sp} \\otimes \\mathsf{Ind}(\\mathcal{C}). \\] The canonical composite is $\\mathcal{C} \\to \\mathcal{C}_+ \\to \\mathsf{SW}(\\mathcal{C})$.\nProposition 9. The canonical functor $\\mathcal{C} \\to \\mathsf{SW}(\\mathcal{C})$ induces an equivalence \\[ \\mathrm{k}(\\mathcal{C}) \\xrightarrow{\\;\\sim\\;} \\mathrm{k}(\\mathsf{SW}(\\mathcal{C})). \\] Proof. Combine Proposition 7 and Proposition 6 . $\\square$ So connective K-theory restricts to a functor \\[ \\mathrm{k}\\colon \\mathsf{Cat}^{\\mathrm{ex}} \\to \\mathsf{Sp}_{\\ge 0}. \\]Dense embeddings and cofinality Definition 10. Let $\\mathcal{C}, \\mathcal{D}$ be categories with finite colimits. A functor $i\\colon \\mathcal{C} \\hookrightarrow \\mathcal{D}$ is a dense embedding if it is fully faithful and every object of $\\mathcal{D}$ is a retract of some $i(c)$. Example 11. The idempotent completion $\\mathcal{C} \\to \\mathcal{C}^{\\mathrm{idem}}$ is a dense embedding. Proposition 12 (Cofinality). Let $\\mathcal{C} \\to \\mathcal{D}$ be a dense embedding. The cofibre of $\\mathrm{k}(\\mathcal{C}) \\to \\mathrm{k}(\\mathcal{D})$ is $0$-truncated: the map $\\mathrm{k}_i(\\mathcal{C}) \\to \\mathrm{k}_i(\\mathcal{D})$ is an isomorphism for $i \u003e 0$ and injective for $i = 0$.\nA class $[D] \\in \\mathrm{k}_0(\\mathcal{D})$ lies in the essential image of $\\mathrm{k}_0(\\mathcal{C})$ iff there exists $n \\ge 0$ with $\\Sigma^n(D_+)$ in the essential image of $\\mathcal{C}_+ \\to \\mathcal{D}_+$.\nIt is natural to ask whether K-theory detects essential surjectivity: given $D \\in \\mathcal{D}$, is $D$ in $\\mathcal{C} \\subseteq \\mathcal{D}$?\nClearly a necessary condition is that $[D] \\in \\mathrm{k}_0(\\mathcal{D})$ lies in the image. When $\\mathcal{C}$ and $\\mathcal{D}$ are both stable, this is also sufficient. In the unstable case, cofinality only guarantees that some suspension of $D$ lies in the image; there are classical counterexamples in topology of finitely dominated but non-finite anima. After two suspensions, however, any finitely dominated anima becomes simply connected (and remains finitely dominated); Wall\u0026rsquo;s insight is that simply connected finitely dominated anima are automatically finite.\nProposition 13 (Wall\u0026#39;s finiteness obstruction). Let $\\mathcal{C} \\hookrightarrow \\mathcal{D}$ be a dense embedding in $\\mathsf{Cat}^{\\mathrm{rex}}$. An object $d \\in \\mathcal{D}$ lies in the essential image of $\\mathcal{C}$ iff the class $[\\mathrm{id}_d] \\in \\mathrm{k}_0(\\mathcal{D}_{/d})$ is in the essential image of \\[ \\mathrm{k}_0(\\mathcal{C}_{/d}) \\to \\mathrm{k}_0(\\mathcal{D}_{/d}). \\] Example 14. Apply this to the embedding $\\mathsf{An}^{\\mathrm{fin}} \\hookrightarrow \\mathsf{An}^{\\aleph_0}$ of finite and finitely dominated anima. For $X \\in \\mathsf{An}^{\\aleph_0}_{\\ge 1}$, $X$ is a finite anima iff $[\\mathrm{id}_X] \\in \\mathrm{k}_0(\\mathsf{An}_{/X}^{\\aleph_0}) = \\mathrm{k}_0((\\mathsf{Sp}^X)^{\\aleph_0})$ lies in the essential image of $\\mathrm{k}_0((\\mathsf{Sp}^X)^{\\mathrm{fin}}) \\to \\mathrm{k}_0((\\mathsf{Sp}^X)^{\\aleph_0})$. For connected $X$, \\[ \\mathrm{k}_0((\\mathsf{Sp}^X)^{\\aleph_0}) = \\mathrm{k}_0(\\mathbb{S}[\\Omega X]) = \\mathrm{k}_0(\\mathbb{Z}[\\pi_1 X]), \\] and $\\mathrm{k}_0((\\mathsf{Sp}^X)^{\\mathrm{fin}}) = \\mathbb{Z}$. The finiteness question reduces to whether $[\\mathrm{id}_X]$ vanishes in the reduced K-group \\[ \\tilde{\\mathrm{k}}_0(\\mathbb{Z}[\\pi_1 X]) = \\mathrm{k}_0(\\mathbb{Z}[\\pi_1 X])/\\mathbb{Z}. \\] Non-connective K-theory We now extend $\\mathrm{k}\\colon \\mathsf{Cat}^{\\mathrm{rex}} \\to \\mathsf{Sp}_{\\ge 0}$ to a functor \\[ \\mathrm{K}\\colon \\mathsf{Cat}^{\\mathrm{rex}} \\to \\mathsf{Sp} \\] landing in all spectra — the non-connective K-theory.\nVerdier sequences The stable case Definition 15. Let \\[ \\mathcal{C} \\xrightarrow{f} \\mathcal{D} \\xrightarrow{p} \\mathcal{E} \\] be a sequence in $\\mathsf{Cat}^{\\mathrm{ex}}$ with $p \\circ f \\simeq 0$. It is a Verdier sequence if it is simultaneously a fibre and cofibre sequence in $\\mathsf{Cat}^{\\mathrm{ex}}$. In this case $f$ is a Verdier embedding and $p$ is a Verdier projection.\nRemark. The condition $p \\circ f \\simeq 0$ is taken inside $\\mathsf{Fun}^{\\mathrm{ex}}(\\mathcal{C}, \\mathcal{E})$. Equivalently, $p \\circ f$ factors as $\\mathcal{C} \\to \\{0\\} \\subset \\mathcal{E}$, i.e. there is a commuting square\nRecall how fibres and cofibres are computed in $\\mathsf{Cat}^{\\mathrm{ex}}$. The fibre of an exact $F\\colon \\mathcal{C} \\to \\mathcal{D}$ is the categorical fibre; the cofibre is subtler.\nDefinition 16 (Verdier quotient). Let $F\\colon \\mathcal{C} \\to \\mathcal{D}$ be exact. A morphism $f\\colon d \\to d'$ in $\\mathcal{D}$ is an isomorphism modulo $\\mathcal{C}$ if $\\mathrm{cofib}(f)$ lies in the smallest stable subcategory containing the essential image of $F$. Let $\\mathcal{D}/\\mathcal{C}$ be the localisation of $\\mathcal{D}$ at the class $W$ of such morphisms — the Verdier quotient of $\\mathcal{D}$ by $\\mathcal{C}$. Proposition 17. Let $F\\colon \\mathcal{C} \\to \\mathcal{D}$ be exact.\n$\\mathcal{D}/\\mathcal{C}$ is stable. For any stable $\\mathcal{E}$, the restriction $\\mathsf{Fun}^{\\mathrm{ex}}(\\mathcal{D}/\\mathcal{C}, \\mathcal{E}) \\to \\mathsf{Fun}^{\\mathrm{ex}}(\\mathcal{D}, \\mathcal{E})$ is fully faithful with essential image the functors whose precomposition with $F$ vanishes. In particular, $\\mathcal{C} \\to \\mathcal{D} \\to \\mathcal{D}/\\mathcal{C}$ is a cofibre sequence in $\\mathsf{Cat}^{\\mathrm{ex}}$. The non-stable case $\\mathsf{Cat}^{\\mathrm{rex}}$ is pointed, with zero object the terminal category $*$, so we can consider cofibre sequences there too. For $F\\colon \\mathcal{C} \\to \\mathcal{D}$ the cofibre is the pushout For concrete computation we lift everything to presentable categories using Gabriel–Ulmer duality. In its strict form, \\[ \\mathsf{Ind}\\colon \\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}} \\xrightarrow{\\;\\simeq\\;} \\mathsf{Pr}_{\\aleph_0}^L \\colon (-)^{\\omega}, \\] where $\\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}}$ is idempotent-complete categories with finite colimits. Idempotent completeness cannot be dropped at $\\kappa = \\aleph_0$: there exist non-idempotent-complete categories with finite colimits, and for such a $\\mathcal{C}$ the category $\\mathsf{Ind}(\\mathcal{C})^{\\omega}$ is idempotent-complete (it is the idempotent completion $\\mathcal{C}^{\\mathrm{idem}}$), not $\\mathcal{C}$ itself.\nRemark. For uncountable $\\kappa$ the picture simplifies. A category with $\\kappa$-small colimits automatically has all countable colimits, in particular sequential colimits. By [lurie-htt, Prop. 5.5.7.8] or [kerodon, Cor. 8.5.4.19] , a category admitting sequential colimits (or limits) is automatically idempotent-complete, since split idempotents are realised as sequential colimits. Hence for $\\kappa \u003e \\aleph_0$, \\[ \\mathsf{Cat}^{\\mathrm{rex}(\\kappa)} \\simeq \\mathsf{Pr}_{\\kappa}^L, \\] with no extra idempotent-completeness hypothesis. We will use this repeatedly when working with $\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1}$.\nReturning to cofibres. Gabriel–Ulmer extends $F$ to a colimit-preserving functor between presentable categories, \\[ \\mathsf{Ind}(F)\\colon \\mathsf{Ind}(\\mathcal{C}) \\to \\mathsf{Ind}(\\mathcal{D}). \\] Colimits in $\\mathsf{Cat}^{\\mathrm{rex}}$ are unwieldy to compute directly; using $\\mathsf{Pr}_{\\aleph_0}^L \\simeq (\\mathsf{Pr}_{\\aleph_0}^R)^{\\mathrm{op}}$, a colimit in $\\mathsf{Cat}^{\\mathrm{rex}}$ is a limit in $\\mathsf{Pr}_{\\aleph_0}^R$, and the forgetful $\\mathsf{Pr}_{\\aleph_0}^R \\to \\mathsf{Cat}$ preserves limits, letting us compute in $\\mathsf{Cat}$.\nConcretely, consider $\\mathsf{Ind}(F)^R\\colon \\mathsf{Ind}(\\mathcal{D}) \\to \\mathsf{Ind}(\\mathcal{C})$, and let $K \\coloneqq \\mathrm{fib}(\\mathsf{Ind}(F)^R)$ — the full subcategory of $\\mathsf{Ind}(\\mathcal{D})$ on those $d$ with $\\mathsf{Ind}(F)^R(d) \\simeq *$. From the right-adjoint side this is exactly the cofibre of the large picture: $K \\simeq \\mathsf{Ind}(\\mathcal{D}/\\mathcal{C})$.\nTo get back to the left-adjoint side, take the left adjoint of the inclusion $K \\hookrightarrow \\mathsf{Ind}(\\mathcal{D})$ to get a localisation $\\mathsf{Ind}(\\mathcal{D}) \\to K$; restricting along $\\mathcal{D} \\subset \\mathsf{Ind}(\\mathcal{D})$ gives $p\\colon \\mathcal{D} \\to K$.\nThis is where the idempotent-completeness issue resurfaces. Because the genuine Gabriel–Ulmer duality is $\\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}} \\simeq \\mathsf{Pr}_{\\aleph_0}^L$, passing from $K \\in \\mathsf{Pr}_{\\aleph_0}^L$ to small categories by taking compact objects gives $K^{\\omega} = (\\mathcal{D}/\\mathcal{C})^{\\mathrm{idem}}$ — the idempotent completion of the cofibre. The cofibre itself is the full subcategory of $K$ generated under finite colimits by the essential image of $p$.\nUnder the further assumptions that\n$F$ is fully faithful (hence so is $\\mathsf{Ind}(F)$), and $\\mathsf{Ind}(F)^R$ preserves pushouts, one obtains the pushout square and mapping spaces are \\[ \\mathrm{Hom}_{\\mathcal{D}/\\mathcal{C}}(pd, pd') \\simeq \\mathrm{Hom}_{\\mathsf{Ind}(\\mathcal{D})}\\!\\left( y(d),\\, y(d') \\sqcup_{\\mathsf{Ind}(F)\\mathsf{Ind}(F)^R(y(d'))} \\mathsf{Ind}(F)(*) \\right). \\] This is the unstable analogue of the Verdier quotient\u0026rsquo;s mapping-space formula; in the stable case both assumptions are automatic. The description is compatible with idempotent completion, since $\\mathsf{Ind}(\\mathcal{C}^{\\mathrm{idem}}) \\simeq \\mathsf{Ind}(\\mathcal{C})$.\nWhat we actually care about are the cofibre sequences in $\\mathsf{Cat}^{\\mathrm{rex}}$ that induce long exact sequences in connective K-theory. This property is not automatic under idempotent completion (recall cofinality: dense embeddings are only injective, not surjective, on $\\mathrm{k}_0$). The following definition assembles all the needed technical assumptions.\nDefinition 18 (Verdier sequence in Cat^rex). A sequence \\[ \\mathcal{C} \\xrightarrow{i} \\mathcal{D} \\xrightarrow{p} \\mathcal{E} \\] in $\\mathsf{Cat}^{\\mathrm{rex}}$ is a Verdier sequence if:\nit is a cofibre sequence; $i$ is fully faithful; $\\mathsf{Ind}(i)^R$ preserves pushouts; the image of $i_+\\colon \\mathcal{C}_+ \\to \\mathcal{D}_+$ is closed under retracts. Remark. It is not clear whether retract-closedness for $i$ itself already implies the same for $i_+$. When $\\mathcal{C}$ is idempotent-complete, $\\mathcal{C}_+$ is too ( ), and fully faithfulness of $i_+$ automatically gives retract-closedness of its image. When $\\mathcal{C}$ and $\\mathcal{D}$ are both stable, the definition reduces to the stable Verdier sequence. Proposition 19. Let $S = (\\mathcal{C} \\xrightarrow{i} \\mathcal{D} \\xrightarrow{p} \\mathcal{E})$ be a sequence in $\\mathsf{Cat}^{\\mathrm{rex}}$. Consider the conditions:\n$S$ is a cofibre sequence, both $\\mathcal{C}$ and $\\mathcal{D}$ have terminal objects, $i$ is fully faithful with retract-closed image, and $i$ preserves the terminal object (then $\\mathcal{E}$ also has a terminal object, and $p$ preserves it). $S$ is a Verdier sequence (Definition 18 ). $(2')$ $S$ is a cofibre sequence, $i$ is fully faithful, $i_+$\u0026rsquo;s image is retract-closed, and $\\mathsf{Ind}(i)$ is a strong left adjoint (its right adjoint has a right adjoint). $S_+ = (\\mathcal{C}_+ \\xrightarrow{i_+} \\mathcal{D}_+ \\xrightarrow{p_+} \\mathcal{E}_+)$ is a cofibre sequence with $i_+$ fully faithful and retract-closed image. $\\mathsf{SW}(S)$ is a Verdier sequence in $\\mathsf{Cat}^{\\mathrm{ex}}$. $\\mathrm{k}(S) = (\\mathrm{k}(\\mathcal{C}) \\to \\mathrm{k}(\\mathcal{D}) \\to \\mathrm{k}(\\mathcal{E}))$ is a fibre sequence in $\\mathsf{Sp}_{\\ge 0}$. Then $(2') \\Rightarrow (2) \\Rightarrow (3) \\Rightarrow (4) \\Rightarrow (5)$ and $(1) \\Rightarrow (3)$. Moreover:\nWhen $\\mathcal{C}, \\mathcal{D}$ (hence $\\mathcal{E}$) are pointed, $(1) \\Leftrightarrow (3)$ and $(2) \\Leftrightarrow (2')$. When they are stable, all implications reverse except $(5) \\Rightarrow (6)$. The functors $(-)^{\\mathrm{idem}}$, $(-)_+$ and $\\mathsf{SW}(-)$ preserve sequences of types $(1)$–$(4)$; the last two also preserve type $(6)$. The classes of sequences satisfying $(3)$, $(4)$, $(6)$ are each closed under filtered colimits in $\\mathsf{Cat}^{\\mathrm{rex}}$. Proof. See [sheaves-on-manifolds, Prop. 3.2.3] .\nWe sketch $(2) \\Rightarrow (3)$, the most instructive unstable implication. We must show $i_+$ is fully faithful. Recall $\\mathsf{Ind}((-)_+) \\simeq \\mathsf{An}_* \\otimes \\mathsf{Ind}(-)$, so $\\mathsf{Ind}(\\mathcal{C}_+) \\simeq \\mathsf{Ind}(\\mathcal{C})_*$ and $\\mathsf{Ind}(i_+) \\simeq \\mathsf{Ind}(i)_*$. In general, if $\\ell\\colon \\mathcal{A} \\hookrightarrow \\mathcal{B}$ is fully faithful in $\\mathsf{Pr}^L$ with $\\ell^R$ preserving pushouts, then $\\ell_*\\colon \\mathcal{A}_* \\to \\mathcal{B}_*$ is fully faithful. Indeed, $(\\ell_*)^R$ is the restriction of $\\ell^R$ to slice categories; fully faithfulness of $\\ell$ and pushout-preservation of $\\ell^R$ together force the counit of the adjunction $\\ell_*$ to be an equivalence.\n$\\square$ Non-connective K-theory Take an idempotent-complete $\\mathcal{C} \\in \\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}}$. Consider the Yoneda embedding $y\\colon \\mathcal{C} \\hookrightarrow \\mathsf{Ind}(\\mathcal{C})$ and its $\\aleph_1$-truncation $j\\colon \\mathcal{C} \\hookrightarrow \\mathsf{Ind}(\\mathcal{C})^{\\aleph_1}$. We get a natural cofibre sequence \\[ \\mathcal{C} \\xrightarrow{\\;j\\;} \\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} \\longrightarrow \\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} / \\mathcal{C}. \\] Idempotent completeness makes $j$ fully faithful with retract-closed image, and $\\mathsf{Ind}(j) = \\hat{y}\\colon \\mathsf{Ind}(\\mathcal{C}) \\to \\mathsf{Ind}(\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1})$ is a strong left adjoint (see the chapter on compactly assembled categories); condition $(2')$ of Proposition 19 is satisfied and the sequence induces a fibre sequence on $\\mathrm{k}$: \\[ \\mathrm{k}(\\mathcal{C}) \\to \\mathrm{k}(\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1}) \\to \\mathrm{k}(\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} / \\mathcal{C}). \\]Next, $\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1}$ has countable colimits — in fact all $\\aleph_1$-small colimits, and is idempotent-complete by . The Eilenberg swindle gives $\\mathrm{k}(\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1}) \\simeq 0$, and the fibre sequence becomes \\[ \\mathrm{k}(\\mathcal{C}) \\simeq \\Omega \\mathrm{k}(\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} / \\mathcal{C}). \\] In particular $\\mathrm{k}_0(\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} / \\mathcal{C}) = 0$.\nThe quotient $\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} / \\mathcal{C}$ is not automatically idempotent-complete, so we cannot iterate directly. Idempotent completion plus cofinality (Proposition 12 ) gives \\[ \\mathrm{k}(\\mathcal{C}) \\simeq \\tau_{\\ge 0} \\Omega \\mathrm{k}\\!\\left((\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} / \\mathcal{C})^{\\mathrm{idem}}\\right). \\] The key point: $\\mathrm{k}_0((\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} / \\mathcal{C})^{\\mathrm{idem}})$ need not vanish, signalling the existence of negative K-groups and the path to extending K-theory to all spectra.\nDefinition 20 (Calkin category). For idempotent-complete $\\mathcal{C}$ with finite colimits, the Calkin category is \\[ \\mathsf{Calk}(\\mathcal{C}) \\coloneqq (\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} / \\mathcal{C})^{\\mathrm{idem}}. \\] For general $\\mathcal{C} \\in \\mathsf{Cat}^{\\mathrm{rex}}$ and $n \\ge 0$, recursively define \\[ \\mathsf{Calk}^0(\\mathcal{C}) = \\mathcal{C}^{\\mathrm{idem}}, \\qquad \\mathsf{Calk}^{n+1}(\\mathcal{C}) = \\mathsf{Calk}(\\mathsf{Calk}^n(\\mathcal{C})). \\] This defines an endofunctor $\\mathsf{Calk}\\colon \\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}} \\to \\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}}$.\nFor all $n \\ge 0$, \\[ \\tau_{\\ge 0} \\Omega \\mathrm{k}(\\mathsf{Calk}^{n+1}(\\mathcal{C})) \\simeq \\mathrm{k}(\\mathsf{Calk}^n(\\mathcal{C})), \\] which leads naturally to the definition:\nDefinition 21 (Non-connective algebraic K-theory). Let $\\mathcal{C} \\in \\mathsf{Cat}^{\\mathrm{rex}}$. The non-connective algebraic K-theory of $\\mathcal{C}$ is the spectrum $\\mathrm{K}(\\mathcal{C})$ characterised by \\[ \\tau_{\\ge -n} \\mathrm{K}(\\mathcal{C}) \\simeq \\Omega^n \\mathrm{k}(\\mathsf{Calk}^n(\\mathcal{C})) \\] together with $\\mathrm{K}(\\mathcal{C}) \\simeq \\operatorname*{colim}_n \\tau_{\\ge -n} \\mathrm{K}(\\mathcal{C})$. For $n \\in \\mathbb{Z}$ set $\\mathrm{K}_n(\\mathcal{C}) \\coloneqq \\pi_n(\\mathrm{K}(\\mathcal{C}))$.\nExample 22. By cofinality, the natural map $\\mathrm{k}_n(\\mathcal{C}) \\to \\mathrm{K}_n(\\mathcal{C})$ is an isomorphism for $n \u003e 0$ and injective for $n = 0$; when $\\mathcal{C}$ is idempotent-complete, it is an isomorphism at $n = 0$ too.\nFor negative degrees, by definition $\\mathrm{K}_{-1}(\\mathcal{C}) = \\mathrm{k}_0(\\mathsf{Calk}(\\mathcal{C}))$, which measures the retract-obstruction to idempotent-completeness of the quotient $\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} / \\mathcal{C}$. In many practical cases this obstruction vanishes: for a regular ring $R$, $\\mathcal{C} = \\mathsf{Perf}(R)$ yields an automatically idempotent-complete quotient, so $\\mathrm{K}_{-1}(R) = 0$; in fact all negative K-groups of a regular ring vanish, and $\\mathrm{K}(R) \\simeq \\mathrm{k}(R)$.\nThe key tool for lifting connective results to the non-connective world is:\nProposition 23 (Properties of Calk). $\\mathsf{Calk}\\colon \\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}} \\to \\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}}$ satisfies:\nGiven a filtered diagram $\\mathcal{C}_{\\bullet}\\colon I \\to \\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}}$, the sequence \\[ \\operatorname*{colim}_i \\mathcal{C}_i \\to \\operatorname*{colim}_i \\mathsf{Ind}(\\mathcal{C}_i)^{\\aleph_1} \\to \\operatorname*{colim}_i \\mathsf{Calk}(\\mathcal{C}_i) \\] at the filtered colimit satisfies condition $(2')$ of Proposition 19 . $\\mathsf{Calk}$ preserves type-$(6)$ sequences: if a cofibre sequence $\\mathcal{C} \\to \\mathcal{D} \\to \\mathcal{E}$ induces a long exact sequence on $\\mathrm{k}$, then so does $\\mathsf{Calk}(\\mathcal{C}) \\to \\mathsf{Calk}(\\mathcal{D}) \\to \\mathsf{Calk}(\\mathcal{E})$. At the level of endofunctors on $\\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}}$, there are natural $\\mathrm{k}$-equivalences \\[ \\mathsf{Calk}(-)_+ \\simeq_{\\mathrm{k}} \\mathsf{Calk}((-)_+), \\qquad \\mathsf{SW}(\\mathsf{Calk}(-)) \\simeq_{\\mathrm{k}} \\mathsf{Calk}(\\mathsf{SW}(-)). \\] $\\mathsf{Calk}$ preserves filtered colimits up to $\\mathrm{k}$-equivalence: \\[ \\mathrm{k}\\!\\left(\\operatorname*{colim}_i \\mathsf{Calk}(\\mathcal{C}_i)\\right) \\xrightarrow{\\;\\sim\\;} \\mathrm{k}\\!\\left(\\mathsf{Calk}\\!\\left(\\operatorname*{colim}_i \\mathcal{C}_i\\right)\\right). \\] Proof. (1) Via $\\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}} \\simeq \\mathsf{Pr}_{\\aleph_0}^L$ and the inclusion $\\mathsf{Pr}_{\\aleph_0}^L \\subset \\mathsf{Pr}_{\\mathrm{ca}}^L$, we prove a more general statement for filtered diagrams in $\\mathsf{Pr}_{\\mathrm{ca}}^L$.\nThe key fact about compactly assembled categories: for $\\mathcal{C} \\in \\mathsf{Pr}_{\\mathrm{ca}}^L$ the colimit functor $k\\colon \\mathsf{Ind}(\\mathcal{C}) \\to \\mathcal{C}$ has both a right adjoint $y$ (Yoneda) and a left adjoint $\\hat{y}$, \\[ \\hat{y} \\dashv k \\dashv y. \\] The embedding $\\hat{y}\\colon \\mathcal{C} \\hookrightarrow \\mathsf{Ind}(\\mathcal{C})$ is fully faithful and colimit-preserving, though it need not preserve limits — this is the extra structure provided by compact assembly.\nCrucially, compactly assembled functors preserve this adjoint structure. For a transition functor $F_{ij}\\colon \\mathcal{C}_i \\to \\mathcal{C}_j$ in our filtered diagram, compact assembly gives a commuting diagram\nConsider the induced functor \\[ \\phi\\colon \\operatorname*{colim}_i \\mathcal{C}_i \\to \\operatorname*{colim}_i \\mathsf{Ind}(\\mathcal{C}_i^{\\aleph_1}) \\simeq \\mathsf{Ind}\\!\\left(\\operatorname*{colim}_i \\mathcal{C}_i^{\\aleph_1}\\right). \\] We show $\\phi$ is a strong left adjoint. Passing to right adjoints (using that $\\hat{y}$\u0026rsquo;s right adjoint is $k$, and $k$\u0026rsquo;s right adjoint is $y$), limits can be computed in $\\mathsf{Cat}$; the limit identifies $y$\u0026rsquo;s limit as the right adjoint of the right adjoint of $\\phi$, establishing strongness.\nFully faithfulness and retract-closedness pass through filtered colimits by below.\n(2) Assume $\\mathcal{C} \\to \\mathcal{D} \\to \\mathcal{E}$ induces a $\\mathrm{k}$-long-exact sequence. Consider the $3 \\times 3$ diagram\nAll columns induce $\\mathrm{k}$-long-exact sequences (by the earlier construction); the top row does by assumption; the middle row is zero by the Eilenberg swindle, hence trivially long-exact. Commutativity of cofibres in $\\mathsf{Sp}$ gives long-exactness of the bottom row.\n(3) For the stabilised version $\\mathsf{SW}(\\mathsf{Calk}(-)) \\simeq_{\\mathrm{k}} \\mathsf{Calk}(\\mathsf{SW}(-))$, consider for $\\mathcal{C} \\in \\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}}$ the diagram\nBoth rows satisfy $(2')$, hence are $\\mathrm{k}$-long-exact; the middle column is zero by the Eilenberg swindle. The five lemma forces the right column to be a $\\mathrm{k}$-equivalence. The pointed case follows similarly, or from the stable case via $\\mathsf{SW}((-)_+) \\simeq \\mathsf{SW}(-)$ and Proposition 6 .\n(4) For a filtered diagram $\\mathcal{C}_{\\bullet}$ with colimit $\\mathcal{C}$, consider\nThe middle vertical is an equivalence because $\\mathsf{Ind}(-)^{\\aleph_1}$ preserves filtered colimits. Both rows are $(2')$-sequences (by part 1 for the top), so they are $\\mathrm{k}$-long-exact. The middle term is zero (Eilenberg), and the five lemma gives the claim.\n$\\square$ Lemma 24. Let $\\mathsf{F}, \\mathsf{R} \\subseteq \\mathrm{Ar}(\\mathsf{Cat}^{\\mathrm{rex}})$ be respectively the full subcategories of fully faithful functors and of functors whose image is retract-closed. Then $\\mathsf{F}$, $\\mathsf{R}$ and their intersection are closed under filtered colimits in $\\mathrm{Ar}(\\mathsf{Cat}^{\\mathrm{rex}})$ (equivalently, in $\\mathrm{Ar}(\\mathsf{Cat})$). Proof. Full faithfulness: mapping spaces in a filtered colimit of categories are filtered colimits of the levelwise mapping spaces.\nRetract-closedness: suppose $d \\in \\mathcal{D}_{\\infty} = \\operatorname*{colim}_i \\mathcal{D}_i$ is a retract of some $\\alpha_{\\infty}(c)$. The minimal diagram witnessing a retract relation is finite, hence compact in $\\mathsf{Cat}$, so the witness already lives at some finite stage $\\mathcal{D}_i$: there is a $d_i \\in \\mathcal{D}_i$ retract of $\\alpha_i(c_i)$. By the levelwise hypothesis $d_i = \\alpha_i(c_i')$, so $d = \\alpha_{\\infty}(\\lambda_i(c_i'))$.\n$\\square$ The main corollary lifts connective properties of $\\mathrm{k}$ to non-connective $\\mathrm{K}$.\nProposition 25 (Properties of K). The non-connective K-theory functor $\\mathrm{K}\\colon \\mathsf{Cat}^{\\mathrm{rex}} \\to \\mathsf{Sp}$ has:\nIdempotent-completion invariance. The natural map $\\mathrm{k}(\\mathcal{C}) \\to \\mathrm{K}(\\mathcal{C})$ is an isomorphism on $\\pi_n$ for $n \\ge 1$ and injective on $\\pi_0$. When $\\mathcal{C}$ is idempotent-complete, $\\mathrm{k}(\\mathcal{C}) \\to \\mathrm{K}(\\mathcal{C})$ is the connective cover. $\\mathrm{K}$ preserves filtered colimits and finite products. Eilenberg swindle. If $F\\colon \\mathcal{C} \\to \\mathcal{C}$ is finite-colimit-preserving with $F \\sqcup \\mathrm{id} \\simeq F$, then $\\mathrm{K}(\\mathcal{C}) \\simeq 0$. If $S = (\\mathcal{C} \\to \\mathcal{D} \\to \\mathcal{E})$ in $\\mathsf{Cat}^{\\mathrm{rex},\\mathrm{idem}}$ induces a $\\mathrm{k}$-long-exact sequence, then it also induces a $\\mathrm{K}$-long-exact sequence. In particular, all sequence types of Proposition 19 are sent to long-exact sequences by $\\mathrm{K}$. $\\mathrm{K}$ sends the canonical maps $\\mathcal{C} \\to \\mathcal{C}_+$ and $\\mathcal{C} \\to \\mathsf{SW}(\\mathcal{C})$ to equivalences. Given connective $\\mathrm{k}$, properties 1 and 4 already determine $\\mathrm{K}$ uniquely.\nProof. (1) Definition plus cofinality.\n(2) Semi-additivity of $\\mathsf{Cat}^{\\mathrm{rex}}$ gives product-preservation for $\\mathsf{Calk}$, hence for $\\mathrm{K} = \\operatorname*{colim} \\Omega^n \\mathrm{k} \\mathsf{Calk}^n$. Filtered colimits: reduce to idempotent-complete $\\mathcal{C}$ via (1), then combine Proposition 7 and Proposition 23 (4).\n(3) Product-preservation yields a map $\\mathrm{K}(F)\\colon \\mathrm{K}(\\mathcal{C}) \\to \\mathrm{K}(\\mathcal{C})$ with $\\mathrm{K}(F) + \\mathrm{id} = \\mathrm{id}$ in $\\pi_0 \\mathrm{Hom}_{\\mathsf{Sp}}(\\mathrm{K}(\\mathcal{C}), \\mathrm{K}(\\mathcal{C}))$. This is a group, so $\\mathrm{id} = 0$ and $\\mathrm{K}(\\mathcal{C}) \\simeq 0$.\n(4) By Proposition 23 (2).\n(5) Combine Proposition 23 (3) with the connective equivalences Proposition 6 and Proposition 9 .\n$\\square$ References A. Krause, T. Nikolaus, P. Pützstück. Sheaves on Manifolds. 2024. PDF. J. Lurie. Higher Topos Theory. Ann. Math. Stud. 170, Princeton Univ. Press, 2009. J. Lurie. Kerodon. Online, 2018–. kerodon.net. ","permalink":"https://ou-liu-red-sugar.github.io/notes/notes/continuous-k-theory/algebraic-k-theory/","summary":"\u003cp\u003eIn this section we first define the \u003cem\u003econnective\u003c/em\u003e K-theory spectrum\n$\\mathrm{k}(\\mathcal{C})$ of a category with finite colimits, then extend it to\nthe \u003cem\u003enon-connective\u003c/em\u003e K-theory spectrum $\\mathrm{K}(\\mathcal{C})$.\u003c/p\u003e\n\u003ch2 id=\"connective-k-theory\"\u003eConnective K-theory\u003c/h2\u003e\n\u003cdiv class=\"thm-block kind-construction has-title\" id=\"cons-cospan\"\u003e\n    \u003cdiv class=\"thm-header\"\u003e\n      \u003cspan class=\"thm-title\"\u003e\n        Construction 1 (Cospan category).\n      \u003c/span\u003e\n    \u003c/div\u003e\n    \u003cdiv class=\"thm-body\"\u003e\u003cp\u003eLet $\\mathcal{C}$ be a category with pushouts. Applied to the saturated triple\n$(\\mathcal{C}^{\\mathrm{op}}, \\mathcal{C}^{\\mathrm{op}}, \\mathcal{C}^{\\mathrm{op}})$,\nthe twisted arrow construction produces a complete Segal anima\n$\\mathrm{N}\\mathsf{Span}(\\mathcal{C}^{\\mathrm{op}})$; denote the corresponding\ncategory by $\\mathsf{coSpan}(\\mathcal{C})$. Concretely:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003eObjects of $\\mathsf{coSpan}(\\mathcal{C})$ are the objects of $\\mathcal{C}$.\u003c/p\u003e","title":"Algebraic K-theory"},{"content":"Definition Definition 1. Let $\\kappa$ be a regular cardinal (for example $\\kappa = \\omega$ or $\\kappa = \\aleph_1$).\nA category $\\mathcal{I}$ is called $\\kappa$-filtered if for any $\\kappa$-small category $\\mathcal{K}$ and any functor $F\\colon \\mathcal{K} \\to \\mathcal{I}$, there exists an extension $F^{\\rhd} \\colon \\mathcal{K}^{\\rhd} \\to \\mathcal{I}$. When $\\kappa = \\omega$, we simply call a $\\kappa$-filtered category a filtered category.\nLet $\\mathcal{C}$ be a category admitting small colimits. An object $X \\in \\mathcal{C}$ is called $\\kappa$-compact if for any $\\kappa$-filtered diagram $(Y_i)_{i \\in \\mathcal{I}}$, there is an isomorphism of mapping spaces \\[ \\operatorname{Hom}_{\\mathcal{C}}(X,\\operatorname{colim}_i Y_i) \\simeq \\operatorname{colim}_i \\operatorname{Hom}_{\\mathcal{C}}(X,Y_i). \\] Denote by $\\mathcal{C}^{\\kappa} \\subset \\mathcal{C}$ the full subcategory spanned by all $\\kappa$-compact objects. When $\\kappa = \\omega$, we simply call $\\kappa$-compact objects compact objects.\nProposition 2. In the anima category $\\mathsf{An}$, $\\kappa$-filtered colimits commute with $\\kappa$-small limits. Proof. The proof can be found in [HTT, Proposition 5.3.3.3] , and a model-independent version of the proof can be found in [Hau25, Corollary 9.9.3] . In fact, this statement also holds for general anima.\nIn fact, the underlying intuition is relatively simple; we take the case of $\\kappa$-small products as an example. Consider \\[ \\operatorname{colim}_{i \\in I} \\prod_{j \\in J} X_{ij} \\to \\prod_{j \\in J} \\operatorname{colim}_{i \\in I} X_i. \\] Here $I$ is a $\\kappa$-filtered category and $J$ is a $\\kappa$-small category. We note that\nA point in the anima on the left-hand side actually consists of the following data: there exists an $i \\in I$, and for each $j \\in J$ a chosen point in $X_{ij}$.\nFor each $j \\in J$, choose some $i(j) \\in I$, and then choose a point in $X_{i(j),j}$.\nSince $I$ is a $\\kappa$-filtered category, for the functor $J \\to I$, we can extend it to $J^{\\rhd} \\to I$. This means that in $I$ we can always choose an upper bound of the $i(j)$. Thus, a point in the anima on the right-hand side is equivalent to choosing some $\\operatorname{sup}_{j \\in J} i(j) \\in I$, and then making the choices for each $j \\in J$, which yields the isomorphism.\n$\\square$ From the proposition above, we can easily obtain the following corollary.\nCorollary 3. Any $\\kappa$-small colimit of $\\kappa$-compact objects and any retract of a $\\kappa$-compact object are $\\kappa$-compact. Proof. Let $(X_i)_{i \\in I}$ be some $\\kappa$-compact objects in $\\mathcal{C}$, where $I$ is a $\\kappa$-small category. We verify that $\\operatorname{colim}_{i \\in I} X_i$ is $\\kappa$-compact. For this, it suffices to note that \\[ \\operatorname{Hom}_{\\mathcal{C}}(\\operatorname{colim}_i X_i,-) \\simeq \\operatorname{lim}_i \\operatorname{Hom}_{\\mathcal{C}}(X_i,-). \\] For the case of retracts, let $X$ be a retract of a $\\kappa$-compact object $X'$. Then $\\operatorname{Hom}_{\\mathcal{C}}(X,-)$ is a retract of $\\operatorname{Hom}_{\\mathcal{C}}(X',-)$. Since a retract of an isomorphism is still an isomorphism, the result follows.\n$\\square$ Now consider the presheaf category \\[ \\mathsf{PShv}(\\mathcal{C}) = \\mathsf{Fun}(\\mathcal{C}^{\\operatorname{op}}, \\mathsf{An}), \\] and denote the Yoneda embedding by \\[ y \\colon \\mathcal{C} \\longrightarrow \\mathsf{PShv}(\\mathcal{C}). \\] Proposition 4. Let $\\mathcal{C}$ be a small category. In the presheaf category $\\mathsf{PShv}(\\mathcal{C})$, an object $F$ is $\\kappa$-compact if and only if it is a retract of a $\\kappa$-small colimit of $y(X_i)$. Proof. Suppose $F$ is $\\kappa$-compact, and decompose $F$ as a colimit \\[ F \\simeq \\operatorname{colim}_i y(X_i). \\] Since any colimit can be written as a combination of a $\\kappa$-filtered colimit and a $\\kappa$-small colimit, we may write the indexing category $I$ as a filtered colimit \\[ I \\simeq \\operatorname{colim}_{k \\in K} \\varphi, \\] where $\\varphi \\colon K \\to \\mathsf{Cat}$ is a diagram of $\\kappa$-small categories. Then \\[ \\operatorname{Hom}_{\\mathsf{PShv}(\\mathcal{C})}(F,F) \\simeq \\operatorname{colim}_k \\operatorname{Hom}_{\\mathsf{PShv}(\\mathcal{C})} \\bigl(F,\\operatorname{colim}_{\\varphi(k)} y(X_i)\\bigr). \\] It is not hard to see that there exists some $k \\in K$ such that $F$ is a retract of $\\operatorname{colim}_{i \\in \\varphi(k)} y(X_i)$.\nConversely, since each $y(X)$ is $\\kappa$-compact, this is exactly Corollary 3 .\n$\\square$ Ind Completion Definition 5. Let $\\mathcal{C}$ be a small category and $\\kappa$ a regular cardinal. Define $\\mathsf{Ind}_{\\kappa}(\\mathcal{C}) \\subset \\mathsf{PShv}(\\mathcal{C})$ to be the smallest full subcategory containing $y(\\mathcal{C})$ and closed under $\\kappa$-filtered colimits. When $\\kappa = \\omega$, we omit the subscript $\\kappa$. Remark. Dually, consider the covariant Yoneda embedding \\[ y' \\colon \\mathcal{C} \\longrightarrow \\mathsf{Fun}(\\mathcal{C},\\mathsf{An})^{\\operatorname{op}}, \\qquad X \\longmapsto \\operatorname{Hom}_{\\mathcal C}(X,-). \\] Define $\\mathsf{Pro}_{\\kappa}(\\mathcal{C})$ to be the full subcategory of $\\mathsf{Fun}(\\mathcal{C},\\mathsf{An})^{\\operatorname{op}}$ that contains $y'(\\mathcal{C})$ and is closed under $\\kappa$-cofiltered limits. It is easy to see that \\[ \\mathsf{Pro}_{\\kappa}(\\mathcal{C}) \\simeq \\mathsf{Ind}_{\\kappa}(\\mathcal{C}^{\\operatorname{op}})^{\\operatorname{op}}. \\] By definition, any object in $\\mathsf{Ind}_{\\kappa}(\\mathcal{C})$ can be written as a $\\kappa$-filtered colimit of the form \\[ \\operatorname{colim}_{i} y(X_i), \\] where the indexing category is $\\kappa$-filtered.\nHowever, in general there is no natural isomorphism \\[ \\operatorname{colim}_{i} y(X_i) \\;\\simeq\\; y\\!\\left(\\operatorname{colim}_{i} X_i\\right). \\] Therefore, even if the corresponding filtered colimit exists in $\\mathcal{C}$, it need not agree with the formally given filtered colimit in $\\mathsf{Ind}_{\\kappa}(\\mathcal{C})$.\nOne can also compute the Hom anima in $\\mathsf{Ind}_{\\kappa}(\\mathcal{C})$ in this way: \\[\\begin{align*} \\operatorname{Hom}_{\\mathsf{Ind}_{\\kappa}(\\mathcal{C})}(X,Y) \u0026 \\simeq \\operatorname{Hom}_{\\mathsf{Ind}_{\\kappa}(\\mathcal{C})} \\bigl(\\operatorname{colim}_i y(X_i),\\operatorname{colim}_j y(Y_j)\\bigr) \\\\ \u0026 \\simeq \\operatorname{lim}_i \\operatorname{Hom}_{\\mathsf{Ind}_{\\kappa}(\\mathcal{C})} \\bigl(y(X_i),\\operatorname{colim}_j y(Y_j)\\bigr)\\\\ \u0026 \\simeq \\operatorname{lim}_i \\operatorname{colim}_{j} y(Y_j)(X_i) = \\operatorname{lim}_i \\operatorname{colim}_{j} \\operatorname{Hom}_{\\mathcal{C}}(X_i,Y_j). \\end{align*}\\] Example 6. Let $\\mathcal I$ be a cofiltered category whose objects are $\\mathbb N$, and such that for $m \\le n$ there is a unique morphism $n \\to m$. Define a functor \\[ X \\colon \\mathcal I \\to \\mathsf{Ab}, \\qquad n \\longmapsto \\mathbb Z, \\] whose structure map $n \\to m$ is given by multiplication by $2^{\\,n-m}$. Then \\[ \\lim_{\\mathcal I} X = 0 . \\]View $X$ as an object of $\\mathsf{Pro}(\\mathsf{Ab})$. Tensor it levelwise with $\\mathbb Z[1/2]$, obtaining a cofiltered diagram $X \\otimes \\mathbb Z[1/2]$. Since multiplication by $2$ is an isomorphism on $\\mathbb Z[1/2]$, this diagram is equivalent to a constant diagram, hence \\[ \\lim_{\\mathcal I} (X \\otimes \\mathbb Z[1/2]) \\cong \\mathbb Z[1/2]. \\]This shows that an object of $\\mathsf{Pro}(\\mathsf{Ab})$ is not determined by its limit in $\\mathsf{Ab}$.\nRemark. Strictly speaking, $\\mathsf{Ab}$ is a large category. One may work inside a sufficiently large Grothendieck universe, or equivalently regard $\\mathsf{Ab}$ as a small category relative to a larger universe. We can summarize the universal property of $\\mathsf{Ind}_{\\kappa}(\\mathcal{C})$ as follows.\nLemma 7. Let $\\mathcal{D}$ be a category admitting $\\kappa$-filtered colimits. Then the Yoneda embedding induces an equivalence of categories \\[ \\mathsf{Fun}^{\\kappa\\text{-fil}}(\\mathsf{Ind}_{\\kappa}(\\mathcal{C}),\\mathcal{D}) \\simeq \\mathsf{Fun}(\\mathcal{C},\\mathcal{D}). \\] Therefore, given a category $\\mathcal{D}$ admitting $\\kappa$-filtered colimits and a functor $F \\colon \\mathcal{C} \\to \\mathcal{D}$, one obtains an induced functor $\\mathsf{Ind}_{\\kappa}(F) \\colon \\mathsf{Ind}_{\\kappa}(\\mathcal{C}) \\to \\mathcal{D}$ such that the following diagram commutes:\nWe also call $\\mathsf{Ind}_{\\kappa} F$ the $\\mathsf{Ind}_{\\kappa}$-extension of $F$. Its concrete construction is nothing but Kan extension.\nMoreover, if $F$ is fully faithful and its image lands in the full subcategory of $\\kappa$-compact objects $\\mathcal{D}^{\\kappa}$, then its $\\mathsf{Ind}_{\\kappa}$-extension is also fully faithful. Furthermore, if $F(\\mathcal{C})$ generates $\\mathcal{D}$ under $\\kappa$-filtered colimits, then $\\mathsf{Ind}_{\\kappa}(F)$ is an equivalence of categories.\nWhen $\\mathcal{C}$ admits $\\kappa$-small colimits, we can identify $\\mathsf{Ind}_{\\kappa}(\\mathcal{C})$ as $\\mathsf{Fun}^{\\kappa\\text{-rex}}(\\mathcal{C}^{\\operatorname{op}},\\mathsf{An})$, that is, the category of all functors $\\mathcal{C}^{\\operatorname{op}} \\to \\mathsf{An}$ preserving $\\kappa$-small limits.\nProposition 8. Let $\\mathcal{C}$ be a small category admitting $\\kappa$-small colimits. Then there is an equivalence of categories \\[ \\mathsf{Ind}_{\\kappa}(\\mathcal{C}) \\simeq \\mathsf{Fun}^{\\kappa\\text{-rex}}(\\mathcal{C}^{\\operatorname{op}},\\mathsf{An}). \\] Since limits in $\\mathsf{Fun}(\\mathcal{C}^{\\operatorname{op}},\\mathsf{An})$ are computed pointwise, and using the commutation of limits, it follows that $\\mathsf{Ind}_{\\kappa}(\\mathcal{C})$ admits all limits.\nProof. For $F \\in \\mathsf{Ind}_{\\kappa}(\\mathcal{C})$, we may write $F$ as a $\\kappa$-filtered colimit \\[ \\operatorname{colim}_i y(X_i) = \\operatorname{colim}_i \\operatorname{Hom}_{\\mathcal{C}}(-,X_i). \\] For $\\kappa$-small limits in $\\mathcal{C}^{\\operatorname{op}}$, these are automatically $\\kappa$-small colimits in $\\mathcal{C}$. Using \\[ \\operatorname{colim}_i \\operatorname{Hom}_{\\mathcal{C}}(\\operatorname{colim}_j Y_j,X_i) \\simeq \\operatorname{colim}_i \\operatorname{lim}_j \\operatorname{Hom}_{\\mathcal{C}}(Y_j,X_i) \\] together with Proposition Proposition 2 , the claim follows.\nNow let $F \\colon \\mathcal{C}^{\\operatorname{op}} \\to \\mathsf{An}$ be a functor preserving $\\kappa$-small limits. The Grothendieck construction $\\mathsf{El}(F)$ is given by the pullback\nSince $F$ preserves $\\kappa$-small limits, $\\mathsf{El}(F)$ is a $\\kappa$-filtered category. Hence $F$, as the colimit of the composite \\[ \\mathsf{El}(F) \\to \\mathcal{C} \\to \\mathsf{PShv}(\\mathcal{C}), \\] is a $\\kappa$-filtered colimit.\n$\\square$ Corollary 9. Let $\\mathcal{C}$ be a category admitting $\\kappa$-small colimits. Then \\[ y \\colon \\mathcal{C} \\to \\mathsf{PShv}(\\mathcal{C}) \\] preserves $\\kappa$-small colimits.\nProof. It suffices to show that for any diagram $(X_i)_i$ indexed by a $\\kappa$-small category $I$, there is an equivalence in $\\mathsf{Ind}_{\\kappa}(\\mathcal{C})$ \\[ \\operatorname{colim}_i y(X_i) \\simeq y(\\operatorname{colim}_i X_i). \\] By the Yoneda lemma, this is equivalent to saying that for any $F \\in \\mathsf{Ind}_{\\kappa}(\\mathcal{C})$, \\[ \\operatorname{lim}_i F(X_i) \\simeq F(\\operatorname{colim}_i X_i), \\] which reduces to Proposition Proposition 8 .\n$\\square$ Since any colimit can be written as a $\\kappa$-filtered colimit together with a $\\kappa$-small colimit, for a small category $\\mathcal{C}$ admitting $\\kappa$-small colimits, $\\mathsf{Ind}_{\\kappa}$ also yields the following universal property.\nProposition 10. Let $\\mathcal{C}$ be a small category admitting $\\kappa$-small colimits. Then for any category $\\mathcal{D}$ admitting small colimits, there is an equivalence of categories \\[ \\mathsf{Fun}^{\\operatorname{colim}}(\\mathsf{Ind}_{\\kappa}(\\mathcal{C}),\\mathcal{D}) \\simeq \\mathsf{Fun}^{\\kappa\\text{-colim}}(\\mathcal{C},\\mathcal{D}). \\] In everyday practice, the vast majority of “large categories” we encounter can naturally be written as the Ind-completion of some small category. In other words, they are often generated by a family of “finite / compact objects” under filtered colimits.\nFor example:\nThe category of sets can be presented as \\[ \\mathsf{Set} \\;\\simeq\\; \\mathsf{Ind}(\\mathsf{FinSet}), \\] that is, any set is a filtered colimit of finite sets.\nThe anima category can be presented as \\[ \\mathsf{An} \\;\\simeq\\; \\mathsf{Ind}(\\mathsf{An}^{\\mathrm{fin}}), \\] where $\\mathsf{An}^{\\mathrm{fin}}$ denotes the full subcategory of finite anima.\nFor a fixed (derived) ring $R$, the category of modules can be presented as \\[ \\mathsf{Mod}_R \\;\\simeq\\; \\mathsf{Ind}(\\mathsf{Perf}_R), \\] that is, any $R$-module is a filtered colimit of perfect modules.\nThese examples show that Ind-completion is not an abstract artificial construction, but a unified language for describing “large objects generated from small data via filtered colimits”. This also leads to the notion of accessible categories.\nAccessible Categories We want to investigate the following question: which categories can be regarded as the $\\mathsf{Ind}_{\\kappa}$-completion of their $\\kappa$-compact objects?\nDefinition 11. Let $\\mathcal{C}$ be a category, and let $\\kappa$ be a regular cardinal. We say that $\\mathcal{C}$ is a $\\kappa$-accessible category if:\n$\\mathcal{C}$ admits $\\kappa$-filtered colimits. The full subcategory $\\mathcal{C}^{\\kappa}$ is small. There is an equivalence $\\mathcal{C} \\simeq \\mathsf{Ind}_{\\kappa}(\\mathcal{C}^{\\kappa})$. We say that $\\mathcal{C}$ is an accessible category if there exists some $\\kappa$ such that $\\mathcal{C}$ is $\\kappa$-accessible.\nExample 12. For any small category $\\mathcal{C}$, the presheaf category $\\mathsf{PShv}(\\mathcal{C})$ is $\\kappa$-accessible for any $\\kappa$. In fact, $\\mathsf{PShv}(\\mathcal{C}) \\simeq \\mathsf{Ind}_{\\kappa}(\\mathsf{PShv}(\\mathcal{C})^{\\kappa})$. Remark. Let $\\lambda \u003e \\kappa$ be a regular cardinal. In general, a $\\kappa$-accessible category need not be $\\lambda$-accessible. From the description of Hom anima in $\\mathsf{Ind}_{\\kappa}(\\mathcal{C})$ above, it follows that accessible categories are locally small categories.\nIn fact, accessible categories can be characterized as idempotent complete categories.\nProposition 13 (Lurie). A small category $\\mathcal{C}$ is accessible if and only if it is idempotent complete. Proof. [HTT, 5.4.3] $\\square$ General Adjoint Functor Theorem In [NRS18] , they prove the following theorem.\nTheorem 14 (adjoint functor theorem). Let $F \\colon \\mathcal{C} \\to \\mathcal{D}$ be a functor between locally small categories.\nAssume that $\\mathcal{C}$ and $\\mathcal{D}$ have all colimits and that $\\mathcal{C}$ is generated under colimits by an essentially small subcategory $\\mathcal{C}_0 \\subset \\mathcal{C}$. Then $F$ admits a right adjoint $G \\colon \\mathcal{D} \\to \\mathcal{C}$ if and only if $F$ preserves colimits.\nAssume that $\\mathcal{C}$ and $\\mathcal{D}$ have all limits, that $\\mathcal{C}$ is accessible, and that for every object $y \\in \\mathcal{D}$ there exists a regular cardinal $\\kappa_y$ such that $y$ is $\\kappa_y$-compact. If there exists a regular cardinal $\\kappa$ such that $F$ preserves limits as well as $\\kappa$-filtered colimits, then $F$ admits a left adjoint $G \\colon \\mathcal{D} \\to \\mathcal{C}$. The converse is true as well provided that $\\mathcal{D}$ is accessible too.\nThus, one can find that in an accessible category, completeness is equivalent to cocompleteness.\nCorollary 15. Let $\\mathcal C$ be a locally small category.\nIf $\\mathcal C$ admits all small colimits and is generated under colimits by an essentially small sub-category, then $\\mathcal C$ admits all small limits.\nIf $\\mathcal C$ is accessible and admits all small limits, then $\\mathcal C$ admits all small colimits.\nProof. Let $\\mathcal I$ be an essentially small category. Then $\\operatorname{Fun}(\\mathcal I,\\mathcal C)$ is locally small, and the constant diagram functor \\[ \\operatorname{const} \\colon \\mathcal C \\longrightarrow \\operatorname{Fun}(\\mathcal I,\\mathcal C) \\] preserves all limits and colimits.\n(a) If $\\mathcal C$ admits all small colimits and is generated under colimits by an essentially small subcategory, then by the adjoint functor theorem, $\\operatorname{const}$ admits a right adjoint \\[ \\lim_{\\mathcal I} \\colon \\operatorname{Fun}(\\mathcal I,\\mathcal C) \\to \\mathcal C. \\] Since $\\mathcal I$ was arbitrary, $\\mathcal C$ admits all small limits.\n(b) Assume that $\\mathcal C$ is accessible and admits all small limits. We show that $\\operatorname{const}$ admits a left adjoint \\[ \\operatorname{colim}_{\\mathcal I} \\colon \\operatorname{Fun}(\\mathcal I,\\mathcal C) \\to \\mathcal C. \\] By accessibility, every object of $\\mathcal C$ is $\\kappa$-compact for some regular cardinal $\\kappa$. For any functor $\\alpha \\colon \\mathcal I \\to \\mathcal C$, choose a sufficiently large regular cardinal $\\tau$ such that $\\mathcal I$ is $\\tau$-small and each value $\\alpha(i)$ is $\\tau$-compact.\nUsing that $\\tau$-small limits commute with $\\tau$-filtered colimits, and that colimits in functor categories are computed pointwise, it follows that $\\alpha$ is $\\tau$-compact. Hence $\\operatorname{Fun}(\\mathcal I,\\mathcal C)$ is generated under filtered colimits by compact objects, and the adjoint functor theorem implies that $\\operatorname{const}$ admits a left adjoint. Therefore $\\mathcal C$ admits all small colimits.\n$\\square$ Presentable Categories Now, we discuss presentable categories.\nDefinition 16. Let $\\mathcal{C}$ be a category and let $\\kappa$ be a regular cardinal. We say that $\\mathcal{C}$ is $\\kappa$-presentable (or $\\kappa$-compactly generated) if:\n$\\mathcal{C}$ is $\\kappa$-accessible. $\\mathcal{C}$ admits small limits. We say that $\\mathcal{C}$ is presentable if there exists some $\\kappa$ such that $\\mathcal{C}$ is $\\kappa$-presentable. In particular, when $\\kappa = \\omega$, we say that $\\mathcal{C}$ is compactly generated.\nRemark. By Corollary 15 , accessibility together with the existence of all small limits already implies the existence of all small colimits, and hence presentability in the sense of Lurie. I thank Marc Hoyois for pointing out this equivalence. Proposition 17. Let $\\mathcal{C}$ be an idempotent complete category. For a regular cardinal $\\kappa$, the following are equivalent:\n$\\mathcal{C}$ admits $\\kappa$-small colimits. $i \\colon \\mathcal{C} \\hookrightarrow \\mathsf{PShv}(\\mathcal{C})^{\\kappa}$ admits a left adjoint $L$. Proof. Suppose that $\\mathcal{C}$ admits $\\kappa$-small colimits. We need to show that the copresheaf $\\operatorname{Hom}_{\\mathsf{PShv}(\\mathcal{C})}(\\mathcal{F},y)$ on $\\mathcal{C}$ is corepresentable for every $\\mathcal{F} \\in \\mathsf{PShv}(\\mathcal{C})^{\\kappa}$. By Proposition 4 , there exists a functor $p \\colon K \\to \\mathcal{C}$ such that $\\mathcal{F}$ is a retract of $\\psi = \\operatorname{colim} y \\circ p$. Observe that by the Yoneda lemma, we have an isomorphism \\[ \\operatorname{Hom}_{\\mathsf{PShv}(\\mathcal{C})}(\\psi,y(-)) \\simeq \\operatorname{lim}_{K^{\\operatorname{op}}}\\operatorname{Hom}_{\\mathcal{C}}(p,-) \\simeq \\operatorname{Hom}_{\\mathcal{C}}(\\operatorname{colim} p,-). \\] Since $\\mathcal{C}$ is idempotent complete, it then follows that the original copresheaf is corepresented by a retract of $\\operatorname{colim} p$. Now, suppose 2. is satisfied, so $i \\colon \\mathcal{C} \\to \\mathsf{PShv}(\\mathcal{C})^{\\kappa}$ admits a left adjoint $L$. For a diagram $p \\colon K \\to \\mathcal{C}$, we know that $i \\circ p$ admits a colimit in $\\mathsf{PShv}(\\mathcal{C})^{\\kappa}$, since $L$ is a Bousfield localization, $p$ has a colimit in $\\mathcal{C}$. $\\square$ Suppose that we have an adjunction $f \\dashv g$. Then we can get an adjunction $g^{\\operatorname{op}} \\dashv f^{\\operatorname{op}}$ (Since $(-)^{\\operatorname{op}}$ is a 2-functor and 2-functors preserve adjunctions), and so an adjunction $f^{\\operatorname{op},*} \\dashv g^{\\operatorname{op},*}$. Since adjunction is unique, we have $f^{\\operatorname{op},*} \\simeq g^{\\operatorname{op}}_!$. So that we also have \\[ f_!^{\\operatorname{op}} \\dashv g_!^{\\operatorname{op}}. \\] Here, both functors restrict to $\\mathsf{Ind}_{\\kappa}(-)$, then we obtain an adjunction \\[ \\mathsf{Ind}_{\\kappa} f \\dashv \\mathsf{Ind}_{\\kappa} g. \\] Since $\\mathsf{PShv}(\\mathcal{C}) \\simeq \\mathsf{Ind}_{\\kappa}(\\mathsf{PShv}(\\mathcal{C})^{\\kappa})$, we have an adjunction \\[ \\mathsf{Ind}_{\\kappa} L \\dashv \\mathsf{Ind}_{\\kappa} i \\] between $\\mathsf{PShv}(\\mathcal{C})$ and $\\mathsf{Ind}_{\\kappa}(\\mathcal{C})$. One can also see that $\\mathsf{Ind}_{\\kappa} i$ is a fully faithful functor, thus $\\mathsf{Ind}_{\\kappa} L$ is still a Bousfield localization. By Proposition 8 , one can find that in this case, $\\mathsf{Ind}_{\\kappa}(\\mathcal{C})$ is a presentable category.\nWe may summarize the preceding discussion in the following equivalent characterizations of presentable categories.\nTheorem 18. Let $\\mathcal{C}$ be a category, the following conditions are equivalent:\n$\\mathcal{C}$ is $\\kappa$-presentable. $\\mathcal{C}$ is a Bousfield localization of $\\mathsf{PShv}(\\mathcal{D})$ for some small category $\\mathcal{D}$ with $\\kappa$-small colimits. There exists a small category $\\mathcal{D}$ with $\\kappa$-small colimits such that $\\mathcal{C} \\simeq \\mathsf{Ind}_{\\kappa}(\\mathcal{D})$. $\\mathcal{C}^{\\kappa}$ is small, and the $\\mathsf{Ind}_{\\kappa}$-extension \\[ \\mathsf{Ind}_{\\kappa}(\\mathcal{C}^{\\kappa}) \\to \\mathcal{C} \\] is an equivalence. The preceding proposition suggests that a presentable category $\\mathcal{C}$ should be regarded as generated, under all small colimits, by its $\\kappa$-compact objects We now make this perspective precise.\nProposition 19. Let $\\mathcal{C}$ be a category admitting all small colimits, and assume that the full subcategory $\\mathcal{C}^{\\kappa}$ of $\\kappa$-compact objects is small. The following conditions are equivalent:\n$\\mathcal{C}$ is $\\kappa$-presentable. Every object of $\\mathcal{C}$ can be expressed as a small colimit of $\\kappa$-compact objects. Proof. (1) $\\Rightarrow$ (2). Assume that $\\mathcal{C}$ is $\\kappa$-presentable. By definition, the canonical functor \\[ \\mathsf{Ind}_{\\kappa}(\\mathcal{C}^{\\kappa}) \\longrightarrow \\mathcal{C} \\] is an equivalence. In particular, it is essentially surjective.\nLet $X \\in \\mathcal{C}$. Since $\\mathcal{C}$ admits all small colimits, $X$ can be written as a small colimit \\[ X \\simeq \\operatorname{colim}_{K} Z_k \\] of objects $Z_k \\in \\mathcal{C}^{\\kappa}$. Any small colimit can be expressed as a $\\kappa$-filtered colimit of $\\kappa$-small colimits, hence we may write \\[ X \\simeq \\operatorname{colim}_{K' \\subset K} \\operatorname{colim}_{K'} Z_k, \\] where $K'$ ranges over the $\\kappa$-small subcategories of $K$. By Corollary 3 , each colimit $\\operatorname{colim}_{K'} Z_k$ is again $\\kappa$-compact. Therefore $X$ is a $\\kappa$-filtered colimit of $\\kappa$-compact objects, and hence lies in the essential image of $\\mathsf{Ind}_{\\kappa}(\\mathcal{C}^{\\kappa})$.\n(2) $\\Rightarrow$ (1). Assume that every object of $\\mathcal{C}$ can be expressed as a small colimit of $\\kappa$-compact objects. Then, in particular, every object of $\\mathcal{C}$ is a $\\kappa$-filtered colimit of objects of $\\mathcal{C}^{\\kappa}$. It follows that the canonical functor \\[ \\mathsf{Ind}_{\\kappa}(\\mathcal{C}^{\\kappa}) \\longrightarrow \\mathcal{C} \\] is essentially surjective. Since it is fully faithful, it is an equivalence, and therefore $\\mathcal{C}$ is $\\kappa$-presentable.\n$\\square$ In presentable category, Theorem 14 admits a substantially simpler formulation.\nTheorem 20 (adjoint functor theorem (presentable case)). Let $\\mathcal C$ and $\\mathcal D$ be presentable categories, and let $F \\colon \\mathcal C \\to \\mathcal D$ be a functor.\nThe functor $F$ admits a right adjoint if and only if it preserves all small colimits.\nThe functor $F$ admits a left adjoint if and only if it preserves all small limits, and there exists a regular cardinal $\\kappa$ such that $F$ preserves $\\kappa$-filtered colimits.\nThe Category of Presentable Categories Now, we define the category of presentable categories.\nDefinition 21. Let $\\mathsf{Pr}^L$ be the category whose objects are presentable categories and whose morphisms are colimit-preserving functors.\nFor a regular cardinal $\\kappa$, let $\\mathsf{Pr}^{L}_{\\kappa} \\subset \\mathsf{Pr}^L$ denote the non-full subcategory whose objects are $\\kappa$-presentable categories and whose morphisms are those colimit-preserving functors that additionally preserve $\\kappa$-compact objects.\nRemark. Every presentable category is necessarily a big category. Consequently, the category $\\mathsf{Pr}^L$ of all presentable categories is a very big category.\nNevertheless, a presentable category $\\mathcal C$ admits a presentation by a pair $(\\kappa, \\mathcal C^{\\kappa})$, consisting of a regular cardinal $\\kappa$ and a small category $\\mathcal C^{\\kappa}$ of $\\kappa$-compact objects. From this point of view, $\\mathsf{Pr}^L$ may be regarded as a big category, in the sense that its objects can be encoded by small data.\nDespite this, $\\mathsf{Pr}^L$ is not locally small. Indeed, for any $\\mathcal C \\in \\mathsf{Pr}^L$, one has \\[ \\operatorname{Hom}_{\\mathsf{Pr}^L}(\\mathsf{An}, \\mathcal C) \\simeq \\mathcal C, \\] since $\\mathsf{An}$ is the free presentable category generated by a single object, and $\\mathcal C$ itself is large.\nBy contrast, the category $\\mathsf{Pr}^{L}_{\\kappa}$ is locally small. Indeed, by definition of morphisms in $\\mathsf{Pr}^{L}_{\\kappa}$, any functor $\\mathsf{An} \\to \\mathcal C$ must preserve $\\kappa$-compact objects, and hence factors through the small category $\\mathcal C^{\\kappa}$. As a result, \\[ \\operatorname{Hom}_{\\mathsf{Pr}^{L}_{\\kappa}}(\\mathsf{An}, \\mathcal C) \\simeq \\mathcal C^{\\kappa}. \\]Thus, we consider the category $\\mathsf{Rex}_{\\kappa}$ defined as follows:\nthe objects of $\\mathsf{Rex}_{\\kappa}$ are small categories $\\mathcal D$ admitting all $\\kappa$-small colimits and closed under retracts; the morphisms of $\\mathsf{Rex}_{\\kappa}$ are functors preserving $\\kappa$-small colimits. There is an equivalence \\[ \\mathsf{Pr}_{\\kappa}^L \\simeq \\mathsf{Rex}_{\\kappa}, \\] sending a $\\kappa$-presentable category $\\mathcal C$ to its full subcategory $\\mathcal C^{\\kappa}$ of $\\kappa$-compact objects, and sending $\\mathcal D \\in \\mathsf{Rex}_{\\kappa}$ to $\\mathsf{Ind}_{\\kappa}(\\mathcal D)$. This equivalence is known as Gabriel\u0026ndash;Ulmer duality [HTT, Proposition 5.5.7.8 and Proposition 5.5.7.10] .\nWe see that $\\mathsf{Pr}^L$ is too large to be an object of itself—it is not even locally small. However, after fixing a regular cardinal $\\kappa$, the situation improves substantially.\nTheorem 22 (Capion). Let $\\kappa$ be a regular cardinal. Then the category $\\mathsf{Pr}^{L}_{\\kappa}$ is itself $\\kappa$-presentable; in particular, \\[ \\mathsf{Pr}^{L}_{\\kappa} \\in \\mathsf{Pr}^{L}_{\\kappa}. \\] Proof. By Gabriel\u0026ndash;Ulmer duality, we may identify $\\mathsf{Pr}_{\\kappa}^L$ with $\\mathsf{Rex}_{\\kappa}$. It therefore suffices to show that $\\mathsf{Rex}_{\\kappa}$ is $\\kappa$-presentable.\nFirst, by the remark above, $\\mathsf{Rex}_{\\kappa}$ is locally small. Consider the forgetful functor \\[ G \\colon \\mathsf{Rex}_{\\kappa} \\longrightarrow \\mathsf{Cat}. \\] This functor admits a left adjoint \\[ F \\colon \\mathsf{Cat} \\longrightarrow \\mathsf{Rex}_{\\kappa}, \\qquad F(S) = \\mathsf{PShv}(S)^{\\kappa}. \\]We claim that the adjunction $F \\dashv G$ is monadic, so that $\\mathsf{Rex}_{\\kappa}$ is equivalent to the category of algebras over the monad $GF$ on $\\mathsf{Cat}$. By the Barr\u0026ndash;Beck\u0026ndash;Lurie theorem, it suffices to verify the following conditions: \\begin{itemize}\nthe functor $G$ is conservative; $G$ preserves geometric realizations of $G$-split simplicial objects. The first condition is immediate. For the second, let $\\mathcal C_\\bullet$ be a simplicial object of $\\mathsf{Rex}_{\\kappa}$ whose geometric realization $\\mathcal C_{-1}$ exists in $\\mathsf{Cat}$, and assume that the augmentation $\\mathcal C_\\bullet \\to \\mathcal C_{-1}$ is split in $\\mathsf{Cat}$. Then for any $\\kappa$-small category $I$, the induced morphism \\[ \\mathsf{Fun}(I, \\mathcal C_\\bullet) \\longrightarrow \\mathsf{Fun}(I, \\mathcal C_{-1}) \\] is again a geometric realization in $\\mathsf{Cat}$. Applying this observation to $I$ and to its right cone $I^{\\rhd}$, one readily verifies that $G$ preserves the required geometric realizations.\nThus, $\\mathsf{Rex}_{\\kappa}$ is monadic over $\\mathsf{Cat}$ and in particular admits all small colimits. Since $\\mathsf{Cat}$ is compactly generated by the finite ordinal categories $[n]$ (and hence $\\kappa$-compactly generated), it remains to observe that the functor $F$ preserves $\\kappa$-compact objects. Equivalently, the right adjoint $G$ preserves $\\kappa$-filtered colimits, which follows from [HTT, Proposition 5.5.7.11] .\n$\\square$ It naturally leads to a question:\nGiven that $\\mathsf{Pr}_{\\kappa}^L$ is $\\kappa$-presentable, it has a large subcategory $(\\mathsf{Pr}_{\\kappa}^L)^{\\kappa}$ of $\\kappa$-compact objects. Is it, in fact, a $\\kappa$-compact object of itself, i.e. \\[ \\mathsf{Pr}_{\\kappa}^L \\in (\\mathsf{Pr}_{\\kappa}^L)^{\\kappa}? \\] This is not full truth. Proposition 23. One has $\\mathsf{Pr}_{\\kappa}^L \\in (\\mathsf{Pr}_{\\kappa}^L)^{\\kappa}$ if and only if $\\kappa$ is not a limit cardinal. Colimits and Limits in $\\mathsf{Pr}^L$ By Theorem 20 , one obtains the following result. We say a functor $F$ is accessible if there is some $\\kappa$ such that $F$ commutes with $\\kappa$-filtered colimits. Corollary 24. Let $\\mathsf{Pr}^R$ be the category whose objects are presentable categories and morphisms are accessible functors that commute with all limits. Then there is an equivalence \\[ (\\mathsf{Pr}^L)^{\\operatorname{op}} \\simeq \\mathsf{Pr}^R \\] giving by passing to adjoint functors.\nSimilarly, let $\\mathsf{Pr}^R_{\\kappa}$ be the category whose objects are $\\kappa$-presentable categories and morphisms are functors commute with $\\kappa$-filtered colimits and all small limits. Then the above equivalence restricts to an equivalence \\[ (\\mathsf{Pr}^L_{\\kappa})^{\\operatorname{op}} \\simeq \\mathsf{Pr}_{\\kappa}^R. \\] Next, we will discuss various operations within $\\mathsf{Pr}^L$.\nProposition 25. $\\mathsf{Pr}^L$ admits all small limits, and the forgetful functor \\[ \\mathsf{Pr}^L \\to \\widehat{\\mathsf{Cat}} \\] preserves small limits. Similarly, for regular cardinal $\\kappa$, $\\mathsf{Pr}_{\\kappa}^L$ admits all small limits, and the composite $\\mathsf{Pr}_{\\kappa}^L \\simeq \\mathsf{Rex}_{\\kappa} \\to \\mathsf{Cat}$ preserves small limits. If $\\kappa$ is uncountable, the inclusion $\\mathsf{Pr}_{\\kappa}^L \\hookrightarrow \\mathsf{Pr}^L$ preserves $\\kappa$-small limits. We omit the proof of this statement. For colimits, by Corollary 24 , the colimits in $\\mathsf{Pr}^L$ is the limits in $\\mathsf{Pr}^R$.\nProposition 26. $\\mathsf{Pr}^R$ admits all small limits, and the forgetful functor \\[ \\mathsf{Pr}^R \\to \\widehat{\\mathsf{Cat}} \\] preserves small limits. Similarly, for regular cardinal $\\kappa$, $\\mathsf{Pr}_{\\kappa}^R$ admits all small limits. The inclusion $\\mathsf{Pr}_{\\kappa}^R \\hookrightarrow \\mathsf{Pr}^R$ preserves all small limits. Let us compute some examples.\nExample 27. Let $I$ be a set, and let $\\mathcal{C}_i \\in \\mathsf{Pr}^L$ be presentable categories, then the product of $\\mathcal{C}_i$ in $\\mathsf{Pr}^L$ agrees with the naive product $\\prod_{i \\in I}\\mathcal{C}_i$. Let $I$ be a set, and let $\\mathcal{C}_i \\in \\mathsf{Pr}^L$ be presentable categories, then the coproduct of $\\mathcal{C}_i$ in $\\mathsf{Pr}^L$ agrees with the naive product $\\prod_{i \\in I}\\mathcal{C}_i$ (by passing to adjoint functor), which means \\[ \\coprod_{i}^{\\mathsf{Pr}^L} \\mathcal{C}_i \\to \\prod_{i}^{\\mathsf{Pr}^L} \\mathcal{C}_i = \\prod_i \\mathcal{C}_i \\] is an equivalence. In other words, $\\mathsf{Pr}^L$ is $\\kappa$-semiadditive for all cardinal $\\kappa$. Let $S$ be a small category. Then we can form the free presentable category $\\langle S \\rangle_{\\text{pres}} \\coloneqq \\mathsf{PShv}(S)$. This can also be seen as an instance of colimits (or free constructions) in $\\mathsf{Pr}^L$ agreeing with limits (or cofree construction). Indeed, giving any presentable category $\\mathcal{C}$ with a functor $i \\colon S \\to \\mathcal{C}$, we get a functor $\\mathcal{C} \\to \\mathsf{PShv}(S)$ via $X \\mapsto \\left(T \\mapsto \\operatorname{Hom}_{\\mathcal{C}}(i(T),X)\\right)$. This functor commutes with all limits and is accessible, thus a left adjoint $\\mathsf{PShv}(S) \\to \\mathcal{C}$, extending the giving functor $i$. Lurie Tensor Product In this section, we will introduce the symmetric monoidal structure on $\\mathsf{Pr}^L$, in fact, we will find that we have the following table of vague analogies\nLinear algebra Presentable category theory ring $\\mathsf{An}$ module presentable category linear map colimit-preserving functor bilinear map functor preserving colimits in each variable tensor product $\\otimes$ Lurie tensor product $\\otimes$ in $\\mathsf{Pr}^L$ Definition 28. Let $\\mathcal{C}$ and $\\mathcal{D}$ be presentable categories. There is a universal presentable category $\\mathcal{C} \\otimes \\mathcal{D}$ equipped with functor \\[ \\mathcal{C} \\times \\mathcal{D} \\to \\mathcal{C} \\otimes \\mathcal{D} \\] that commutes with small colimits in each variable. If $\\mathcal{C}$ and $\\mathcal{D}$ are $\\kappa$-presentable, then $\\mathcal{C} \\otimes \\mathcal{D}$ are $\\kappa$-presentable, and $\\mathcal{C} \\times \\mathcal{D} \\to \\mathcal{C} \\otimes \\mathcal{D}$ preserves $\\kappa$-compact objects. This endows $\\mathsf{Pr}^L$ and $\\mathsf{Pr}_{\\kappa}^L$ with compactible symmetric monoidal structres, with unit object $\\mathsf{An}$. This symmetric monoidal structure is called Lurie tensor product.\nsketch of construction. It suffices to prove existence. By Theorem 18 , one can choose small categories $S$ and $T$ such that \\[ \\mathcal{C} = \\langle S \\rangle_{\\text{pres}}[R^{-1}] \\quad \\text{and} \\quad \\mathcal{D} = \\langle T \\rangle_{\\text{pres}}[Q^{-1}]. \\] Then it is easy to see that \\[ \\mathcal{C} \\otimes \\mathcal{D} \\coloneqq \\langle S \\times T \\rangle_{\\text{pres}}[(R \\times Q)^{-1}] \\] $\\square$ By construction of $\\mathcal{C} \\otimes \\mathcal{D}$, one can easily find that for small category $\\mathcal{C}$, we have \\[ \\mathsf{PShv}(\\mathcal{C}) \\otimes \\mathcal{D} \\simeq \\mathsf{Fun}(\\mathcal{C}^{\\operatorname{op}},\\mathcal{D}) \\]Now, one can introduce the notion of a symmetric monoidal presentable category.\nDefinition 29. A presentable symmetric monoidal category is a commutative algebra object in $\\mathsf{Pr}^L$ with respect to the Lurie tensor product. A $\\kappa$-presentable symmetric monoidal category is a commutative algebra object in $\\mathsf{Pr}^L_{\\kappa}$ with respect to the Lurie tensor product. In other words, if $\\mathcal{C}$ is a $\\kappa$-presentable symmetric monoidal category, then its tensor product commutes with colimits in each variable, preserves $\\kappa$-compact objects, and its unit object is $\\kappa$-compact.\nReferences [Hau25] Rune Haugseng. Yet Another Introduction to ∞-Categories. 2025. PDF. [HTT] Jacob Lurie. Higher Topos Theory. Princeton University Press, 2009. PDF. [NRS18] Hoang\u0026nbsp;Kim Nguyen, George\u0026nbsp;Raptis, and Christoph\u0026nbsp;Schrade. Adjoint functor theorems for ∞-categories. arXiv:1803.01664, 2018. ","permalink":"https://ou-liu-red-sugar.github.io/notes/notes/gestalten/presentable-categories/","summary":"\u003ch2 id=\"definition\"\u003eDefinition\u003c/h2\u003e\n\u003cdiv class=\"thm-block kind-definition\" id=\"main-1\"\u003e\n    \u003cdiv class=\"thm-header\"\u003e\n      \u003cspan class=\"thm-title\"\u003e\n        Definition 1.\n      \u003c/span\u003e\n    \u003c/div\u003e\n    \u003cdiv class=\"thm-body\"\u003e\u003cp\u003eLet $\\kappa$ be a regular cardinal (for example $\\kappa = \\omega$ or $\\kappa = \\aleph_1$).\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eA category $\\mathcal{I}$ is called \u003cstrong\u003e$\\kappa$-filtered\u003c/strong\u003e if for any $\\kappa$-small category $\\mathcal{K}$ and any functor\n$F\\colon \\mathcal{K} \\to \\mathcal{I}$,\nthere exists an extension $F^{\\rhd} \\colon \\mathcal{K}^{\\rhd} \\to \\mathcal{I}$.\nWhen $\\kappa = \\omega$, we simply call a $\\kappa$-filtered category a filtered category.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eLet $\\mathcal{C}$ be a category admitting small colimits.\nAn object $X \\in \\mathcal{C}$ is called \u003cstrong\u003e$\\kappa$-compact\u003c/strong\u003e\nif for any $\\kappa$-filtered diagram $(Y_i)_{i \\in \\mathcal{I}}$, there is an isomorphism of mapping spaces\n\u003c/p\u003e","title":"Presentable Categories"},{"content":"The goal of this note is to study the dualizable stable categories — the compactly assembled categories. The reasons to care about them are several:\nIn practice, many categories we meet are not compactly generated ($\\aleph_0$-presentable), but compactly assembled (compactly generated categories are a fortiori compactly assembled). A typical example is the category of sheaves $\\mathsf{Shv}(X)$ on a locally compact Hausdorff space $X$.\nGiven a topos $\\mathcal{X}$ and a category $\\mathcal{C}$, set $\\mathsf{Shv}_{\\mathcal{C}}(\\mathcal{X}) \\coloneqq \\mathsf{Fun}^{\\lim}(\\mathcal{X}^{\\mathrm{op}}, \\mathcal{C})$ (sensible because every colimit in a topos is van Kampen, i.e. of descent type). One would like a notion of $\\mathcal{C}$-valued structure sheaf on $\\mathcal{X}$. One route is the classifying topos: a topos $\\mathcal{E}$ equipped with a universal object $\\mathcal{F} \\in \\mathsf{Shv}_{\\mathcal{C}}(\\mathcal{E})$ such that for every topos $\\mathcal{X}$ the assignment $f^* \\mapsto f^*\\mathcal{F}$ gives an equivalence \\[ \\mathsf{Fun}^*(\\mathcal{E}, \\mathcal{X}) \\xrightarrow{\\;\\sim\\;} \\mathsf{Shv}_{\\mathcal{C}}(\\mathcal{X}); \\] then $f^*\\mathcal{F} \\in \\mathsf{Shv}_{\\mathcal{C}}(\\mathcal{X})$ is the universal $\\mathcal{C}$-valued sheaf $\\mathcal{O}_{\\mathcal{X}}$. Such a classifying topos need not exist — but when $\\mathcal{C}$ is compactly assembled, it always does, and is given by $\\mathsf{Fun}^{\\omega}(\\mathcal{C}, \\mathsf{An})$.\nConsider $\\mathsf{Pr}_{\\mathbb{S}}^L = \\mathsf{Pr}_{\\mathrm{st}}^L$, the category of $\\mathbb{S}$-linear presentable categories. The compactly assembled stable categories are exactly the dualizable objects inside it; this gives them a status analogous to finite-dimensional vector spaces. From the K-theoretic standpoint this is our main reason to study them.\nDualizability in $R$-linear categories For a commutative ring spectrum $R$, $\\mathsf{Mod}_R$ is automatically presentable and a commutative algebra in $(\\mathsf{Pr}^L, \\otimes)$ under the Lurie tensor product, so we can form the category of modules $\\mathsf{Mod}_{\\mathsf{Mod}_R}(\\mathsf{Pr}^L)$.\nDefinition 1. Let $R$ be a commutative ring spectrum. A presentable category $\\mathcal{C}$ is $R$-linear if $\\mathcal{C} \\in \\mathsf{Mod}_{\\mathsf{Mod}_R}(\\mathsf{Pr}^L)$. Write $\\mathsf{Pr}_R^L$ for the category of $R$-linear categories. Since $\\mathsf{Sp} \\simeq \\mathsf{Mod}_{\\mathbb{S}}$, we have $\\mathsf{Pr}_{\\mathbb{S}}^L \\simeq \\mathsf{Pr}_{\\mathrm{st}}^L$: on every presentable stable $\\mathcal{C}$ there is a canonical action $\\mathsf{Sp} \\otimes \\mathcal{C} \\to \\mathcal{C}$ preserving colimits in each variable.\nDualizable objects Definition 2. Let $\\mathcal{C}$ be a symmetric monoidal category and $X, Y \\in \\mathcal{C}$. A map $\\mathrm{ev}\\colon Y \\otimes X \\to \\mathbb{1}$ exhibits $Y$ as a dual of $X$ if there is $\\mathrm{coev}\\colon \\mathbb{1} \\to X \\otimes Y$ satisfying the triangle identities:\nWe then call $\\mathrm{ev}$ the evaluation and $\\mathrm{coev}$ the coevaluation, and call $(Y, \\mathrm{ev})$ dualizing data for $X$; $X$ is a dualizable object.\nLemma 3. In an idempotent-complete symmetric monoidal category $\\mathcal{C}$, dualizable objects are closed under retracts. Proof. Let $X \\in \\mathcal{C}$ be dualizable with dual $X^{\\vee}$, and let $i\\colon A \\rightleftarrows X \\colon r$ with $r \\circ i = \\operatorname{id}_A$. Set $e \\coloneqq i \\circ r\\colon X \\to X$.\nDualizing produces an idempotent $e^{\\vee}\\colon X^{\\vee} \\to X^{\\vee}$, \\[ e^{\\vee} \\coloneqq X^{\\vee} \\xrightarrow{\\operatorname{id} \\otimes \\mathrm{coev}} X^{\\vee} \\otimes X \\otimes X^{\\vee} \\xrightarrow{\\operatorname{id} \\otimes e \\otimes \\operatorname{id}} X^{\\vee} \\otimes X \\otimes X^{\\vee} \\xrightarrow{\\mathrm{ev} \\otimes \\operatorname{id}} X^{\\vee}. \\] By idempotent completeness it splits: there are $A^{\\vee} \\in \\mathcal{C}$ and $i'\\colon A^{\\vee} \\rightleftarrows X^{\\vee} \\colon r'$ with $r' \\circ i' = \\operatorname{id}_{A^{\\vee}}$ and $i' \\circ r' = e^{\\vee}$.\nDefine \\begin{align*} \\mathrm{ev}_A \u0026amp;\\colon A^{\\vee} \\otimes A \\xrightarrow{i\u0026rsquo; \\otimes i} X^{\\vee} \\otimes X \\xrightarrow{\\mathrm{ev}_X} \\mathbb{1}, \\ \\mathrm{coev}_A \u0026amp;\\colon \\mathbb{1} \\xrightarrow{\\mathrm{coev}_X} X \\otimes X^{\\vee} \\xrightarrow{r \\otimes r\u0026rsquo;} A \\otimes A^{\\vee}. \\end{align*}\nWe verify the first triangle identity; the second is analogous. The composite $(\\operatorname{id}_A \\otimes \\mathrm{ev}_A) \\circ (\\mathrm{coev}_A \\otimes \\operatorname{id}_A)$ unfolds to \\[ A \\xrightarrow{i} X \\xrightarrow{\\mathrm{coev}_X \\otimes \\operatorname{id}} X \\otimes X^{\\vee} \\otimes X \\xrightarrow{\\operatorname{id} \\otimes e^{\\vee} \\otimes \\operatorname{id}} X \\otimes X^{\\vee} \\otimes X \\xrightarrow{\\operatorname{id} \\otimes \\mathrm{ev}_X} X \\xrightarrow{r} A. \\] Using the adjunction identity $(\\operatorname{id} \\otimes e^{\\vee}) \\circ \\mathrm{coev}_X = (e \\otimes \\operatorname{id}) \\circ \\mathrm{coev}_X$, this rewrites as $r \\circ e \\circ \\big[(\\operatorname{id} \\otimes \\mathrm{ev}_X) \\circ (\\mathrm{coev}_X \\otimes \\operatorname{id})\\big] \\circ i$. The bracketed term is $\\operatorname{id}_X$ by $X$\u0026rsquo;s triangle identity, and $r \\circ e \\circ i = r \\circ i \\circ r \\circ i = \\operatorname{id}_A$.\n$\\square$ Compactly generated categories are dualizable Let $\\mathcal{C}$ be a compactly generated stable category. As a compactly generated category we have $\\mathcal{C} \\simeq \\mathsf{Ind}(\\mathcal{C}^{\\aleph_0})$, and as a stable category we have the mapping-spectrum functor $\\mathrm{hom}_{\\mathcal{C}}\\colon \\mathcal{C}^{\\mathrm{op}} \\times \\mathcal{C} \\to \\mathsf{Sp}$, determined by Currying gives $\\rho_{\\mathcal{C}}\\colon \\mathcal{C}^{\\mathrm{op}} \\to \\mathsf{Fun}(\\mathcal{C}, \\mathsf{Sp})$, $\\rho_{\\mathcal{C}}(D) = \\mathrm{hom}_{\\mathcal{C}}(D, -)$. Since $\\mathrm{hom}_{\\mathcal{C}}$ is the internal-Hom functor, it preserves limits in the first variable, so $\\rho_{\\mathcal{C}}$ preserves limits.\nRestricting to $(\\mathcal{C}^{\\aleph_0})^{\\mathrm{op}}$ gives an exact functor $\\rho_{\\mathcal{C}}^{\\aleph_0}\\colon (\\mathcal{C}^{\\aleph_0})^{\\mathrm{op}} \\to \\mathsf{Fun}(\\mathcal{C}, \\mathsf{Sp})$ (left exactness plus the stable target gives exactness). For a compact $D \\in \\mathcal{C}^{\\aleph_0}$ the functor $\\mathrm{hom}_{\\mathcal{C}}(D, -)$ preserves filtered colimits (by compactness) and is exact, hence preserves all colimits. So $\\rho_{\\mathcal{C}}^{\\aleph_0}$ lands in $\\mathsf{Fun}^L(\\mathcal{C}, \\mathsf{Sp})$. By the Ind-extension universal property, \\[ \\mathsf{Fun}^L\\!\\left(\\mathsf{Ind}\\!\\left((\\mathcal{C}^{\\aleph_0})^{\\mathrm{op}}\\right),\\, \\mathsf{Fun}^L(\\mathcal{C}, \\mathsf{Sp})\\right) \\simeq \\mathsf{Fun}^{\\mathrm{rex}}\\!\\left((\\mathcal{C}^{\\aleph_0})^{\\mathrm{op}},\\, \\mathsf{Fun}^L(\\mathcal{C}, \\mathsf{Sp})\\right), \\] so $\\rho_{\\mathcal{C}}^{\\aleph_0}$ uniquely extends to a colimit-preserving $\\mathrm{P}\\colon \\mathsf{Ind}((\\mathcal{C}^{\\aleph_0})^{\\mathrm{op}}) \\to \\mathsf{Fun}^L(\\mathcal{C}, \\mathsf{Sp})$. Since $\\mathsf{Fun}^L(\\mathcal{C}, \\mathsf{Sp})$ is the internal-Hom in $\\mathsf{Pr}^L$, $\\mathrm{P}$ corresponds to a morphism in $\\mathsf{Pr}^L$, \\[ \\mathsf{Fun}^L\\!\\left(\\mathsf{Ind}((\\mathcal{C}^{\\aleph_0})^{\\mathrm{op}}),\\, \\mathsf{Fun}^L(\\mathcal{C}, \\mathsf{Sp})\\right) \\simeq \\mathsf{Fun}^L\\!\\left(\\mathsf{Ind}((\\mathcal{C}^{\\aleph_0})^{\\mathrm{op}}) \\otimes \\mathcal{C},\\, \\mathsf{Sp}\\right), \\] and we obtain the evaluation \\[ \\mathrm{ev}\\colon \\mathsf{Ind}((\\mathcal{C}^{\\aleph_0})^{\\mathrm{op}}) \\otimes \\mathcal{C} \\to \\mathsf{Sp}. \\]Similarly $\\mathrm{hom}_{\\mathcal{C}^{\\aleph_0}}\\colon (\\mathcal{C}^{\\aleph_0})^{\\mathrm{op}} \\times \\mathcal{C}^{\\aleph_0} \\to \\mathsf{Sp}$ is exact in both variables and gives an object of $\\mathsf{Ind}((\\mathcal{C}^{\\aleph_0})^{\\mathrm{op}}) \\otimes \\mathsf{Ind}(\\mathcal{C}^{\\aleph_0})$, hence the coevaluation \\[ \\mathrm{coev}\\colon \\mathsf{Sp} \\to \\mathcal{C} \\otimes \\mathsf{Ind}((\\mathcal{C}^{\\aleph_0})^{\\mathrm{op}}). \\] The triangle identities check out, so $\\mathcal{C}$ is dualizable in $\\mathsf{Pr}^L_{\\mathrm{st}}$ with $\\mathcal{C}^{\\vee} \\simeq \\mathsf{Ind}((\\mathcal{C}^{\\aleph_0})^{\\mathrm{op}})$.\nCompactly assembled categories The natural question is: what does a general dualizable object in $\\mathsf{Pr}^L_{\\mathrm{st}}$ look like? This is the compactly assembled setting.\nBy what we just established, compactly generated stable categories are dualizable. $\\mathsf{Pr}_{\\mathrm{st}}^L$ is idempotent-complete, so Lemma 3 tells us retracts of compactly generated categories are still dualizable.\nConversely, every dualizable object is a retract of a compactly generated one. Let $\\mathcal{C} \\in \\mathsf{Pr}^L_{\\mathrm{st}}$ be dualizable with dual $\\mathcal{C}^{\\vee}$. Pick a regular cardinal $\\kappa$ such that $\\mathcal{C}$ is $\\kappa$-presentable; the canonical functor \\[ \\varphi\\colon \\mathcal{D} \\coloneqq \\mathsf{Ind}(\\mathcal{C}^{\\kappa}) \\to \\mathcal{C} \\] is a left Bousfield localization ([sheaves-on-manifolds, Cor. 2.1.27] ). Localizations in $\\mathsf{Pr}^L$ are preserved by tensor product ([sheaves-on-manifolds, Ex. 2.8.4] ), so $\\varphi \\otimes \\mathrm{id}_{\\mathcal{C}^{\\vee}}\\colon \\mathcal{D} \\otimes \\mathcal{C}^{\\vee} \\to \\mathcal{C} \\otimes \\mathcal{C}^{\\vee}$ is also a localization, in particular essentially surjective. Dualizability identifies these tensor products with $\\mathsf{Fun}^L(\\mathcal{C}, \\mathcal{D})$ and $\\mathsf{Fun}^L(\\mathcal{C}, \\mathcal{C})$, so $\\mathrm{id}_{\\mathcal{C}} \\in \\mathsf{Fun}^L(\\mathcal{C}, \\mathcal{C})$ has a preimage $\\psi\\colon \\mathcal{C} \\to \\mathcal{D}$ with $\\varphi \\circ \\psi \\simeq \\mathrm{id}_{\\mathcal{C}}$. So $\\mathcal{C}$ is a retract of the compactly generated $\\mathcal{D}$ in $\\mathsf{Pr}^L$.\nDualizable objects in $\\mathsf{Pr}_{\\mathrm{st}}^L$ are therefore exactly the retracts of compactly generated stable categories. This is an external characterisation; we now give an intrinsic one.\nDefinition 4 (Compact and compactly exhaustible). Let $\\mathcal{C}$ be presentable.\nA morphism $f\\colon X \\to Y$ in $\\mathcal{C}$ is a compact morphism if, for every filtered colimit $Z \\simeq \\operatorname*{colim}_i Z_i$, the square\nis a pullback. Equivalently, the fibre $\\operatorname{fib}(\\operatorname{Hom}(Y,Z) \\xrightarrow{f^*} \\operatorname{Hom}(X,Z))$ preserves filtered colimits in $Z$.\nAn object $X$ is compactly exhaustible if \\[ X \\simeq \\operatorname*{colim}\\left(X_0 \\to X_1 \\to X_2 \\to \\cdots\\right) \\] with every $X_i \\to X_{i+1}$ a compact morphism.\n$\\mathcal{C}$ is compactly assembled if it is generated under colimits by compactly exhaustible objects.\nRemark. When $X$ is compact, the map $0 \\to X$ is a compact morphism, hence every compact object is compactly exhaustible (take the constant sequence). In particular compactly generated categories are compactly assembled. We now identify compactly assembled categories intrinsically.\nLemma 5. Let $\\mathcal{C}$ be $\\kappa$-presentable. Then $\\mathcal{C}$ is compactly assembled iff the colimit functor \\[ k\\colon \\mathsf{Ind}(\\mathcal{C}^{\\kappa}) \\to \\mathcal{C} \\] has a left adjoint. The property \u0026ldquo;$k$ has a left adjoint\u0026rdquo; is closed under retracts in $\\mathsf{Pr}^L$.\nProof. ($\\Rightarrow$) Suppose $\\mathcal{C}$ is compactly assembled. Then $\\mathcal{C}$ is $\\aleph_1$-presentable (every compactly exhaustible object is $\\aleph_1$-compact, and conversely every $\\aleph_1$-compact object is compactly exhaustible). Construct the left adjoint $\\hat{y}\\colon \\mathcal{C} \\to \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1})$ explicitly: for $X \\in \\mathcal{C}$, write $X \\simeq \\operatorname*{colim}_n X_n$ with $X_n \\to X_{n+1}$ compact, and set \\[ \\hat{y}(X) \\coloneqq \\operatorname*{colim}_n y(X_n), \\] the colimit computed in $\\mathsf{Ind}(\\mathcal{C}^{\\aleph_1})$. This $\\hat{y}$ is fully faithful. To verify the adjunction, use that for $Z \\simeq \\operatorname*{colim}_i Z_i$ filtered, \\[ \\operatorname{Hom}_{\\mathcal{C}}(X, Z) \\simeq \\lim_n \\operatorname{Hom}(X_n, Z) \\simeq \\lim_n \\operatorname*{colim}_i \\operatorname{Hom}(X_n, Z_i) \\simeq \\operatorname{Hom}_{\\mathsf{Ind}}(\\operatorname*{colim}_n y(X_n), \\operatorname*{colim}_i y(Z_i)), \\] where the second equivalence uses that $(\\operatorname{Hom}(X_n, Z))_n$ and $(\\operatorname*{colim}_i \\operatorname{Hom}(X_n, Z_i))_n$ are isomorphic in $\\mathsf{Pro}(\\mathsf{An})$ — a consequence of the compact-morphism condition. The fully faithful embedding $\\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) \\hookrightarrow \\mathsf{Ind}(\\mathcal{C}^{\\kappa})$ transfers the adjoint to the general $\\kappa$ case.\n($\\Leftarrow$) If $\\hat{y}$ exists, write $\\hat{y}(X) \\simeq \\operatorname*{colim}_i y(X_i)$ in $\\mathsf{Ind}(\\mathcal{C}^{\\kappa})$ with $X_i \\in \\mathcal{C}^{\\kappa}$. Full faithfulness of $\\hat{y}$ and pullbacks against filtered colimits of $Z$ give the compact-morphism pullback square at each $y(X_i) \\to \\hat{y}(X)$, showing $X$ is compactly exhaustible.\nRetract closure: view $\\mathsf{Pr}^L$ as a $2$-category with internal $\\mathsf{Fun}^L$. By [ramzi-dualizable, Lem. 1.47] , a $1$-morphism $f$ in a $2$-category $\\mathbb{B}$ has a left adjoint provided that $\\mathsf{Hom}_{\\mathbb{B}}(X, Z)$ is idempotent-complete for every $Z$ and $f$ is a retract (in $\\mathsf{Fun}([1], \\mathbb{B})$) of some $g$ with a left adjoint. For presentable $\\mathcal{C}, \\mathcal{D}$ the category $\\mathsf{Fun}^L(\\mathcal{C}, \\mathcal{D})$ is always idempotent-complete, and for compactly generated $\\mathcal{D}$ the canonical $k$ is an equivalence, with a trivial left adjoint; retracts inherit the left adjoint.\n$\\square$ Theorem 6. For $\\mathcal{C} \\in \\mathsf{Pr}^L$, $\\mathcal{C}$ is compactly assembled iff it is a retract in $\\mathsf{Pr}^L$ of a compactly generated category. In $\\mathsf{Pr}^L_{\\mathrm{st}}$, $\\mathcal{C}$ is compactly assembled iff it is dualizable. Proof. ($\\Rightarrow$) If $\\mathcal{C}$ is compactly assembled, Lemma 5 gives a left adjoint $\\hat{y}$ to $k\\colon \\mathcal{D} \\coloneqq \\mathsf{Ind}(\\mathcal{C}^{\\kappa}) \\to \\mathcal{C}$. As a localization, $k$ has fully faithful right adjoint $y$, so the counit $\\varepsilon\\colon ky \\xrightarrow{\\sim} \\operatorname{id}_{\\mathcal{C}}$ is an equivalence. Using the two adjunctions, \\[ \\operatorname{Hom}_{\\mathcal{C}}(k(\\hat{y}(c)), c') \\simeq \\operatorname{Hom}_{\\mathcal{D}}(\\hat{y}(c), y(c')) \\simeq \\operatorname{Hom}_{\\mathcal{C}}(c, ky(c')) \\simeq \\operatorname{Hom}_{\\mathcal{C}}(c, c'), \\] naturally in $c'$, so $k \\circ \\hat{y} \\simeq \\operatorname{id}_{\\mathcal{C}}$ by Yoneda. Since $\\hat{y}$ is a left adjoint it is a $\\mathsf{Pr}^L$-map, exhibiting $\\mathcal{C}$ as a retract of the compactly generated $\\mathcal{D}$.\n($\\Leftarrow$) A retract of a compactly generated category has $k\\colon \\mathsf{Ind}((\\cdot)^{\\omega}) \\to (\\cdot)$ an equivalence in the compactly generated case, so the retract inherits a left adjoint to its own $k$, and by Lemma 5 it is compactly assembled.\n$\\square$ Combining with the earlier characterisation: in $\\mathsf{Pr}^L_{\\mathrm{st}}$, dualizable objects are exactly retracts of compactly generated stable categories, so the intrinsic notion (compactly assembled) and the external notion (dualizable) coincide in the stable world.\nA finer statement inside general $\\mathsf{Pr}^L$:\nTheorem 7 (Lurie–Clausen). For presentable $\\mathcal{C}$, the following are equivalent:\n$\\mathcal{C}$ is compactly assembled. $\\mathcal{C}$ is $\\aleph_1$-presentable and the colimit functor $k\\colon \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) \\to \\mathcal{C}$ has a left adjoint $\\hat{y}$. There is a regular cardinal $\\kappa$ for which $\\mathcal{C}$ is $\\kappa$-presentable and $\\mathsf{Ind}(\\mathcal{C}^{\\kappa}) \\to \\mathcal{C}$ has a left adjoint. $\\mathcal{C}$ is a retract in $\\mathsf{Pr}^L$ of a compactly generated category. Filtered colimits in $\\mathcal{C}$ distribute over all small limits: for every small $K$ and filtered $I$, \\[ \\operatorname*{colim}_{I^K} \\operatorname*{lim}_K F \\xrightarrow{\\;\\sim\\;} \\operatorname*{lim}_K \\operatorname*{colim}_I F. \\] Proof. See [sheaves-on-manifolds, §2.3] . $\\square$ The canonical example:\nProposition 8. Let $X$ be Hausdorff. The following are equivalent:\n$\\mathsf{Shv}(X)$ is compactly assembled. $\\mathsf{Open}(X)$ is compactly assembled. $X$ is locally compact. Properties We record some properties; proofs are omitted.\nProposition 9. In Theorem 7 , the left adjoint $\\hat{y}\\colon \\mathcal{C} \\to \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1})$ is fully faithful, i.e. the unit $\\eta^{\\hat{y}}\\colon \\operatorname{id} \\Rightarrow k \\circ \\hat{y}$ is an equivalence. With $\\hat{y} \\dashv k \\dashv y$ at hand we may construct a natural transformation $\\hat{y} \\Rightarrow y$. Contemplate where $\\hat{y}\\varepsilon^y$ is an equivalence (since $\\varepsilon^y\\colon ky \\xrightarrow{\\sim} \\operatorname{id}$ and $y$ is fully faithful) and $y\\eta^{\\hat{y}}$ is an equivalence (since $\\hat{y}$ is fully faithful). The dashed arrow supplies the required $\\hat{y} \\Rightarrow y$.\nProposition 10. For a morphism $f\\colon X \\to Y$ in a compactly assembled $\\mathcal{C}$, the following are equivalent:\n$f$ is a compact morphism. $y(f)\\colon y(X) \\to y(Y)$ factors through $\\hat{y}(Y)$. $\\hat{y}(f)\\colon \\hat{y}(X) \\to \\hat{y}(Y)$ factors through $y(X)$. $\\hat{y}(f)$ is a compact morphism in $\\mathsf{Ind}(\\mathcal{C}^{\\aleph_1})$. So a compact morphism $f\\colon X \\to Y$ is recorded by a lift Equivalently, compact morphisms $X \\to Y$ correspond to arrows $y(X) \\to \\hat{y}(Y)$.\nDefinition 11. Let $\\mathcal{C}$ be compactly assembled. A compactly assembled morphism $X \\to Y$ is a compact morphism together with a choice of lift $y(X) \\to \\hat{y}(Y)$. Set \\[ \\operatorname{Hom}_{\\mathcal{C}}^{\\mathrm{ca}}(X,Y) \\coloneqq \\operatorname{Hom}_{\\mathsf{Ind}(\\mathcal{C})}(y(X), \\hat{y}(Y)). \\] The map $\\operatorname{Hom}_{\\mathcal{C}}^{\\mathrm{ca}}(X,Y) \\to \\operatorname{Hom}_{\\mathcal{C}}(X,Y)$ (via $\\hat{y}(Y) \\to y(Y)$) is not a subspace inclusion: a compactly assembled morphism carries strictly more information (the choice of lift), expressed as higher-homotopy data.\nInd-extension and assembly For compactly assembled $\\mathcal{C}$ and a category $\\mathcal{D}$ with filtered colimits, any functor $F\\colon \\mathcal{C} \\to \\mathcal{D}$ has an Ind-extension: and $k_{\\mathcal{D}} \\circ \\mathsf{Ind}(F)$ preserves filtered colimits. Writing $\\mathsf{Fun}^{\\mathrm{filt}}$ for the full subcategory of filtered-colimit-preserving functors, Ind-extension is the equivalence \\[ y^*\\colon \\mathsf{Fun}^{\\mathrm{filt}}(\\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}), \\mathcal{D}) \\xrightarrow{\\;\\sim\\;} \\mathsf{Fun}(\\mathcal{C}, \\mathcal{D}). \\]There is also $\\hat{y}\\colon \\mathcal{C} \\to \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1})$, prompting the question: how do $y^*$ and $\\hat{y}^*$ relate?\nLemma 12. Let $\\mathcal{C}$ be compactly assembled and $\\mathcal{D}$ a category with filtered colimits. A functor $F\\colon \\mathcal{C} \\to \\mathcal{D}$ preserves filtered colimits iff its Ind-extension sends $\\hat{y} \\Rightarrow y$ to an equivalence. Explicitly, there is an equivalence \\[ y^* \\simeq \\hat{y}^*\\colon \\mathsf{Fun}_{\\hat{y} \\Rightarrow y}^{\\mathrm{filt}}(\\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}), \\mathcal{D}) \\xrightarrow{\\;\\sim\\;} \\mathsf{Fun}^{\\mathrm{filt}}(\\mathcal{C}, \\mathcal{D}), \\] where the left side consists of filtered-colimit-preserving functors sending $\\hat{y} \\Rightarrow y$ to an equivalence.\nProposition 13. $\\mathsf{Fun}^{\\mathrm{filt}}(\\mathcal{C}, \\mathcal{D})$ is a left Bousfield localization of $\\mathsf{Fun}(\\mathcal{C}, \\mathcal{D})$ with localization \\[ \\mathrm{asm}_{\\mathrm{filt}}\\colon \\mathsf{Fun}(\\mathcal{C}, \\mathcal{D}) \\to \\mathsf{Fun}^{\\mathrm{filt}}(\\mathcal{C}, \\mathcal{D}), \\qquad F \\mapsto k_{\\mathcal{D}} \\circ \\mathsf{Ind}(F) \\circ \\hat{y}. \\] The counit \\[ k_{\\mathcal{D}} \\circ \\mathsf{Ind}(F) \\circ \\hat{y} \\Rightarrow F \\] is the assembly map: the terminal object of $\\mathsf{Fun}^{\\mathrm{filt}}(\\mathcal{C}, \\mathcal{D})_{/F}$.\nIntuitively, the assembled functor $\\mathrm{asm}_{\\mathrm{filt}}(F)$ is $F$ restricted to compactly exhaustible objects, reassembled by filtered colimits.\nThe category $\\mathsf{Pr}^L_{\\mathrm{ca}}$ The relevant functors between compactly assembled categories are those that preserve the defining structure — compact morphisms.\nDefinition 14. Let $\\mathcal{C}$ and $\\mathcal{D}$ be compactly assembled. A left adjoint $F\\colon \\mathcal{C} \\to \\mathcal{D}$ is a compactly assembled functor if it preserves compact morphisms. Let \\[ \\mathsf{Pr}^L_{\\mathrm{ca}} \\subset \\mathsf{Pr}^L \\] be the non-full subcategory of compactly assembled categories and compactly assembled functors.\nProposition 15. A left adjoint $F\\colon \\mathcal{C} \\to \\mathcal{D}$ between compactly assembled categories preserves compact morphisms iff it commutes with $\\hat{y}$:\nLet $F\\colon \\mathcal{C} \\to \\mathcal{D}$ be a morphism in $\\mathsf{Pr}^L$ with $\\mathcal{C}$ compactly assembled. Then $F$ preserves compact morphisms iff $F^R$ preserves filtered colimits.\nVia Gabriel–Ulmer $\\mathsf{Pr}_{\\aleph_1}^L \\simeq \\mathsf{Cat}^{\\mathrm{rex}(\\aleph_1)}$ we can recognise $\\mathsf{Pr}^L_{\\mathrm{ca}}$ intrinsically. Since $\\mathsf{Pr}^L_{\\mathrm{ca}} \\subset \\mathsf{Pr}_{\\aleph_1}^L$, only an extra condition on $\\mathsf{Cat}^{\\mathrm{rex}(\\aleph_1)}$ is needed.\nDefinition 16. Define $\\mathsf{Cat}^{\\mathrm{ca}}$:\nObjects are small categories $\\mathcal{C}$ with countable colimits in which every object is compactly exhaustible (an $\\mathbb{N}$-colimit along compact morphisms). Morphisms are functors preserving $\\aleph_1$-small colimits and compact morphisms. Proposition 17. There is an equivalence $\\mathsf{Pr}^L_{\\mathrm{ca}} \\simeq \\mathsf{Cat}^{\\mathrm{ca}}$. Proof. ($\\Rightarrow$) For compactly assembled $\\mathcal{C}$, $\\mathcal{C}^{\\aleph_1}$ has countable colimits, and every object is compactly exhaustible, so $\\mathcal{C}^{\\aleph_1} \\in \\mathsf{Cat}^{\\mathrm{ca}}$.\n($\\Leftarrow$) For $\\mathcal{C} \\in \\mathsf{Cat}^{\\mathrm{ca}}$, set $\\mathcal{D} = \\mathsf{Ind}_{\\aleph_1}(\\mathcal{C})$. By Lemma 5 , it suffices to construct a left adjoint to $k\\colon \\mathsf{Ind}(\\mathcal{C}) \\to \\mathcal{D}$. For $X = \\operatorname*{colim}_n X_n \\in \\mathcal{C}$ compactly exhaustible and any $Y \\in \\mathsf{Ind}(\\mathcal{C})$, \\[ \\operatorname{Hom}_{\\mathsf{Ind}(\\mathcal{C})}(\\operatorname*{colim}_n y(X_n), Y) \\simeq \\operatorname{Hom}_{\\mathcal{D}}(\\operatorname*{colim}_n X_n, kY) = \\operatorname{Hom}_{\\mathcal{D}}(X, kY), \\] exhibiting $\\hat{y}(X) = \\operatorname*{colim}_n y(X_n)$ as the left adjoint.\n$\\square$ Proposition 18. $\\mathsf{Pr}^L_{\\mathrm{ca}}$ has all colimits, and the inclusion $\\mathsf{Pr}^L_{\\mathrm{ca}} \\to \\mathsf{Pr}^L$ preserves colimits. For any regular cardinal $\\kappa$, the functor $(-)^{\\kappa}\\colon \\mathsf{Pr}^L_{\\mathrm{ca}} \\to \\mathsf{Cat}^{\\mathrm{rex,idem}}$ preserves $\\kappa$-filtered colimits. $\\mathsf{Pr}^L_{\\mathrm{ca}}$ is $\\aleph_1$-presentable. $\\mathsf{Pr}^L_{\\mathrm{ca}} \\to \\mathsf{Pr}^L$ preserves finite limits; $\\mathsf{Pr}^L_{\\mathrm{ca}}$ has finite products, so is semi-additive. Symmetric monoidal structure Finally, we sketch the symmetric monoidal structure on $\\mathsf{Pr}^L_{\\mathrm{ca}}$. In $\\mathsf{Pr}^L$ the Lurie tensor product is classified by bifunctors $F\\colon \\mathcal{C} \\times \\mathcal{D} \\to \\mathcal{E}$ preserving colimits in each variable. To descend to $\\mathsf{Pr}^L_{\\mathrm{ca}}$ we add a compact-morphism condition: for compact $f \\in \\mathcal{C}$ and $g \\in \\mathcal{D}$, $F(f, g)$ must be a compact morphism in $\\mathcal{E}$.\nThe universal property is then: for such $F$, there is a unique (up to homotopy) factorisation To make this concrete we use the Gabriel–Ulmer construction of $\\mathsf{Pr}_{\\kappa}^L$\u0026rsquo;s monoidal structure from [lurie-ha, §4.8.1] . Take $\\kappa = \\aleph_1$. The idea: equip $\\mathsf{Cat}^{\\mathrm{rex}(\\aleph_1)}$ (small categories with countable colimits) with the monoidal structure induced from $\\mathsf{Cat}^{\\times}$, then transport to $\\mathsf{Pr}_{\\aleph_1}^L$ via duality. At this level, \\[ \\mathsf{Ind}_{\\aleph_1}(\\mathcal{C}_0) \\otimes \\mathsf{Ind}_{\\aleph_1}(\\mathcal{D}_0) \\simeq \\mathsf{Ind}_{\\aleph_1}(\\mathcal{C}_0 \\otimes \\mathcal{D}_0). \\]Restricting to $\\mathsf{Pr}^L_{\\mathrm{ca}}$: every object of $\\mathcal{C}_0 \\otimes \\mathcal{D}_0$ is a countable colimit, and cofinality lets us reduce such a colimit to an $\\mathbb{N}$-colimit of compactly exhaustible generators. This shows $\\mathcal{C} \\otimes \\mathcal{D}$ remains in $\\mathsf{Pr}^L_{\\mathrm{ca}}$. The fact that the induced $\\mathcal{C} \\otimes \\mathcal{D} \\to \\mathcal{E}$ is a valid morphism of $\\mathsf{Pr}^L_{\\mathrm{ca}}$ likewise reduces to a small-category statement via Gabriel–Ulmer.\nFor any compactly assembled $\\mathcal{C}$ and any locally compact Hausdorff $X$ there is a lax symmetric monoidal equivalence \\[ \\mathsf{Shv}(X, \\mathcal{C}) \\simeq \\mathsf{Shv}(X) \\otimes \\mathcal{C}. \\] Since both factors are compactly assembled, so is their tensor product; $\\mathsf{Sp}$ is compactly generated (hence compactly assembled), so $\\mathsf{Shv}(X, \\mathsf{Sp})$ is compactly assembled.\nNote also that $\\mathsf{Sp}$ is a commutative algebra in $\\mathsf{Pr}_{\\mathrm{ca}}^L$, so we may form $\\mathsf{Mod}_{\\mathsf{Sp}}(\\mathsf{Pr}_{\\mathrm{ca}}^L)$ — the category of compactly assembled stable categories, inheriting the symmetric monoidal structure. Denote it $\\mathsf{Pr}^L_{\\mathrm{dual}}$.\nDefinition 19. Let $\\mathcal{C}$ be a compactly assembled stable category.\n$\\mathcal{C}$ is smooth if $\\mathsf{Sp} \\to \\mathcal{C} \\otimes \\mathcal{C}^{\\vee}$ is a strong left adjoint (its right adjoint has a right adjoint). $\\mathcal{C}$ is proper if $\\mathcal{C} \\otimes \\mathcal{C}^{\\vee} \\to \\mathsf{Sp}$ is a strong left adjoint. References A. Krause, T. Nikolaus, P. Pützstück. Sheaves on Manifolds. 2024. PDF. M. Ramzi. Dualizable presentable $\\infty$-categories. 2024. arXiv:2410.21537. A. I. Efimov. K-theory and localizing invariants of large categories. 2025. arXiv:2405.12169. J. Lurie. Higher Algebra. PDF. ","permalink":"https://ou-liu-red-sugar.github.io/notes/notes/continuous-k-theory/compactly-assembled-categories/","summary":"\u003cp\u003eThe goal of this note is to study the dualizable stable categories — the\n\u003cstrong\u003ecompactly assembled categories\u003c/strong\u003e. The reasons to care about them are several:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003eIn practice, many categories we meet are not compactly generated\n($\\aleph_0$-presentable), but compactly assembled (compactly generated\ncategories are a fortiori compactly assembled). A typical example is the\ncategory of sheaves $\\mathsf{Shv}(X)$ on a locally compact Hausdorff space\n$X$.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eGiven a topos $\\mathcal{X}$ and a category $\\mathcal{C}$, set\n$\\mathsf{Shv}_{\\mathcal{C}}(\\mathcal{X}) \\coloneqq \\mathsf{Fun}^{\\lim}(\\mathcal{X}^{\\mathrm{op}}, \\mathcal{C})$\n(sensible because every colimit in a topos is van Kampen, i.e. of\ndescent type). One would like a notion of $\\mathcal{C}$-valued structure\nsheaf on $\\mathcal{X}$. One route is the \u003cstrong\u003eclassifying topos\u003c/strong\u003e: a topos\n$\\mathcal{E}$ equipped with a universal object\n$\\mathcal{F} \\in \\mathsf{Shv}_{\\mathcal{C}}(\\mathcal{E})$ such that for every\ntopos $\\mathcal{X}$ the assignment $f^* \\mapsto f^*\\mathcal{F}$ gives an\nequivalence\n\u003c/p\u003e","title":"Compactly assembled categories"},{"content":"What is \\( n\\mathsf{Pr} \\) Recall that in the previous note we reviewed the basic theory of presentable categories. In this section, we introduce the notion of presentable \\(n\\)-categories.\nLet \\(\\mathcal{C} \\in \\mathsf{CAlg}(\\mathsf{Pr}^L)\\) be a commutative algebra object in \\(\\mathsf{Pr}^L\\) with respect to the Lurie tensor product, and let \\(A \\in \\mathsf{Alg}(\\mathcal{C})\\) be a commutative algebra object. We may then consider the category of \\(A\\)-modules \\[ \\mathsf{Mod}_A(\\mathcal{C}). \\]It is well known that \\(\\mathsf{Mod}_A(\\mathcal{C})\\) carries a natural symmetric monoidal structure, defined via the colimit of the bar construction. More precisely, for \\(A\\)-modules \\(M\\) and \\(N\\), consider the simplicial object \\[ \\operatorname{Bar}_A(M,N)_\\bullet, \\] defined as follows:\n\\(\\operatorname{Bar}_A(M,N)_n \\coloneqq M \\otimes A^{\\otimes n} \\otimes N\\). The face maps \\[ d_i \\colon \\operatorname{Bar}_A(M,N)_n \\to \\operatorname{Bar}_A(M,N)_{n-1} \\] are given by: \\(d_0 = \\operatorname{act}_M \\otimes \\operatorname{id}_{A^{\\otimes n-1}} \\otimes \\operatorname{id}_N\\). For \\(0 \u003c i \u003c n\\), \\(d_i\\) is induced by the multiplication map \\[ m \\colon A \\otimes A \\to A \\] on the \\(i\\)-th and \\((i+1)\\)-th factors of \\(A^{\\otimes n}\\). \\(d_n = \\operatorname{id}_M \\otimes \\operatorname{id}_{A^{\\otimes n-1}} \\otimes \\operatorname{act}_N\\). The relative tensor product is then defined by \\[ M \\otimes_A N \\coloneqq \\operatorname{colim}\\, \\operatorname{Bar}_A(M,N)_\\bullet. \\] This construction equips \\(\\mathsf{Mod}_A(\\mathcal{C})\\) with a symmetric monoidal structure.\nMoreover, an object \\(B \\in \\mathsf{CAlg}(\\mathsf{Mod}_A(\\mathcal{C}))\\) is equivalently a commutative algebra object in \\(\\mathcal{C}\\) together with a morphism from \\(A\\).\nIn other words, there is a canonical equivalence \\[ \\mathsf{CAlg}(\\mathcal{C})_{/A} \\simeq \\mathsf{CAlg}(\\mathsf{Mod}_A(\\mathcal{C})). \\]Now, we regard \\(\\mathcal{C}\\) as a category linear over itself. Since \\(\\mathcal{C} \\in \\mathsf{CAlg}(\\mathsf{Pr}^L)\\), the tensor product on \\(\\mathcal{C}\\) preserves colimits separately in each variable. In particular, for any commutative algebra object \\(A \\in \\mathsf{CAlg}(\\mathcal{C})\\), the category \\(\\mathsf{Mod}_A(\\mathcal{C})\\) exists in \\(\\mathsf{Pr}^L\\) and the forgetful functor \\[ \\mathsf{Mod}_A(\\mathcal{C}) \\to \\mathcal{C} \\] preserves colimits. If \\(\\mathcal{C}\\) is \\(\\kappa\\)-presentable, then \\(\\mathsf{Mod}_A(\\mathcal{C})\\) is again \\(\\kappa\\)-presentable (by [LurieHA, Corollary 4.2.3.7] ).\nRecall that \\(\\mathsf{Pr}^L_\\kappa\\) is \\(\\kappa\\)-presentable. Thus, one can consider \\(\\mathsf{Mod}_{\\mathcal{C}}(\\mathsf{Pr}^L_\\kappa)\\). And now, one can regard \\(\\mathsf{Mod}_A(\\mathcal{C})\\) as a commutative algebra in \\(\\mathsf{Mod}_{\\mathcal{C}}(\\mathsf{Pr}^L_{\\kappa})\\), and hence we have \\(\\mathsf{Mod}_A(\\mathcal{C}) \\in \\mathsf{CAlg}(\\mathsf{Pr}^L_{\\kappa})_{\\mathcal{C}/}\\).\nDefinition 1. Let \\(\\mathcal{C} \\in \\mathsf{CAlg}(\\mathsf{Pr}^L_{\\kappa})\\), the symmetric monoidal category of \\(\\mathcal{C}\\)-linear categories is \\[ \\mathsf{Mod}_{\\mathcal{C}}(\\mathsf{Pr}_{\\kappa}^L). \\] Theorem 2. The functor \\(A \\mapsto \\mathsf{Mod}_A(\\mathcal{C})\\) yields a fully faithful functor \\[ \\mathsf{CAlg}(\\mathcal{C}) \\hookrightarrow \\mathsf{CAlg}(\\mathsf{Mod}_{\\mathcal{C}}(\\mathsf{Pr}_{\\kappa}^L)) \\] whose right adjoint is given by \\(\\mathcal{D} \\mapsto \\mathsf{End}_{\\mathcal{D}}(1)\\). The displayed map is a map in \\(\\mathsf{CAlg}(\\mathsf{Pr}_{\\kappa}^L)\\).\nProof. Fully faithfulness in [LurieHA, Corollary 4.8.5.21] . The identification of the right adjoint in [LurieHA, Remark 4.8.5.12] . Everything lives in \\(\\mathsf{CAlg}(\\mathsf{Pr}_{\\kappa}^L)\\) is checked in [Aok25] , by checking that \\(\\mathcal{D} \\mapsto \\mathsf{End}_{\\mathcal{D}}(1)\\) commutes with \\(\\kappa\\)-filtered colimits. $\\square$ Example 3. Let \\(\\mathcal{C} = \\mathsf{Mod}_A \\coloneqq \\mathsf{Mod}_A(\\mathsf{Sp})\\) for some commutative ring spectra \\(A\\). Then it follows that for any commutative \\(A\\)-algebra \\(B\\) and \\(C\\), one has \\[ \\mathsf{Mod}_B \\otimes_{\\mathsf{Mod}_A} \\mathsf{Mod}_C = \\mathsf{Mod}_{B \\otimes_A C}. \\] Indeed, this is a special case of the commutation of the above functor with colimits. More generally, the theorem says that the algebraic operations of colimits and tensor products as performed on the level of categories faithfully reflect the same operations at the level of rings and modules.\nAt this point, we have succeeded in pushing all the important algebraic operations to the level of categories. Now, we replace \\(\\mathcal{C}\\) by \\(\\mathsf{Pr}^L_{\\kappa}\\), then we have a fully faithful functor \\[ \\mathsf{CAlg}(\\mathsf{Pr}^L_{\\kappa}) \\hookrightarrow \\mathsf{CAlg}\\!\\left( \\mathsf{Mod}_{\\mathsf{Pr}^L_{\\kappa}}(\\mathsf{Pr}_{\\kappa}^L) \\right), \\quad \\mathcal{C} \\mapsto \\mathsf{Mod}_{\\mathcal{C}}(\\mathsf{Pr}_{\\kappa}^L). \\]Note that \\(\\mathsf{Pr}_{\\kappa}^L\\) with the Lurie tensor product is also a commutative algebra object in \\(\\mathsf{Pr}^L_{\\kappa}\\), so we can consider \\[ \\mathsf{Mod}_{\\mathsf{Pr}_{\\kappa}^L}(\\mathsf{Pr}_{\\kappa}^L) \\in \\mathsf{CAlg}\\!\\left( \\mathsf{Mod}_{\\mathsf{Pr}^L_{\\kappa}}(\\mathsf{Pr}_{\\kappa}^L) \\right). \\]We denote \\(\\mathsf{Mod}_{\\mathsf{Pr}_{\\kappa}^L}(\\mathsf{Pr}_{\\kappa}^L)\\) by \\(2\\mathsf{Pr}_{\\kappa}^L\\), since it is a \\(\\mathsf{Pr}_{\\kappa}^L\\)-enriched category: for every \\(\\mathcal{C}, \\mathcal{D} \\in 2\\mathsf{Pr}_{\\kappa}^L\\), we have \\[ \\mathsf{Hom}_{2\\mathsf{Pr}_{\\kappa}^L}(\\mathcal{C},\\mathcal{D}) \\in \\mathsf{Pr}_{\\kappa}^L \\] determined as the universal object equipped with a morphism \\[ \\mathsf{Hom}_{2\\mathsf{Pr}_{\\kappa}^L}(\\mathcal{C},\\mathcal{D}) \\otimes \\mathcal{C} \\to \\mathcal{D}, \\] which endows \\(2\\mathsf{Pr}_{\\kappa}^L\\) with an enrichment over \\(\\mathsf{Pr}_{\\kappa}^L\\), and hence with the structure of a \\(2\\)-category. We thus get a full embedding \\[ \\mathsf{CAlg}(\\mathsf{Pr}_{\\kappa}^L) \\hookrightarrow \\mathsf{CAlg}(2\\mathsf{Pr}_{\\kappa}^L). \\]Note that we have \\[ 2\\mathsf{Pr}_{\\kappa}^L \\in \\mathsf{CAlg}(\\mathsf{Pr}_{\\kappa}^L). \\] Hence, one can replace \\(\\mathcal{C}\\) in Theorem 2 by \\(2\\mathsf{Pr}_{\\kappa}^L\\), and obtain a full embedding \\[ 2\\mathsf{Pr}_{\\kappa}^L \\hookrightarrow \\mathsf{CAlg}\\!\\left( \\mathsf{Mod}_{2\\mathsf{Pr}_{\\kappa}^L}(\\mathsf{Pr}_{\\kappa}^L) \\right). \\]We denote \\[ 3\\mathsf{Pr}_{\\kappa}^L \\coloneqq \\mathsf{Mod}_{2\\mathsf{Pr}_{\\kappa}^L}(\\mathsf{Pr}_{\\kappa}^L). \\] More generally, one defines inductively \\[ n\\mathsf{Pr}_{\\kappa}^L \\coloneqq \\mathsf{Mod}_{(n-1)\\mathsf{Pr}_{\\kappa}^L}(\\mathsf{Pr}_{\\kappa}^L). \\] Each of these is itself a symmetric monoidal \\(\\kappa\\)-presentable category. Consequently, we obtain a chain of fully faithful embeddings \\[ \\mathsf{CAlg}(\\mathsf{Pr}_{\\kappa}^L) \\hookrightarrow \\mathsf{CAlg}(2\\mathsf{Pr}_{\\kappa}^L) \\hookrightarrow \\cdots \\hookrightarrow \\mathsf{CAlg}(n\\mathsf{Pr}_{\\kappa}^L) \\hookrightarrow \\cdots . \\]From now on, we take \\(\\kappa = \\aleph_1\\) and abbreviate \\[ n\\mathsf{Pr} \\coloneqq n\\mathsf{Pr}_{\\aleph_1}^L. \\] Thus, we obtain a sequence \\[ \\mathsf{CAlg}(1\\mathsf{Pr}) \\hookrightarrow \\mathsf{CAlg}(2\\mathsf{Pr}) \\hookrightarrow \\cdots \\hookrightarrow \\mathsf{CAlg}(n\\mathsf{Pr}) \\hookrightarrow \\cdots . \\]Taking the colimit of this sequence in \\(1\\mathsf{Pr} = \\mathsf{Pr}_{\\omega}^L\\), we obtain a presentable category, denoted \\(\\mathsf{StRing}\\), which will be studied in the next note.\nReferences [Aok25] Ko Aoki. Higher presentable categories and limits. 2025. arXiv:2510.13503. [Lur17] Jacob Lurie. Higher Algebra. 2017. PDF. ","permalink":"https://ou-liu-red-sugar.github.io/notes/notes/gestalten/presentable-n-categories/","summary":"\u003ch2 id=\"what-is\"\u003eWhat is \\( n\\mathsf{Pr} \\)\u003c/h2\u003e\n\u003cp\u003eRecall that in the previous note we reviewed the basic theory of presentable categories.\nIn this section, we introduce the notion of presentable \\(n\\)-categories.\u003c/p\u003e\n\u003cp\u003eLet \\(\\mathcal{C} \\in \\mathsf{CAlg}(\\mathsf{Pr}^L)\\) be a commutative algebra object in \\(\\mathsf{Pr}^L\\) with respect to the Lurie tensor product, and let\n\\(A \\in \\mathsf{Alg}(\\mathcal{C})\\) be a commutative algebra object.\nWe may then consider the category of \\(A\\)-modules\n\u003c/p\u003e\n\\[\n  \\mathsf{Mod}_A(\\mathcal{C}).\n\\]\u003cp\u003eIt is well known that \\(\\mathsf{Mod}_A(\\mathcal{C})\\) carries a natural symmetric monoidal structure,\ndefined via the colimit of the bar construction.\nMore precisely, for \\(A\\)-modules \\(M\\) and \\(N\\), consider the simplicial object\n\u003c/p\u003e","title":"Presentable n-Categories"},{"content":"Algebraic Pattern Algebraic pattern is a blueprint for a notion of functors on a fixed category satisfying a Segal condition, suitable for formalizing homotopy-coherent algebra in the Cartesian setting.\nInformally, algebraic pattern generalizes the active and inert morphisms in operads and chooses certain objects to control the Segal condition.\nDefinition 1. An algebraic pattern is a category $\\mathcal{O}$ equipped with:\nA collection of objects called elementary objects. A factorization system $(\\mathcal{O}^{\\text{inv}}, \\mathcal{O}^{\\text{act}})$ where every morphism factors uniquely (up to equivalence) as an inert morphism followed by an active morphism. We let $\\mathcal{O}^{\\mathrm{el}}$ denote the full subcategory of $\\mathcal{O}$ spanned by the elementary objects and the inert morphisms between them. For any object $X \\in \\mathcal{O}$, we also write \\[ \\mathcal{O}^{\\mathrm{el}}_{X/} := \\mathcal{O}^{\\mathrm{el}} \\times_{\\mathcal{O}^{\\text{inv}}} \\mathcal{O}^{\\text{inv}}_{X/}. \\] for the category of inter morphisms $X \\to E$ with $E \\in \\mathcal{O}^{\\text{el}}$.\nA morphism of algebraic patterns from $\\mathcal{O}$ to $\\mathcal{P}$ is a functor $f \\colon \\mathcal{O} \\to \\mathcal{P}$ that preserves inert and active morphisms and elementary objects.\nWe will use $\\mathsf{AlgPatt}$ to denote the category of algebraic patterns.\nRemark. A factorization system on a category $\\mathcal{C}$ is a pair of subcategories $(\\mathcal{L}, \\mathcal{R})$ that contain all objects, such that for any morphism $f \\colon X \\to X'$, the anima of factorizations $X \\xrightarrow{l} Y \\xrightarrow{r} X'$ with $l \\in \\mathcal{L}$ and $r \\in \\mathcal{R}$ is contractible. Proposition 2. $\\mathsf{AlgPatt}$ admits limits and filtered colimits, and the forgetful functor \\[ \\mathsf{AlgPatt} \\to \\mathsf{Cat} \\] preserves these.\nProof. [Chu_Haugseng_2021, Corollary 5.5] $\\square$ Definition 3 (Trivial Pattern). Trivial Pattern $\\mathsf{Triv}$ is the final object of $\\mathsf{AlgPatt}$. The underlying category of trivial pattern is the final category $*$. Definition 4 (Empty Pattern). Empty Pattern $\\varnothing$ is the initial object of $\\mathsf{AlgPatt}$. The underlying category of empty pattern is the initial category $\\varnothing$. Definition 5 (Commutative Pattern). Consider the category of pointed finite sets $\\mathsf{Fin}_*$ with $\\langle n \\rangle \\coloneqq (\\{0,1,\\cdots,n\\},0)$. We say a morphism $f \\colon \\langle n \\rangle \\to \\langle m \\rangle$ is:\nInert, if $f$ restricts to an isomorphism $\\langle n \\rangle \\setminus f^{-1}(0) \\to \\langle m \\rangle \\setminus \\{0\\}$. Active, if $f^{-1}(0) = \\{0\\}$. We make this an algebraic pattern by taking $\\langle 1 \\rangle$ to be the single elementary object and denote it by $\\mathsf{Comm}$. We refer to this pattern as commutative pattern.\nRemark. In Definition 5 , if we take $\\langle 0 \\rangle$ and $\\langle 1 \\rangle$ to be the elementary objects, we can get a new algebraic pattern $\\mathsf{Fin}_*^{\\natural}$. Definition 6 (Associative Pattern). Consider the opposite of the simplex category, $\\Delta^{\\operatorname{op}}$. A morphism $f \\colon [n] \\to [m]$ in $\\Delta^{\\operatorname{op}}$ is:\nInert, if its corresponding map $g \\colon [m] \\to [n]$ in $\\Delta$ is an interval inclusion. Active, if its corresponding map $g \\colon [m] \\to [n]$ preserves endpoints. We choose the object $[1]$ as the unique elementary object. This algebraic pattern, denoted $\\mathsf{Assoc}$, is called the associative pattern.\nDefinition 7 ($\\mathbb{A}_n$-Pattern). For $1 \\le n \\le \\infty$, let $\\mathbb{A}_n$ denote the full subcategory of $\\Delta^{\\operatorname{op}}$ spanned by the objects $[m]$ for $0 \\le m \\le n$.\nThe category $\\mathbb{A}_n$ can be endowed with an algebraic pattern structure inherited from $\\mathsf{Assoc}$, in which the inert and active morphisms are precisely those that are inert or active in $\\mathsf{Assoc}$, and the elementary object is given by $[1]$. We will refer to $\\mathbb{A}_n$ as the $\\mathbb{A}_n$-pattern.\nRemark. One can find that the $\\mathbb{A}_{\\infty}$-pattern is the associative pattern. Definition 8 (Nonunital $\\mathbb{A}_n$-Pattern). Let $\\Delta_{\\operatorname{inj}} \\subseteq \\Delta$ denote the subcategory whose morphisms are strictly increasing maps. For $0 \\le n \\le \\infty$, let $\\mathbb{A}_n^{\\operatorname{nu}}$ denote the full subcategory of $\\Delta_{\\operatorname{inj}}^{\\operatorname{op}}$ spanned by the objects $[m]$ for $1 \\le m \\le n$ (so $\\mathbb{A}_0^{\\operatorname{nu}} \\simeq \\varnothing$).\nThe category $\\mathbb{A}_n^{\\operatorname{nu}}$ can be endowed with an algebraic pattern structure inherited from $\\mathsf{Assoc}$, in which the inert and active morphisms are precisely those that are inert or active in $\\mathsf{Assoc}$, and the elementary object is given by $[1]$. We will refer to $\\mathbb{A}_n^{\\operatorname{un}}$ as the nonunital $\\mathbb{A}_n$-pattern.\nDefinition 9 ($\\mathsf{E}_k$-Pattern). Consider the $k$-copies of the opposite of simplex categories, $\\Delta^{k,\\operatorname{op}} \\coloneqq (\\Delta^{\\operatorname{op}})^{\\times k}$, equipped with the factorization system where the inert and active maps are those that are inert or active in $\\mathsf{Assoc}$ in each component. We choose the object $([1],\\cdots,[1])$ to be the unique elementary object. This algebraic pattern, denoted $\\mathsf{E}_k$, is called the $\\mathsf{E}_k$-pattern. Next, we introduce patterns related to modules.\nDefinition 10 (Commutative Modules Pattern). The underlying category of commutative modules pattern $\\mathsf{CM}$ is the category $\\mathsf{Fin}_{*,\\langle 1 \\rangle/}$. Its factorzation system is lifted from $\\mathsf{Comm}$ along the canonical left fibration $\\mathsf{Fin}_{*,\\langle 1 \\rangle/} \\to \\mathsf{Fin}_*$. And the elementary objects are given by $\\langle 1 \\rangle \\to \\{0\\} \\subseteq \\langle 1 \\rangle$ and $\\operatorname{id}_{\\langle 1 \\rangle}$.\nBy construction, an object $\\langle 1 \\rangle \\to \\langle n \\rangle$ in $\\mathsf{Fin}_{*,\\langle 1 \\rangle/}$ can be regarded as a pair $(\\langle n \\rangle,i)$, where $i$ is the image of $1 \\in \\langle 1 \\rangle$. The morphism in $\\mathsf{CM}$, which is form $f \\colon (\\langle n \\rangle,i) \\to (\\langle h \\rangle,j)$, refers to the pointed map $\\langle n \\rangle \\to \\langle h \\rangle$ with $f(i) = j$.\nDefinition 11 (Bimodule Pattern). The underlying category of bimodule pattern $\\mathsf{BM}$ is the category $(\\Delta_{/[1]})^{\\operatorname{op}}$. Its factorization system is lifted from $\\mathsf{Assoc}$ along the canonical left fibration $(\\Delta_{/[1]})^{\\operatorname{op}} \\to \\Delta^{\\operatorname{op}}$. And the elementary objects are given by $[1] \\simeq \\{0\\} \\to [1]$, $[1] \\simeq \\{1\\} \\to [1]$ and $\\operatorname{id}_{[1]}$.\nBy construction, an object $[n] \\to [1]$ in $\\Delta_{/[1]}$ can be viewed as an ordered sequence $(i_0, \\dots, i_n)$ where $0 \\le i_0 \\le \\dots \\le i_n \\le 1$. The elementary objects correspond to the sequences $(0,0)$, $(0,1)$, and $(1,1)$.\nDefinition 12 (Left Module Pattern, Simplicial model version). The underlying category of left module pattern $\\mathsf{LM}$ is the category $\\Delta^{\\operatorname{op}} \\times [1]$. Consider the functor $T \\colon \\Delta^{\\operatorname{op}} \\to \\Delta^{\\operatorname{op}}$, sending $[n]$ to $[n] \\star [0] \\simeq [n+1]$. Then the functor induces a functor \\[ (T \\to \\operatorname{id}) \\colon \\Delta^{\\operatorname{op}} \\times [1] \\to \\Delta^{\\operatorname{op}}. \\] The algebraic pattern structure of $\\mathsf{LM}$ is lifted from $\\mathsf{Assoc}$ along $(T \\to \\operatorname{id})$. More precisely,\n$\\Delta^{\\operatorname{op}} \\times \\{1\\}$ is precisely $\\mathsf{Assoc}$. For each $[n] \\in \\Delta$, the induced morphism $([n],0) \\to ([n],1)$ is an inert morphism in $\\mathsf{LM}$. If $f \\colon [h] \\to [n]$ is an inert morphism in $\\Delta$ such that $f(h) = n$, then the corresponding morphism $([n],0) \\to ([h],0)$ is an inert morphism in $\\mathsf{LM}$. If $g \\colon [h] \\to [n]$ is a morphism in $\\Delta$ such that $g(0) = 0$, then the corresponding morphism $([n],0) \\to ([h],0)$ is an active morphism in $\\mathsf{LM}$. The elementary objects of $\\mathsf{LM}$ are $([0],0)$ (often denoted by $\\mathfrak{m}$) and $([1],1)$ (often denoted by $\\mathfrak{a}$). Remark. Saying \u0026ldquo;left\u0026rdquo; versus \u0026ldquo;right\u0026rdquo; is just a convention; the same algebraic pattern also encodes right actions. Segal Object The algebra represented by the algebraic pattern is called Segal object.\nDefinition 13. Let $\\mathcal{O}$ be an algebraic pattern. A functor $X \\colon \\mathcal{O} \\to \\mathcal{C}$ is called Segal $\\mathcal{O}$-object of category $\\mathcal{C}$ if for every $O \\in \\mathcal{O}$ the induced functor \\[ \\left(\\mathcal{O}_{O/}^{\\text{el}}\\right)^{\\lhd} \\to \\mathcal{O} \\xrightarrow{X} \\mathcal{C} \\] is a limit diagram. If $\\mathcal{C}$ has limit for diagrams indexed by $\\mathcal{O}_{O/}^{\\text{el}}$ for all $O \\in \\mathcal{O}$ in which case we say that $\\mathcal{C}$ is $\\mathcal{O}$-complete, then this condition is equivalent to the canonical morphisms \\[ X(O) \\to \\underset{E \\in \\mathcal{O}_{O/}^{\\text{el}}}{\\operatorname{lim}}\\, X(E). \\] Now, we will provide some examples to explain how algebraic patterns work.\nExample 14 (Segal $\\mathsf{Trivial}$-Objects). Let $\\mathcal{C}$ be a category. Then the Segal $\\mathsf{Trivial}$-object in $\\mathcal{C}$ is just an object in $\\mathcal{C}$. Example 15 (Segal $\\mathsf{Comm}$-Objects). Let $\\mathcal{C}$ be a category with finite products, and let $X \\colon \\mathsf{Comm} \\to \\mathcal{C}$ be a functor. The Segal condition on $X$ is \\[ X(\\langle n \\rangle) \\simeq \\underset{(\\langle n \\rangle \\to \\langle 1 \\rangle) \\in \\mathsf{Comm}^{\\text{inv}}}{\\operatorname{lim}}\\, X(\\langle 1 \\rangle). \\] We can identify the category $\\mathsf{Comm}^{\\text{el}}_{\\langle n \\rangle/}$ with the set of inert morphisms $\\{\\rho_i \\colon i = 1, \\dots, n\\}$, where $\\rho_i \\colon \\langle n \\rangle \\to \\langle 1 \\rangle$ is given by \\[ \\rho_i(j) = \\begin{cases} 1 \u0026 \\text{if } j=i, \\\\ 0 \u0026 \\text{if } j \\neq i. \\end{cases} \\] Then, the Segal condition says that the canonical map \\[ (\\rho_i^*)_{i = 1}^n \\colon X(\\langle n \\rangle) \\to \\prod_{i=1}^n X(\\langle 1 \\rangle) \\] is an equivalence. This means that for each non-basepoint element $i \\in \\langle n \\rangle$ (where $i \\neq 0$), we can specify a corresponding object $x_i \\in X(\\langle 1 \\rangle)$. Therefore, we can describe an object in $X(\\langle n \\rangle)$ as a sequence $(x_1, \\dots, x_n)$.\nNext, we will show how inert and active morphisms work:\nLet $f \\colon \\langle n \\rangle \\to \\langle m \\rangle$ be an inert morphism in $\\mathsf{Comm}$. Then $f$ corresponds to the projection \\[ X(f) \\colon X(\\langle n \\rangle) \\simeq X(\\langle 1 \\rangle)^n \\to X(\\langle 1 \\rangle)^m \\simeq X(\\langle m \\rangle), \\quad (x_1, \\dots, x_n) \\mapsto (x_{f^{-1}(1)}, \\dots, x_{f^{-1}(m)}). \\] Let $g \\colon \\langle n \\rangle \\to \\langle m \\rangle$ be an active morphism in $\\mathsf{Comm}$, and let $I_j \\coloneqq g^{-1}(j)$ for $j \\in \\{1, \\dots, m\\}$. Then $g$ corresponds to the morphism \\[ X(g) \\colon X(\\langle n \\rangle) \\simeq X(\\langle 1 \\rangle)^n \\to X(\\langle 1 \\rangle)^m \\simeq X(\\langle m \\rangle), \\quad (x_1, \\dots, x_n) \\mapsto \\left(\\prod_{i_1 \\in I_1} x_{i_1}, \\dots, \\prod_{i_m \\in I_m} x_{i_m}\\right). \\] In particular, the active morphism $s \\colon \\langle 2 \\rangle \\to \\langle 2 \\rangle$ that swaps $1$ and $2$ corresponds to the map $(x_1, x_2) \\mapsto (x_2, x_1)$, which enforces the commutativity. Active morphisms represent the “commutative multiplication”. When we set $\\mathcal{C} = \\mathsf{Set}$, we find that $\\mathsf{Comm}$-Segal objects are precisely commutative monoids.\nExample 16 (Segal $\\mathsf{Assoc}$-Objects). Let $\\mathcal{C}$ be a category with finite products, and let $X \\colon \\mathsf{Assoc} \\to \\mathcal{C}$ be a simplicial object. The Segal condition on $X$ is \\[ X([n]) \\simeq \\underset{([n] \\to [1]) \\in \\mathsf{Assoc}^{\\text{inv}}}{\\operatorname{lim}}\\, X([1]). \\] Now, let\u0026rsquo;s analyze the limit above. An inert morphism $e_i \\colon [n] \\to [1]$ in $\\Delta^{\\operatorname{op}}$ corresponds to an inclusion $[1] \\hookrightarrow [n]$ in $\\Delta$ with image $\\{i-1, i\\}$. Notice that $[n]$ is a linearly ordered set, and one can think of it as being cut into $n$ pieces: \\[ [n] = \\left\\{0 \\xrightarrow{e_1} 1 \\xrightarrow{e_2} \\cdots \\xrightarrow{e_n} n \\right\\}, \\] where the segment $i-1 \\to i$ corresponds to the inert map $e_i$. The Segal condition says that the canonical map \\[ (e_i^*)_{i=1}^n \\colon X([n]) \\to \\prod_{i=1}^n X([1]) \\] is an equivalence. This means we can associate each arrow $i-1 \\to i$ in $[n]$ with a corresponding object $x_i \\in X([1])$. Therefore, we can describe an object in $X([n])$ as a sequence $(x_1, \\dots, x_n)$.\nNext, we will show how inert and active morphisms work:\nLet $f \\colon [n] \\to [m]$ be an inert morphism in $\\mathsf{Assoc}$, and let $f^{\\operatorname{op}} \\colon [m] \\to [n]$ be the corresponding morphism in $\\Delta$. In the Segal object $X$, the morphism $X(f)$ corresponds to a projection: \\[ X([n]) \\simeq X([1])^n \\to X([1])^m \\simeq X([m]), \\quad (x_1, \\dots, x_n) \\mapsto (x_{f^{\\operatorname{op}}(1)}, \\dots, x_{f^{\\operatorname{op}}(m)}). \\] Let $g \\colon [n] \\to [m]$ be an active morphism in $\\mathsf{Assoc}$, and let \\[ I_j \\coloneqq \\{(g^{\\operatorname{op}})^{-1}(j-1)+1,\\cdots,(g^{\\operatorname{op}})^{-1}(j)\\} \\] for $j \\in \\{1, \\dots, m\\}$. In the Segal object $X$, the morphism $X(g)$ corresponds to: \\[ X([n]) \\simeq X([1])^n \\to X([1])^m \\simeq X([m]), \\quad (x_1, \\dots, x_n) \\mapsto \\left(\\prod_{i_1 \\in I_1} x_{i_1}, \\dots, \\prod_{i_m \\in I_m} x_{i_m}\\right), \\] which represents “multiplication”. When we set $\\mathcal{C} = \\mathsf{Set}$, we find that $\\mathsf{Assoc}$-Segal objects are precisely monoids.\nRemark. Note that:\nthe morphism $[0] \\to [1]$ in $\\mathsf{Assoc}$ corresponds to the morphism $1 \\colon * \\to X([1])$, which is the unit of the associative algebra. the degenerate morphism $\\operatorname{s}^i\\colon [n] \\to [n-1]$ corresponds to the morphism \\[ X([n-1]) \\to X([n]), \\quad (x_1,\\cdots,x_{n-1}) \\mapsto (x_1,\\cdots,\\underset{i\\text{-th}}{1},\\cdots, x_{n-1}). \\] Therefore, if we remove these data, we will get the Segal $\\mathbb{A}_{\\infty}^{\\operatorname{nu}}$-object (Definition~\\ref{def-nonunital-An-pattern}). Example 17 (Segal $\\mathsf{CM}$-Objects). Let $\\mathcal{C}$ be a category with finite products, and $X \\colon \\mathsf{CM} \\to \\mathcal{C}$ be a functor. The Segal condition on $X$ is \\[ X(\\langle n \\rangle,i) \\simeq \\underset{(\\langle h \\rangle,j) \\in \\mathsf{CM}^{\\text{el}}_{(\\langle n \\rangle,i)/}}{\\operatorname{lim}}\\, X(\\langle h \\rangle,j). \\] Now, let\u0026rsquo;s analyze the limit above. When $i = 0$, the Segal condition is equivalent to saying that \\[ X(\\langle n \\rangle,0) \\simeq \\prod_{j = 1}^n X(\\langle 1 \\rangle,0). \\]When $i \\neq 0$, the Segal condition is equivalent to saying that \\[ X(\\langle n \\rangle,i) \\simeq \\prod_{j = 1}^{i-1} X(\\langle 1 \\rangle,0) \\times X(\\langle 1 \\rangle,1) \\times \\prod_{j = i+1}^n X(\\langle 1 \\rangle,0). \\]We will refer to $X(\\langle 1 \\rangle,0)$ as $A$ and $X(\\langle 1 \\rangle,1)$ as $M$. Next, we will show how inert and active morphisms work:\nLet $f \\colon (\\langle n \\rangle,i) \\to (\\langle h \\rangle,j)$ be an inert morphism in $\\mathsf{CM}$. If $i = 0$, we have $j = 0$ by construction of $\\mathsf{CM}$, in this case, $f$ corresponds to the projection \\[ \\begin{aligned} A^n \u0026\\to A^h\\\\ (a_1,\\cdots,a_n) \u0026\\mapsto (a_{f^{-1}(1)},\\cdots, a_{f^{-1}(h)}). \\end{aligned} \\] If $i \\neq 0$ with $j = 0$, then $f$ corresponds to the projection \\[ \\begin{aligned} A^{i-1} \\times M \\times A^{n-i} \u0026\\to A^h\\\\ (a_1,\\cdots,m,\\cdots,a_n) \u0026\\mapsto (a_{f^{-1}(1)},\\cdots,a_{f^{-1}(h)}). \\end{aligned} \\] If $i \\neq 0$ with $j \\neq 0$, then $f$ corresponds to the projection \\[ \\begin{aligned} A^{i-1} \\times M \\times A^{n-i} \u0026\\to A^{j-1} \\times M \\times A^{h-j}\\\\ (a_1,\\cdots,m,\\cdots,a_n) \u0026\\mapsto (a_{f^{-1}(1)},\\cdots,m,\\cdots,a_{f^{-1}(h)}). \\end{aligned} \\] Let $g \\colon (\\langle n \\rangle,i) \\to (\\langle h \\rangle,j)$ be an active morphism in $\\mathsf{CM}$. If $i = 0$, then we have $j = 0$ by construction of $\\mathsf{CM}$, in this case, let $I_j \\coloneqq g^{-1}(j)$ for $j \\in \\{1, \\dots, m\\}$, we have $g$ corresponds to the morphism \\[ \\begin{aligned} A^n \u0026\\to A^h\\\\ (a_1,\\cdots,a_n) \u0026\\mapsto \\left(\\prod_{i_1 \\in I_1}a_{i_1},\\cdots, \\prod_{i_h \\in I_h}a_{i_h} \\right). \\end{aligned} \\] If $i \\neq 0$, then by the description of the active morphism, we know that $j \\neq 0$. Let $I_j \\coloneqq g^{-1}(j)$ for $j \\in \\{1, \\dots, m\\}$, we have $g$ corresponds to the morphism \\[ \\begin{aligned} A^{i-1} \\times M \\times A^{n-i} \u0026\\to A^{j-1} \\times M \\times A^{h-j}\\\\ (a_1,\\cdots,a_n) \u0026\\mapsto \\left(\\prod_{i_1 \\in I_1}a_{i_1},\\cdots, \\left(\\prod_{i_j \\in I_j} a_{i_j}\\right)\\cdot m, \\cdots, \\prod_{i_h \\in I_h}a_{i_h} \\right). \\end{aligned} \\] which represents “action”. Example 18 (Segal $\\mathsf{LM}$-Objects). Let $\\mathcal{C}$ be a category with finite products, and $X \\colon \\mathsf{LM} \\to \\mathcal{C}$ be a functor. The Segal condition on $X$ is \\[ X([n],i) \\simeq \\underset{([h],j) \\in \\mathsf{LM}^{\\text{el}}_{([n],i)/}}{\\operatorname{lim}}\\, X([h],j). \\] Now, let\u0026rsquo;s analyze the limit above. Consider the canonical projection $p \\colon ([n],0) \\to ([0],0)$ in $\\mathsf{LM}$ corresponds to an inclusion $[0] \\simeq \\{n\\} \\hookrightarrow [n]$.\nWhen $i = 1$, the Segal condition is equivalent to saying that \\[ \\left((e_i,1)^*\\right)_{i = 1}^n \\colon X([n],1) \\simeq \\prod_{i = 1}^n X([1],1). \\]When $i = 0$, the Segal condition is equivalent to saying that \\[ \\left((e_i,0)^*\\right)_{i = 1}^{n} \\times p \\colon X([n],0) \\to \\left(\\prod_{i = 1}^n X([1],1)\\right) \\times X([0],0). \\]We will refer to $X([1],1)$ as $A$ and $X([0],0)$ as $M$.\nThat is, the Segal $\\mathsf{LM}$-objects in $\\mathcal{C}$ consists of those natural transformations \\[ M_{\\bullet} \\Rightarrow A_{\\bullet} \\] of simplicial objects $M_{\\bullet},A_{\\bullet} \\colon \\Delta^{\\operatorname{op}} \\to \\mathcal{C}$ such that $A_{\\bullet}$ is a Segal $\\mathsf{Assoc}$-object in $\\mathcal{C}$ and for all $n \\ge 0$, we have \\[ M_n = A^n \\times M. \\]Now we will show how inert and active morphisms work:\n$\\{1\\} \\times \\Delta^{\\operatorname{op}}$ consistent with $\\mathsf{Assoc}$. For each $[n] \\in \\Delta$, the induced morphism $([n],0) \\to ([n],1)$ corresponds to the projection \\[ \\begin{aligned} M_n = A^n \\times M \u0026\\to A^n\\\\ (a_1,\\cdots,a_n,m) \u0026\\mapsto (a_1,\\cdots,a_n). \\end{aligned} \\] Let $f \\colon ([n],0) \\to ([h],0)$ be an inert morphism in $\\mathsf{LM}$, denote its image under $(T \\to \\operatorname{id})$ as $\\tilde{f} \\colon [n+1] \\to [h+1]$. Then $f$ corresponds to the projection \\[ \\begin{aligned} M_n = A^n \\times M \u0026\\to A^h \\times M = M_h \\\\ (a_1,\\cdots,a_n,m) \u0026\\mapsto \\left(a_{\\tilde{f}^{\\operatorname{op}}(1)},\\cdots,a_{\\tilde{f}^{\\operatorname{op}}(h)},m\\right). \\end{aligned} \\] Let $g \\colon ([n],0) \\to ([h],0)$ be an active morphism in $\\mathsf{LM}$, denote its image under $(T \\to \\operatorname{id})$ as $\\tilde{g} \\colon [n+1] \\to [h+1]$. Let \\[ I_j \\coloneqq \\{(\\tilde{g}^{\\operatorname{op}})^{-1}(j-1)+1,\\cdots,(\\tilde{g}^{\\operatorname{op}})^{-1}(j)\\} \\] for $j \\in \\{1, \\dots, h+1\\}$. Then $g$ corresponds to the morphism \\[ \\begin{aligned} M_n = A^n \\times M \u0026\\to A^h \\times M = M_h\\\\ (a_1,\\cdots,a_n,m) \u0026\\mapsto \\left(\\prod_{i_1 \\in I_1} a_{i_1},\\cdots,\\prod_{i_h \\in I_h} a_{i_h},\\left(\\prod_{i_{h+1} \\in I_{h+1}} a_{i_{h+1}}\\right)\\cdot m \\right) \\end{aligned} \\] which represents “left action”. Operad Over An Algebraic Pattern In this section, we introduce operads, which are mathematical structures designed to encode the abstract properties of algebraic operations. Building on the notion of algebraic patterns, operads allow us to describe entire algebraic theories, such as the theory of homotopy-coherence algebras.\nFor more heuristics of operads, we refer to the excellent overview in [Cno25, Chapter 10] .\nAn operad $\\mathcal{O}$ over an algebraic pattern $\\mathcal{P}$ can be regarded as a category of “$\\mathcal{P}$-type” operation, where the “$\\mathcal{P}$-type” means Segal condition of $\\mathcal{P}$.\nDefinition 19 (Operad). Let $\\mathcal{P}$ be an algebraic pattern. An $\\mathcal{P}$-operad is a functor \\[ p \\colon \\mathcal{O} \\to \\mathcal{P} \\] with the algebraic pattern structure lifted from $\\mathcal{P}$ such that:\n$\\mathcal{O}$ has $p$-coCartesian lifts of inert morphisms in $\\mathcal{P}$. For $P \\in \\mathcal{P}$, let $\\mathcal{O}_{P}$ denote the fiber of $P$. For $X \\in \\mathcal{O}_{P}$, if \\[ \\xi \\colon \\left( \\mathcal{P}_{P/}^{\\text{el}} \\right)^{\\lhd} \\to \\mathcal{C} \\] is a diagram of coCartesian morphisms over the object of $\\mathcal{P}_{P/}^{\\text{el}}$, then for $Y \\in \\mathcal{O}_{P'}$, the commutative square is Cartesian. 3. The functor \\[ \\mathcal{O}_{P} \\to \\underset{O \\in \\mathcal{P}_{P/}^{\\text{el}}}{\\operatorname{lim}}\\, \\mathcal{O}_{O} \\] is an equivalence.\nWe refer to $\\mathsf{Op}(\\mathcal{P})$ as the category of $\\mathcal{P}$-operads and functors over $\\mathcal{P}$ that preserve inert coCartesian morphisms.\nRemark. When we consider the category (without algebraic pattern structure) of a $\\mathcal{P}$-operad $\\mathcal{O}$, we will denote it as $\\mathcal{O}^{\\otimes}$. Example 20 (Operad). $\\mathsf{Comm}$-operad is precisely the operad in the sense of [HA, Definition 2.1.1.10] . Unless specified otherwise, we will use the term “operad” to mean a $\\mathsf{Comm}$-operad and denote its category by $\\mathsf{Op}$. Example 21 (Generalized Operad). By , one can consider $\\mathsf{Fin}_*^{\\natural}$-operad, which is precisely the generalized operad in the sense of [HA, Definition 2.3.2.1] . Unless specified otherwise, we will use the term “generalized operad” to mean a $\\mathsf{Fin}_*^{\\natural}$-operad and denote its category by $\\mathsf{Op}^{\\operatorname{gen}}$. Remark. Consider the inclusion \\[ \\mathsf{Op} \\subseteq \\mathsf{Op}^{\\operatorname{gen}}. \\] By [HA, Proposition 2.1.4.6] , [HA, Remark 2.3.2.4] , and [HTT, Proposition A.3.7.6] , one can find that both $\\mathsf{Op}$ and $\\mathsf{Op}^{\\operatorname{gen}}$ are presentable. And $\\mathsf{Op} \\hookrightarrow \\mathsf{Op}^{\\operatorname{gen}}$ preserves limits. Therefore, it admits a left adjoint \\[ \\operatorname{L}_{\\operatorname{gen}} \\colon \\mathsf{Op}^{\\operatorname{gen}} \\to \\mathsf{Op}. \\] Example 22 (Planar Operad). $\\mathsf{Assoc}$-operad is the planar operad in the sense of [HA, Definition 4.1.3.2] . Unless specified otherwise, we will use the term “planar operad” to mean a $\\mathsf{Assoc}$-operad and denote its category by $\\mathsf{Op}^{\\operatorname{ns}}$. If \\[ f \\colon \\mathcal{O} \\to \\mathcal{P} \\] is a functor between algebraic patterns that preserves the factorization system and elementary objects, and moreover, the induced functor \\[ \\mathcal{O}_{X/}^{\\text{el}} \\to \\mathcal{P}_{f(X)/}^{\\text{el}} \\] is initial, that is, let $F \\colon \\mathcal{P}_{f(X)/}^{\\text{el}} \\to \\mathcal{C}$, then \\[ \\operatorname{lim} F \\simeq \\operatorname{lim} F \\circ f. \\] Then the pullback functor \\[ f^* \\colon \\mathsf{Op}(\\mathcal{P}) \\to \\mathsf{Op}(\\mathcal{O}) \\] admits a left adjoint under mild assumptions.\nExample 23 (Cut). Define a functor \\[ \\operatorname{Cut} \\colon \\Delta^{\\operatorname{op}} \\to \\mathsf{Fin}_* \\] that takes $[n]$ to $\\langle n \\rangle$ and a morphism $\\varphi \\colon [n] \\to [m]$ in $\\Delta$ to the map \\[ \\operatorname{Cut}(\\varphi) \\colon \\langle m \\rangle \\to \\langle n \\rangle \\] given by \\[ \\operatorname{Cut}(\\varphi)(i) = \\begin{cases} j, \u0026 \\varphi(j-1) \u003c i \\leq \\varphi(j), \\\\ 0, \u0026 \\text{otherwise}. \\end{cases} \\] Pulling back along this gives a functor \\[ \\mathsf{Op} \\to \\mathsf{Op}^{\\operatorname{ns}} \\] that informally “forgets symmetric group actions”. Its left adjoint is a “symmetrization” functor \\[ \\mathsf{Op}^{\\operatorname{ns}} \\to \\mathsf{Op}. \\] In fact, [BHS24, Theorem 5.1.1] proves a comparison result that gives conditions for certain functors as above to induce equivalences.\nLet us examine the structure of a $\\mathsf{Comm}$-operad $\\mathcal{O}$ in more detail. We refer to the fiber \\[ \\mathcal{O}_{\\langle 1 \\rangle} \\coloneqq p^{-1}(\\langle 1 \\rangle) \\] as the underlying category of the operad $\\mathcal{O}$. We denote its groupoid core by \\[ \\mathcal{O}^{\\simeq} \\coloneqq \\left(\\mathcal{O}_{\\langle 1 \\rangle}^{\\otimes}\\right) \\] and refer to it as the anima of colors of $\\mathcal{O}$.\nFor $\\langle n \\rangle \\in \\mathsf{Fin}_*$, condition~(3) guarantees that every object in $\\mathcal{O}_{I}^{\\otimes}$ may be uniquely written as a product \\[ \\prod_{i \\in I} x_i \\] for some colors $x_i \\in \\mathcal{O}^{\\simeq}$. We also denote such a product as an unordered tuple $\\{x_i\\}_{i \\in I}$.\nGiven another color $y \\in \\mathcal{O}^{\\simeq}$, we define the anima of multimorphisms (or anima of operations) in $\\mathcal{O}$ from $\\{x_i\\}_{i \\in I}$ to $y$ as the anima of morphisms in $\\mathcal{O}^{\\otimes}$ that map to the active morphism $\\langle n \\rangle \\to \\langle 1 \\rangle$: A multimorphism in an operad $\\mathcal{O}$ represents an abstract operation with multiple inputs. When interpreted in a symmetric monoidal category $\\mathcal{C}$, this corresponds to a morphism of the form \\[ x_1 \\otimes \\cdots \\otimes x_n \\to y. \\]At the end of this section, we will give the notion weakly enrichment.\nDefinition 24. Let $\\mathcal{C} \\to \\mathsf{Assoc}$ be a planar operad, and let $\\mathcal{M}$ be a category. We say $\\mathcal{M}$ is weakly enriched over $\\mathcal{C}$ if there exists a $\\mathsf{LM}$-operad $\\mathcal{O}$ such that \\[ \\mathcal{O}_{\\mathfrak{a}}^{\\otimes} \\simeq \\mathcal{C} \\quad\\text{and}\\quad \\mathcal{O}_{\\mathfrak{m}}^{\\otimes} \\simeq \\mathcal{M}. \\] $\\mathcal{O}$-Monoidal Category And $\\mathcal{O}$-Algebra In this section, we will consider $\\mathcal{O}$-monoidal category and $\\mathcal{O}$-algebra for some algebraic pattern $\\mathcal{O}$.\nBy the discussion above, it is suitable to consider algebraic objects in the Cartesian setting.\nDefinition 25 (Cartesian pattern). A Cartesian pattern is an algebraic pattern $\\mathcal{O}$ equipped with a morphism of algebraic patterns \\[ |-| \\colon \\mathcal{O} \\to \\mathsf{Comm} \\] such that for every $O \\in \\mathcal{O}$, the induced functor \\[ \\mathcal{O}_{O/}^{\\text{el}} \\to \\mathsf{Comm}_{|O|/}^{\\text{el}} \\] is an equivalence.\nRemark. All of the examples we considered before are Cartesian patterns. Now, one can define the $\\mathcal{O}$-monoidal categories and $\\mathcal{O}$-algebras in them.\nDefinition 26. Let $\\mathcal{O}$ be a Cartesian pattern. An $\\mathcal{O}$-monoidal category is a coCartesian fibration \\[ p_{\\mathcal{C}} \\colon \\mathcal{C}^{\\otimes} \\to \\mathcal{O} \\] whose associated functor $\\mathcal{O} \\to \\mathsf{Cat}$ is an $\\mathcal{O}$-Segal object.\nRemark. One can find that every $\\mathcal{O}$-monoidal category is an $\\mathcal{O}$-operad, if we lift the algebraic pattern structure along $p_{\\mathcal{C}}$. Definition 27. Let $\\mathcal{O}$ be a Cartesian pattern and $\\mathcal{V}^{\\otimes}$, $\\mathcal{W}^{\\otimes}$ be $\\mathcal{O}$-monoidal categories. A lax $\\mathcal{O}$-monoidal functor between them is a commutative triangle\nsuch that $F$ preserves inert morphisms. Equivalently, a lax $\\mathcal{O}$-monoidal functor is a morphism of algebraic patterns over $\\mathcal{O}$.\nIf the functor $F \\colon \\mathcal{V}^{\\otimes} \\to \\mathcal{W}^{\\otimes}$ preserves all coCartesian morphisms over $\\mathcal{O}$, we call it an $\\mathcal{O}$-monoidal functor.\nExample 28. A monoidal category is a $\\mathsf{Assoc}$-monoidal category $\\mathcal{C}^{\\otimes}$. We will denote the image of $[1]$ by $\\mathbb{1}_{\\mathcal{C}}$ and refer to it as the unit of the monoidal structure. We also let $\\mathcal{C}$ denote the fiber of $[1]$. In this context, $\\mathcal{C}^{\\otimes}$ will be referred to as the monoidal structure on $\\mathcal{C}$. By Example 16 , the active morphism $[2] \\to [1]$ in $\\mathsf{Assoc}$ corresponds to a functor \\[ -\\otimes- \\colon \\mathcal{C} \\times \\mathcal{C} \\to \\mathcal{C}, \\] which we will refer to as the tensor product functor.\nExample 29. A symmetric monoidal category is a $\\mathsf{Comm}$-monoidal category $\\mathcal{C}^{\\otimes}$. We will denote the image of $\\langle 1 \\rangle$ by $\\mathbb{1}_{\\mathcal{C}}$ and refer to it as the unit of the monoidal structure. We also let $\\mathcal{C}$ denote the fiber of $\\langle 1 \\rangle$. In this context, $\\mathcal{C}^{\\otimes}$ will be referred to as the symmetric monoidal structure on $\\mathcal{C}$. By Example 15 , the active morphism $\\langle 2 \\rangle \\to \\langle 1 \\rangle$ corresponds to a functor \\[ -\\otimes- \\colon \\mathcal{C} \\times \\mathcal{C} \\to \\mathcal{C}, \\] which we will refer to as the tensor product functor.\nLet $X \\colon \\mathsf{LM} \\to \\mathsf{Cat}$ be a $\\mathsf{LM}$-monoidal category. Using Grothendieck–Lurie construction, one can get a coCartesian fibration \\[ p \\colon \\mathcal{O} \\to \\mathsf{LM}. \\] Let $\\mathcal{O}_{\\mathfrak{a}}$ and $\\mathcal{O}_{\\mathfrak{m}}$ be the fiber of $\\mathfrak{a}$ and $\\mathfrak{m}$, respectively. One can imply the existence of the following structures:\nThe fiber $\\mathcal{O}_{\\mathfrak{a}}$ is a monoidal category. The fiber $\\mathcal{O}_{\\mathfrak{m}}$ is a category that is a left module over $\\mathcal{O}_{\\mathfrak{a}}$, meaning there is an action functor \\[ \\otimes \\colon \\mathcal{O}_{\\mathfrak{a}} \\times \\mathcal{O}_{\\mathfrak{m}} \\to \\mathcal{O}_{\\mathfrak{m}} \\] which is well-defined up to homotopy. Definition 30. Let $\\mathcal{C}$ be a monoidal category. We say a category $\\mathcal{M}$ is $\\mathcal{C}$-linear if there exists an $\\mathsf{LM}$-Segal object in $\\mathsf{Cat}$ given by a coCartesian fibration \\[ p \\colon \\mathcal{O} \\to \\mathsf{LM} \\] satisfying the following two properties:\n$\\mathcal{O}_{\\mathfrak{a}} \\simeq \\mathcal{C}$; $\\mathcal{O}_{\\mathfrak{m}} \\simeq \\mathcal{M}$. Remark. One can find that if $\\mathcal{M}$ is linear over $\\mathcal{C}$, then $\\mathcal{M}$ is weakly enriched over $\\mathcal{C}$. Example 31. For every category $\\mathcal{C}$, $\\mathcal{C}$ can be regarded as a $\\mathsf{Fun}(\\mathcal{C},\\mathcal{C})$-linear category (where the monoidal structure is composition). The action is given by \\[ (F,c) \\mapsto F(c). \\] The required higher coherence data is provided by the natural associativity of functor composition.\nExample 32. For any pair of categories $\\mathcal{C}$ and $\\mathcal{D}$, the category $\\mathsf{Fun}(\\mathcal{C},\\mathcal{D})$ can be regarded as being left tensored over the monoidal category $\\mathsf{Fun}(\\mathcal{C},\\mathcal{C})$ (where the monoidal structure is composition). The action is given by precomposition: \\[ \\otimes \\colon \\mathsf{Fun}(\\mathcal{C},\\mathcal{C}) \\times \\mathsf{Fun}(\\mathcal{C},\\mathcal{D}) \\to \\mathsf{Fun}(\\mathcal{C},\\mathcal{D}), \\quad (T,G) \\mapsto G \\circ T. \\] The required higher coherence data is provided by the natural associativity of functor composition.\nNow, we define the algebra object in a $\\mathcal{O}$-monoidal category $\\mathcal{C}$.\nDefinition 33. Let $\\mathcal{P}$ and $\\mathcal{P}'$ be Cartesian patterns with a morphism \\[ f \\colon \\mathcal{P} \\to \\mathcal{P}' \\] over $\\mathsf{Comm}$ and let $\\mathcal{O}$ be a $\\mathcal{P}$-operad. An $\\mathcal{O}$-algebra in $\\mathcal{O}$ is a commutative triangle\nsuch that $A$ takes inert morphisms in $\\mathcal{P}$ to coCartesian morphisms in $\\mathcal{O}$. We write $\\mathsf{Alg}_{\\mathcal{P}/\\mathcal{P}'}(\\mathcal{O})$ for the full subcategory of $\\mathsf{Fun}_{/\\mathcal{P}'}(\\mathcal{P},\\mathcal{O})$ spanned by the $\\mathcal{P}$-algebras. If $f = \\operatorname{id}_{\\mathcal{P}'}$, we will denote $\\mathsf{Alg}_{\\mathcal{P}/\\mathcal{P}'}(\\mathcal{O})$ by $\\mathsf{Alg}_{/\\mathcal{P}'}(\\mathcal{C})$. If $\\mathcal{P}' = \\mathsf{Comm}$, then we will omit $\\mathcal{P}'$ in $\\mathsf{Alg}_{\\mathcal{P}/\\mathcal{P}'}(\\mathcal{O})$.\nMoreover, if $\\mathcal{O} = \\mathcal{C}^{\\otimes}$ is a $\\mathcal{P}'$-monoidal category, then we will omit the notation $\\otimes$ in $\\mathsf{Alg}_{\\mathcal{P}/\\mathcal{P}'}(\\mathcal{C}^{\\otimes})$.\nExample 34. Let $\\mathcal{C}^{\\otimes}$ be a symmetric monoidal category. Consider the morphism \\[ |-| \\colon \\mathsf{Trivial} \\to \\mathsf{Comm} \\] sending $*$ to the elementary object $\\langle 1 \\rangle$. Then, $\\mathsf{Trivial}$-algebra in $\\mathcal{C}^{\\otimes}$ is just an object in \\[ \\mathcal{C} \\simeq \\mathcal{C}_{\\langle 1 \\rangle}^{\\otimes}. \\] Now, let $\\mathcal{P} = \\mathsf{Comm}$, we try to describe what an $\\mathcal{O}$-algebra is in a symmetric monoidal category $\\mathcal{C}$.\nDefinition 35. Let $\\mathcal{O}$ be an operad. Then:\n$\\mathsf{Comm}$-algebra in $\\mathcal{O}$ is called commutative algebra in $\\mathcal{O}$, and we denote $\\mathsf{Alg}_{\\mathsf{Comm}}(\\mathcal{O})$ by $\\mathsf{CAlg}(\\mathcal{O})$. $\\mathsf{Assoc}$-algebra in $\\mathcal{O}$ is called associative algebra in $\\mathcal{O}$, and we denote $\\mathsf{Alg}_{\\mathsf{Assoc}}(\\mathcal{O})$ by $\\mathsf{Alg}(\\mathcal{O})$. $\\mathsf{CM}$-algebra in $\\mathcal{O}$ is called modules over commutative algebra in $\\mathcal{O}$, and we denote $\\mathsf{Alg}_{\\mathsf{CM}}(\\mathcal{O})$ by $\\mathsf{Mod}(\\mathcal{O})$. $\\mathsf{LM}$-algebra in $\\mathcal{O}$ is called left modules in $\\mathcal{O}$, and we denote $\\mathsf{Alg}_{\\mathsf{LM}}(\\mathcal{O})$ by $\\mathsf{LMod}(\\mathcal{O})$. $\\mathsf{BM}$-algebra in $\\mathcal{O}$ is called bimodule in $\\mathcal{O}$, and we denote $\\mathsf{Alg}_{\\mathsf{BM}}(\\mathcal{O})$ by $\\mathsf{BMod}(\\mathcal{O})$. Remark. By , one can also use $\\mathsf{LM}$ to define right modules in $\\mathcal{O}$ (in this case, we will use $\\mathsf{RM}$ to denote $\\mathsf{LM}$) and use \\[ \\mathsf{RMod}(\\mathcal{O}) \\coloneqq \\mathsf{Alg}_{\\mathsf{RM}}(\\mathcal{O}) \\] to denote the category of right modules.\nDefinition 36. Let $\\mathcal{C}$ be a monoidal category and let \\[ q \\colon \\mathcal{O} \\to \\mathsf{LM} \\] exhibit $\\mathcal{M}$ weakly enriched over $\\mathcal{C}$. We let $\\mathsf{LMod}(\\mathcal{M})$ denote the category $\\mathsf{Alg}_{/\\mathsf{LM}}(\\mathcal{O})$. We will refer to $\\mathsf{LMod}(\\mathcal{M})$ as the category of left module objects of $\\mathcal{M}$. If $A$ is an associative algebra in $\\mathcal{C}$, we let $\\mathsf{LMod}_A(\\mathcal{M})$ denote the pullback\nRemark. One can analogously define $\\mathsf{Mod}_A$, $\\mathsf{RMod}_A$, and $_B\\mathsf{BMod}_A$ as pullbacks. References [CH21] Hongyi\u0026nbsp;Chu and Rune\u0026nbsp;Haugseng. Homotopy-coherent algebra via Segal conditions. Advances in Mathematics 385, 2021. DOI. [Cno25] Bastiaan\u0026nbsp;Cnossen. An ∞-categorical introduction to Stable Homotopy Theory and Higher Algebra. 2025. PDF. [HA] Jacob\u0026nbsp;Lurie. Higher Algebra. 2017. PDF. [HTT] Jacob\u0026nbsp;Lurie. Higher Topos Theory. Princeton University Press, 2009. PDF. [BHS24] Shaul\u0026nbsp;Barkan, Rune\u0026nbsp;Haugseng, and Jan\u0026nbsp;Steinebrunner. Envelopes for Algebraic Patterns. arXiv:2208.07183, 2024. arXiv. ","permalink":"https://ou-liu-red-sugar.github.io/notes/notes/basic-concepts-on-higher-algebra/","summary":"\u003ch2 id=\"algebraic-pattern\"\u003eAlgebraic Pattern\u003c/h2\u003e\n\u003cp\u003eAlgebraic pattern is a blueprint for a notion of functors on a fixed category satisfying a Segal condition, suitable for formalizing homotopy-coherent algebra in the Cartesian setting.\u003c/p\u003e\n\u003cp\u003eInformally, algebraic pattern generalizes the active and inert morphisms in operads and chooses certain objects to control the Segal condition.\u003c/p\u003e\n\u003cdiv class=\"thm-block kind-definition\" id=\"main-1\"\u003e\n    \u003cdiv class=\"thm-header\"\u003e\n      \u003cspan class=\"thm-title\"\u003e\n        Definition 1.\n      \u003c/span\u003e\n    \u003c/div\u003e\n    \u003cdiv class=\"thm-body\"\u003e\u003cp\u003eAn \u003cem\u003ealgebraic pattern\u003c/em\u003e is a category $\\mathcal{O}$ equipped with:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eA collection of objects called \u003cem\u003eelementary objects\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eA factorization system $(\\mathcal{O}^{\\text{inv}}, \\mathcal{O}^{\\text{act}})$ where every morphism factors uniquely (up to equivalence) as an \u003cem\u003einert\u003c/em\u003e morphism followed by an \u003cem\u003eactive\u003c/em\u003e morphism.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eWe let $\\mathcal{O}^{\\mathrm{el}}$ denote the full subcategory of $\\mathcal{O}$ spanned by the elementary objects and the inert morphisms between them. For any object $X \\in \\mathcal{O}$, we also write\n\u003c/p\u003e","title":"Basic Concepts on Higher Algebra"},{"content":"Recall that for a category with finite colimits and idempotent-completeness, $\\mathcal{C} \\in \\mathsf{Cat}^{\\mathrm{rex,idem}}$, we defined the Calkin category $\\mathsf{Calk}(\\mathcal{C}) = (\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1}/\\mathcal{C})^{\\mathrm{idem}}$ and used the relation $\\mathrm{k}(\\mathsf{Calk}^n(\\mathcal{C})) \\simeq \\tau_{\\ge 0}\\Omega\\mathrm{k}(\\mathsf{Calk}^{n+1}(\\mathcal{C}))$ to construct non-connective algebraic K-theory $\\mathrm{K}$.\nThe aim of this note is to use the inclusion $\\mathsf{Cat}^{\\mathrm{rex}} \\subset \\mathsf{Cat}^{\\mathrm{ca}} \\simeq \\mathsf{Pr}^L_{\\mathrm{ca}}$ to extend algebraic K-theory to compactly assembled categories. The result is continuous (Efimov) K-theory.\nThe continuous Calkin category First we extend the Calkin construction from small categories to compactly assembled categories, i.e. we want to produce a dashed arrow making commute.\nFor compactly assembled $\\mathcal{C}$, consider the Verdier cofibre sequence in $\\mathsf{Pr}^L_{\\mathrm{ca}}$, \\[ \\mathcal{C} \\xhookrightarrow{\\;\\hat{y}\\;} \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) \\longrightarrow \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) / \\mathcal{C}. \\] Since $\\mathsf{Pr}^L_{\\mathrm{ca}} \\to \\mathsf{Pr}^L$ preserves colimits and colimits in $\\mathsf{Pr}^L$ correspond to limits in $\\mathsf{Pr}^R$ under right-adjoint duality, this cofibre can be computed as the fibre of the right adjoint $k\\colon \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) \\to \\mathcal{C}$ over the terminal object: Here $* \\to \\mathcal{C}$ picks out the terminal object; fully faithfulness of this map identifies $\\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) / \\mathcal{C}$ with the full subcategory of $\\mathsf{Ind}(\\mathcal{C}^{\\aleph_1})$ on objects sent to $*$ by $k$ — the Ind-objects that \u0026ldquo;vanish in $\\mathcal{C}$\u0026rdquo;.\nProposition 1. Let $\\mathcal{C}$ be compactly assembled. The functor $\\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) \\to \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) / \\mathcal{C}$ is a Bousfield localization, and the target $\\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) / \\mathcal{C}$ is compactly generated. Proof. The right adjoint is the inclusion $k^{-1}(*) \\hookrightarrow \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1})$, which is fully faithful, so we have a Bousfield localization.\nFor compact generation, it suffices to show the inclusion preserves filtered colimits. Let $(X_i)_{i \\in I}$ be a filtered diagram in $k^{-1}(*)$. Since $k$ is a left adjoint, $k(\\operatorname*{colim}_i X_i) \\simeq \\operatorname*{colim}_i k(X_i) \\simeq \\operatorname*{colim}_i * \\simeq *$, so $\\operatorname*{colim}_i X_i \\in k^{-1}(*)$. The inclusion hence preserves filtered colimits, so the localization preserves compact objects, and their image generates the quotient. Since $\\mathsf{Ind}(\\mathcal{C}^{\\aleph_1})$ is compactly generated, so is the quotient.\n$\\square$ By Gabriel–Ulmer duality, the compact objects $(\\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) / \\mathcal{C})^{\\aleph_0}$ of a compactly generated category form an idempotent-complete category with finite colimits.\nDefinition 2 (Continuous Calkin category). Let $\\mathcal{C}$ be compactly assembled. The continuous Calkin category of $\\mathcal{C}$ is \\[ \\mathsf{Calk}^{\\mathrm{cont}}(\\mathcal{C}) \\coloneqq \\left( \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) / \\mathcal{C} \\right)^{\\aleph_0}. \\] Proposition 3. For compactly assembled $\\mathcal{C}$, there is a right-exact $p\\colon \\mathcal{C}^{\\aleph_1} \\to \\mathsf{Calk}^{\\mathrm{cont}}(\\mathcal{C})$ such that \\[ \\mathsf{Ind}(p)\\colon \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) \\to \\mathsf{Ind}(\\mathsf{Calk}^{\\mathrm{cont}}(\\mathcal{C})) \\] is a Bousfield localization. Moreover, there is a natural cofibre sequence \\[ \\mathcal{C} \\xrightarrow{\\;\\hat{y}\\;} \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) \\xrightarrow{\\;\\mathsf{Ind}(p)\\;} \\mathsf{Ind}(\\mathsf{Calk}^{\\mathrm{cont}}(\\mathcal{C})), \\] and $\\mathsf{Ind}(p)^R$ preserves pushouts.\nProof. $\\mathsf{Ind}(p)$ is the Bousfield localization of Proposition 1 , and it preserves compact objects, so by Gabriel–Ulmer we recover a right-exact $p$ on the small-category side.\nFor the pushout claim: $\\mathsf{Ind}(p)^R$ is the inclusion $k^{-1}(*) \\hookrightarrow \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1})$, which sits inside the pullback\nGiven a pushout diagram $A, B, C \\in k^{-1}(*)$, the pushout $A \\cup_C B$ taken in $\\mathsf{Ind}(\\mathcal{C}^{\\aleph_1})$ satisfies $k(A \\cup_C B) \\simeq k(A) \\cup_{k(C)} k(B) \\simeq * \\cup_* * \\simeq *$, since $k$ preserves colimits. So $A \\cup_C B \\in k^{-1}(*)$, and the inclusion preserves pushouts.\n$\\square$ Proposition 4. For an idempotent-complete $\\mathcal{C}$ with finite colimits, \\[ \\mathsf{Calk}^{\\mathrm{cont}}(\\mathsf{Ind}(\\mathcal{C})) \\simeq \\mathsf{Calk}(\\mathcal{C}). \\] Proof. Plugging the compactly generated $\\mathsf{Ind}(\\mathcal{C})$ into the previous cofibre sequence, \\[ \\mathsf{Ind}(\\mathcal{C}) \\xhookrightarrow{\\;\\hat{y}\\;} \\mathsf{Ind}(\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1}) \\xrightarrow{\\;\\mathsf{Ind}(p)\\;} \\mathsf{Ind}(\\mathsf{Calk}^{\\mathrm{cont}}(\\mathsf{Ind}(\\mathcal{C}))). \\] Compactly-generated makes $\\hat{y} = \\mathsf{Ind}(y)$ (idempotent completeness gives $\\mathcal{C} \\simeq \\mathsf{Ind}(\\mathcal{C})^{\\aleph_0} \\subset \\mathsf{Ind}(\\mathcal{C})^{\\aleph_1}$). So the full sequence is obtained by applying $\\mathsf{Ind}$ to $\\mathcal{C} \\xrightarrow{y} \\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} \\xrightarrow{p} \\mathsf{Calk}^{\\mathrm{cont}}(\\mathsf{Ind}(\\mathcal{C}))$. Gabriel–Ulmer equivalence $\\mathsf{Cat}^{\\mathrm{rex,idem}} \\simeq \\mathsf{Pr}^L_{\\aleph_0}$ lets us recover \\[ \\mathsf{Calk}^{\\mathrm{cont}}(\\mathsf{Ind}(\\mathcal{C})) \\simeq \\left( \\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} / \\mathcal{C} \\right)^{\\aleph_0} \\simeq \\left( \\mathsf{Ind}(\\mathcal{C})^{\\aleph_1} / \\mathcal{C} \\right)^{\\mathrm{idem}} = \\mathsf{Calk}(\\mathcal{C}), \\] the second equivalence because compact objects of a compactly generated category are automatically idempotent-complete.\n$\\square$ Verdier cofibre sequences in $\\mathsf{Pr}^L_{\\mathrm{ca}}$ For a more thorough study of $\\mathsf{Calk}^{\\mathrm{cont}}$ we need an appropriate notion of Verdier sequence in the large setting.\nDefinition 5 (Verdier cofibre sequence in Pr^L_ca). A cofibre sequence \\[ \\mathcal{C} \\xrightarrow{\\;i\\;} \\mathcal{D} \\xrightarrow{\\;p\\;} \\mathcal{E} \\] in $\\mathsf{Pr}^L_{\\mathrm{ca}}$ is a Verdier cofibre sequence if\n$i$ is fully faithful; $i^R$ preserves pushouts. The natural sequence $\\mathcal{C} \\xrightarrow{\\hat{y}} \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) \\to \\mathsf{Ind}(\\mathsf{Calk}^{\\mathrm{cont}}(\\mathcal{C}))$ from Proposition 3 is a Verdier cofibre sequence.\nContinuous (Efimov) K-theory We now have all the pieces to define continuous K-theory.\nDefinition 6 (Continuous K-theory). Let $\\mathcal{C}$ be a compactly assembled category. Its continuous K-theory is \\[ \\mathrm{K}^{\\mathrm{cont}}(\\mathcal{C}) \\coloneqq \\Omega \\mathrm{K}(\\mathsf{Calk}^{\\mathrm{cont}}(\\mathcal{C})). \\] Iteration reduces $\\mathrm{K}^{\\mathrm{cont}}$ to connective $\\mathrm{k}$ just as before. For compactly assembled $\\mathcal{C}$ and $n \\ge 1$, define \\[ \\mathsf{Calk}^{\\mathrm{cont}, n}(\\mathcal{C}) \\coloneqq \\mathsf{Calk}^{n-1}(\\mathsf{Calk}^{\\mathrm{cont}}(\\mathcal{C})). \\] Then $\\mathrm{K}^{\\mathrm{cont}}(\\mathcal{C})_n = \\mathrm{k}(\\mathsf{Calk}^{\\mathrm{cont},n}(\\mathcal{C}))$ for $n \\ge 1$. The $n = 0$ layer has no such formula: the first Calkin step already passes from the large world to the small.\nBasic properties Theorem 7 (Properties of continuous K-theory). The functor $\\mathrm{K}^{\\mathrm{cont}}\\colon \\mathsf{Pr}^L_{\\mathrm{ca}} \\to \\mathsf{Sp}$ has the following properties.\nAgreement with classical $\\mathrm{K}$. There is a natural map $\\mathrm{K}(\\mathcal{C}^{\\omega}) \\to \\mathrm{K}^{\\mathrm{cont}}(\\mathcal{C})$ that is an equivalence when $\\mathcal{C}$ is compactly generated. $\\mathrm{K}^{\\mathrm{cont}}$ preserves filtered colimits and finite products. Eilenberg swindle. If there is a compactly assembled functor $F\\colon \\mathcal{C} \\to \\mathcal{C}$ with $F \\sqcup \\operatorname{id} \\simeq F$, then $\\mathrm{K}^{\\mathrm{cont}}(\\mathcal{C}) \\simeq 0$. If $S = (\\mathcal{C} \\to \\mathcal{D} \\to \\mathcal{E})$ is a sequence in $\\mathsf{Pr}^L_{\\mathrm{ca}}$ whose stabilization $\\mathsf{Sp} \\otimes S$ is a Verdier sequence (in particular, if $S$ is a Verdier cofibre sequence), then $\\mathrm{K}^{\\mathrm{cont}}(S)$ is a cofibre sequence. $\\mathrm{K}^{\\mathrm{cont}}$ inverts the canonical maps $\\mathcal{C} \\to \\mathsf{An}_* \\otimes \\mathcal{C}$ and $\\mathcal{C} \\to \\mathsf{Sp} \\otimes \\mathcal{C}$. Given $\\mathrm{K}$, properties (1) and (2) already determine $\\mathrm{K}^{\\mathrm{cont}}$ uniquely.\nThe proof flows from a structural theorem on how the Calkin construction interacts with various cofibre sequences in $\\mathsf{Pr}^L_{\\mathrm{ca}}$ — the large-scale analogue of from the previous section.\nProposition 8 (Calk^cont vs cofibre sequences). Let $S = (\\mathcal{C} \\xrightarrow{i} \\mathcal{D} \\xrightarrow{p} \\mathcal{E})$ be a sequence in $\\mathsf{Pr}^L_{\\mathrm{ca}}$. Consider the conditions:\n$S$ is a cofibre sequence, $i$ is fully faithful, and $i$ preserves the terminal object. $S$ is a Verdier cofibre sequence (Definition 5 ). $(2')$ $S$ is a cofibre sequence and $i$ is a fully faithful strong left adjoint. $\\mathsf{An}_* \\otimes S = (\\mathcal{C}_* \\to \\mathcal{D}_* \\to \\mathcal{E}_*)$ is a cofibre sequence with $\\mathsf{An}_* \\otimes i$ fully faithful. $\\mathsf{Sp} \\otimes S$ is a Verdier sequence in the stable setting, i.e. a cofibre sequence with $\\mathsf{Sp} \\otimes i$ fully faithful. $\\mathrm{K}^{\\mathrm{cont}}(S)$ is a cofibre sequence. Then $(2') \\Rightarrow (2)$, $(1) \\Rightarrow (3) \\Rightarrow (4) \\Rightarrow (5)$ and $(2) \\Rightarrow (3)$.\nMoreover:\nWhen $\\mathcal{C}, \\mathcal{D}, \\mathcal{E}$ are pointed, $(1) \\Leftrightarrow (3)$ and $(2) \\Leftrightarrow (2')$; when they are stable, all implications except $(4) \\Rightarrow (5)$ become equivalences. Accordingly, $\\mathsf{An}_* \\otimes -$ and $\\mathsf{Sp} \\otimes -$ preserve sequences of all listed types. The natural cofibre sequence $\\mathcal{C} \\xhookrightarrow{\\hat{y}} \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) \\to \\mathsf{Ind}(\\mathsf{Calk}^{\\mathrm{cont}}(\\mathcal{C}))$ is of type $(2')$. Types $(3)$, $(4)$ and $(5)$ are closed under filtered colimits; filtered colimits of the natural type-$(2')$ sequences above are again type-$(2')$. $\\mathsf{Ind}\\colon \\mathsf{Cat}^{\\mathrm{rex,idem}} \\to \\mathsf{Pr}^L_{\\mathrm{ca}}$ preserves sequences of all types (idempotent-completeness is needed only for type $(5)$), and $\\mathsf{Calk}^{\\mathrm{cont}}\\colon \\mathsf{Pr}^L_{\\mathrm{ca}} \\to \\mathsf{Cat}^{\\mathrm{rex,idem}}$ preserves Verdier cofibre sequences. As a consequence $\\mathsf{Calk}$ also preserves Verdier cofibre sequences. Proof. See [sheaves-on-manifolds, Prop. 3.4.3] . The key non-trivial implication is $(4) \\Rightarrow (5)$: for a type-$(4)$ sequence $\\mathcal{C} \\to \\mathcal{D} \\to \\mathcal{E}$, natural equivalences \\[ \\Sigma \\mathrm{K}^{\\mathrm{cont}} \\simeq \\mathrm{K}(\\mathsf{Calk}^{\\mathrm{cont}}(-)) \\simeq \\mathrm{K}(\\mathsf{SW}(\\mathsf{Calk}^{\\mathrm{cont}}(-))) \\simeq \\mathrm{K}(\\mathsf{Calk}^{\\mathrm{cont}}(\\mathsf{Sp} \\otimes -)) \\] reduce the question to: tensoring with $\\mathsf{Sp}$ turns the sequence into a Verdier cofibre sequence, $\\mathsf{Calk}^{\\mathrm{cont}}$ preserves these, and $\\mathrm{K}$ sends Verdier cofibre sequences in $\\mathsf{Cat}^{\\mathrm{rex,idem}}$ to cofibre sequences.\n$\\square$ Proof of Theorem on K^cont. Given Proposition 8 , we can extract each property of Theorem 7 :\n(1) When $\\mathcal{C}$ is compactly generated, $\\mathcal{C} \\simeq \\mathsf{Ind}(\\mathcal{C}^{\\omega})$ and Proposition 4 gives $\\mathsf{Calk}^{\\mathrm{cont}}(\\mathcal{C}) \\simeq \\mathsf{Calk}(\\mathcal{C}^{\\omega})$, so \\[ \\mathrm{K}^{\\mathrm{cont}}(\\mathcal{C}) = \\Omega \\mathrm{K}(\\mathsf{Calk}^{\\mathrm{cont}}(\\mathcal{C})) = \\Omega \\mathrm{K}(\\mathsf{Calk}(\\mathcal{C}^{\\omega})) \\simeq \\mathrm{K}(\\mathcal{C}^{\\omega}). \\](2) Filtered colimits and finite products: $\\mathsf{Calk}^{\\mathrm{cont}}$ preserves them ($\\mathsf{Pr}^L_{\\mathrm{ca}}$ is semi-additive and $\\mathsf{Calk}^{\\mathrm{cont}}$ is computed via $(-)^{\\aleph_0}$ which preserves both operations on compactly generated quotients), and $\\mathrm{K}$ does too.\n(3) Product-preservation gives a map $\\mathrm{K}^{\\mathrm{cont}}(F)$ with $\\mathrm{K}^{\\mathrm{cont}}(F) + \\operatorname{id} = \\operatorname{id}$ in $\\pi_0 \\mathrm{End}(\\mathrm{K}^{\\mathrm{cont}}(\\mathcal{C}))$; this group is abelian, so $\\operatorname{id} = 0$.\n(4) By Proposition 8 , $(4) \\Rightarrow (5)$.\n(5) By part of Proposition 8 , since $\\mathcal{C} \\to \\mathsf{An}_* \\otimes \\mathcal{C}$ and $\\mathcal{C} \\to \\mathsf{Sp} \\otimes \\mathcal{C}$ are inverted under $\\mathrm{K}^{\\mathrm{cont}}$.\nUniqueness. Given $\\mathrm{K}$ and properties (1) and (2), $\\mathrm{K}^{\\mathrm{cont}}$ is determined. Indeed, every compactly assembled $\\mathcal{C}$ is a filtered colimit of compactly generated categories (via $\\mathsf{Ind}(-)^{\\aleph_1}$-style approximations and cofinality), and (1) pins $\\mathrm{K}^{\\mathrm{cont}}$ on the compactly generated ones while (2) propagates the definition along the filtered colimit.\n$\\square$ Finite products vs. products A common thread in the proofs is that finite-product preservation combined with a stable/additive target forces additional properties automatically. Concretely:\nRemark. The type-$(2')$ cofibre sequence $\\mathcal{C} \\hookrightarrow \\mathcal{C} \\times \\mathcal{D} \\to \\mathcal{D}$ admits a splitting, so any localizing (or continuous localizing) invariant preserves finite products. The Eilenberg swindle follows formally once products are preserved, because the target is additive. The universal property of $\\mathrm{K}^{\\mathrm{cont}}$ The construction $F \\mapsto F^{\\mathrm{cont}} \\coloneqq \\Omega F \\circ \\mathsf{Calk}^{\\mathrm{cont}}$ makes sense for any localizing invariant $F\\colon \\mathsf{Cat}^{\\mathrm{rex,idem}} \\to \\mathcal{D}$, not just $\\mathrm{K}$. Write $\\mathrm{Loc}(\\mathcal{D})$ for the category of such $F$, and $\\mathrm{Loc}^{\\mathrm{cont}}(\\mathcal{D})$ for the category of continuous localizing invariants $\\mathsf{Pr}^L_{\\mathrm{ca}} \\to \\mathcal{D}$ (those preserving Verdier cofibre sequences and filtered colimits).\nTheorem 9 (Efimov\u0026#39;s universal property). Restriction along $\\mathsf{Ind}\\colon \\mathsf{Cat}^{\\mathrm{rex,idem}} \\to \\mathsf{Pr}^L_{\\mathrm{ca}}$ gives an equivalence \\[ \\mathrm{Loc}^{\\mathrm{cont}}(\\mathcal{D}) \\xrightarrow{\\;\\sim\\;} \\mathrm{Loc}(\\mathcal{D}), \\] restricting to an equivalence between the finitary subcategories $\\mathrm{Loc}^{\\mathrm{cont}}_{\\omega}(\\mathcal{D}) \\xrightarrow{\\sim} \\mathrm{Loc}_{\\omega}(\\mathcal{D})$. Write $F^{\\mathrm{cont}}$ for the inverse image of $F \\in \\mathrm{Loc}(\\mathcal{D})$.\nProof. For $F \\in \\mathrm{Loc}(\\mathcal{D})$, the natural cofibre sequence $\\mathcal{C} \\hookrightarrow \\mathsf{Ind}(\\mathcal{C}^{\\aleph_1}) \\to \\mathsf{Ind}(\\mathsf{Calk}^{\\mathrm{cont}}(\\mathcal{C}))$ and the Eilenberg swindle force \\[ F^{\\mathrm{cont}}(\\mathcal{C}) \\coloneqq \\Omega F(\\mathsf{Calk}^{\\mathrm{cont}}(\\mathcal{C})). \\] This gives $F^{\\mathrm{cont}}(\\mathsf{Ind}(-)) = F$ by Proposition 4 , and $F^{\\mathrm{cont}}$ is determined by this. That $\\mathsf{Calk}^{\\mathrm{cont}}$ preserves Verdier cofibre sequences (part of Proposition 8 ) together with $F$\u0026rsquo;s localizing property gives localizingness of $F^{\\mathrm{cont}}$. The finitary case is analogous, using that $\\mathsf{Calk}^{\\mathrm{cont}}$ preserves filtered colimits up to $\\mathrm{k}$-equivalence.\n$\\square$ In particular this produces $\\mathrm{K}^{\\mathrm{cont}}$ as the unique continuous extension of $\\mathrm{K}$ — the universal status claimed on the index page of this series.\nReferences A. Krause, T. Nikolaus, P. Pützstück. Sheaves on Manifolds. 2024. PDF. A. I. Efimov. K-theory and localizing invariants of large categories. 2025. arXiv:2405.12169. M. Ramzi. Dualizable presentable $\\infty$-categories. 2024. arXiv:2410.21537. ","permalink":"https://ou-liu-red-sugar.github.io/notes/notes/continuous-k-theory/continuous-efimov-k-theory/","summary":"\u003cp\u003eRecall that for a category with finite colimits and idempotent-completeness,\n$\\mathcal{C} \\in \\mathsf{Cat}^{\\mathrm{rex,idem}}$, we defined the Calkin\ncategory $\\mathsf{Calk}(\\mathcal{C}) = (\\mathsf{Ind}(\\mathcal{C})^{\\aleph_1}/\\mathcal{C})^{\\mathrm{idem}}$\nand used the relation $\\mathrm{k}(\\mathsf{Calk}^n(\\mathcal{C})) \\simeq \\tau_{\\ge 0}\\Omega\\mathrm{k}(\\mathsf{Calk}^{n+1}(\\mathcal{C}))$\nto construct non-connective algebraic K-theory $\\mathrm{K}$.\u003c/p\u003e\n\u003cp\u003eThe aim of this note is to use the inclusion\n$\\mathsf{Cat}^{\\mathrm{rex}} \\subset \\mathsf{Cat}^{\\mathrm{ca}} \\simeq \\mathsf{Pr}^L_{\\mathrm{ca}}$\nto extend algebraic K-theory to compactly assembled categories. The result is\n\u003cstrong\u003econtinuous (Efimov) K-theory\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"the-continuous-calkin-category\"\u003eThe continuous Calkin category\u003c/h2\u003e\n\u003cp\u003eFirst we extend the Calkin construction from small categories to compactly\nassembled categories, i.e. we want to produce a dashed arrow making\n\u003cfigure class=\"tikzcd-fig tikzcd\" data-tikzcd-hash=\"a4e80fe27e59dc76bf074773322ab01ff3298f3b\"\u003e\u003cimg class=\"tikzcd-img\" src=\"/generated/tikzcd/a4e80fe27e59dc76bf074773322ab01ff3298f3b.svg\" alt=\"tikzcd diagram\" loading=\"lazy\" decoding=\"async\"\u003e\u003c/figure\u003e\n\ncommute.\u003c/p\u003e","title":"Continuous (Efimov) K-theory"},{"content":"Stefanich Rings In the last note, we define \\[ n\\mathsf{Pr} \\coloneqq \\mathsf{Mod}_{(n-1)\\mathsf{Pr}}(1\\mathsf{Pr}), \\qquad 1\\mathsf{Pr} \\coloneqq \\mathsf{Pr}_{\\aleph_1}^{L}. \\] Now, we let $0\\mathsf{Pr} \\coloneqq \\mathsf{An}$.\nSince we have \\[ \\mathsf{CAlg}(\\mathcal{C}) \\hookrightarrow \\mathsf{CAlg}(\\mathsf{Mod}_{\\mathcal{C}}(\\mathsf{Pr}_{\\aleph_1}^L)), \\quad A \\mapsto \\mathsf{Mod}_A(\\mathcal{C}), \\] and $1\\mathsf{Pr} \\in \\mathsf{Pr}_{\\aleph_1}^L$, we obtain a sequence \\[ \\mathsf{CAlg}(0\\mathsf{Pr}) \\hookrightarrow \\mathsf{CAlg}(1\\mathsf{Pr}) \\hookrightarrow \\cdots \\hookrightarrow \\mathsf{CAlg}(n\\mathsf{Pr}) \\hookrightarrow \\cdots . \\]In Presentable Categories, we know that $1\\mathsf{Pr}$ admits all small colimits, which can be computed in $\\widehat{\\mathsf{Cat}}$ by passing to adjoint functors. Thus, we obtain the following definition.\nDefinition 1. The $\\aleph_1$-presentable category $\\mathsf{StRing}$ of Stefanich rings is \\[ \\begin{aligned} \\mathsf{StRing} \u0026= \\operatorname{colim}_{1\\mathsf{Pr}} \\bigl( \\mathsf{CAlg}(\\mathsf{An}) \\hookrightarrow \\mathsf{CAlg}(1\\mathsf{Pr}) \\hookrightarrow \\cdots \\bigr) \\\\ \u0026= \\operatorname{lim}_{\\widehat{\\mathsf{Cat}}} \\bigl( \\mathsf{CAlg}(\\mathsf{An}) \\xleftarrow{\\mathsf{End}_{(-)}(1)} \\mathsf{CAlg}(1\\mathsf{Pr}) \\xleftarrow{\\mathsf{End}_{(-)}(1)} \\cdots \\bigr). \\end{aligned} \\] We often write $A = (A_0, A_1, \\dots) \\in \\mathsf{StRing}$, where $A_n \\in \\mathsf{CAlg}(n\\mathsf{Pr})$ and $A_{n} \\simeq \\mathsf{End}_{A_{n+1}}(1)$.\nRemark. Stefanich rings are also known as $\\aleph_1$-compactly generated categorical spectra, originally introduced by [ACS, Remark 3.3.6] .\nIn [Aok25] , the author uses $\\mathsf{PrSp}^{\\aleph_1}$ to denote the category of Stefanich rings, and also considers the construction \\[ \\mathsf{PrSp} \\coloneqq \\operatorname{colim}_{\\kappa} \\mathsf{PrSp}^{\\kappa}. \\] Unfortunately, this notion is no longer stable: as shown in [Aok25, Theorem C] , there exists an object which is not given by a sequence $(A_n)_n \\in \\prod_{n} \\mathsf{CAlg}(n\\mathsf{Pr}^L)$, where $n\\mathsf{Pr}^L \\coloneqq \\operatorname{colim}_{\\kappa} n\\mathsf{Pr}^{L}_{\\kappa}$.\nThe unit object in $\\mathsf{StRing}$ is given by $(*,\\mathsf{Ani},1\\mathsf{Pr},\\cdots)$. Consequently, any Stefanich ring $A$ is equipped with compatible symmetric monoidal maps $(n-1)\\mathsf{Pr} \\to A_n$. This map admits a lax symmetric monoidal right adjoint, which induces a forgetful functor $$ \\mathsf{CAlg}(A_n) \\to \\mathsf{CAlg}((n-1)\\mathsf{Pr}), $$ allowing us to regard an $A_n$-algebra as a $(n-1)\\mathsf{Pr}$-algebra. For $n \\geq 1$, these structures naturally arise from the identification $$ A_n \\in \\mathsf{CAlg}(n\\mathsf{Pr}) \\simeq \\mathsf{CAlg}\\bigl(\\mathsf{Mod}_{(n-1)\\mathsf{Pr}}(1\\mathsf{Pr})\\bigr). $$ A morphism $f \\colon A \\to B$ of Stefanich rings consists of compatible maps $f_{n-1}^* \\colon A_n \\to B_n$ in $\\mathsf{CAlg}(n\\mathsf{Pr})$ for all $n \\geq 1$. Since $f_{n-1}^*$ is a morphism in $1\\mathsf{Pr}$, it admits a lax symmetric monoidal right adjoint $f_{n-1,*} \\colon B_n \\to A_n$, which induces a functor $$ f_{n-1,*} \\colon \\mathsf{CAlg}(B_n) \\to \\mathsf{CAlg}(A_n). $$ In particular, the unit $\\mathbb{1}_{B_n} \\in \\mathsf{CAlg}(B_n)$ determines a commutative algebra $$ (B/A)_{n-1} \\coloneqq f_{n-1,*}(\\mathbb{1}_{B_n}) \\in \\mathsf{CAlg}(A_n). $$ Under the forgetful functor $\\mathsf{CAlg}(A_n) \\to \\mathsf{CAlg}((n-1)\\mathsf{Pr})$, this object maps to $B_{n-1} \\in \\mathsf{CAlg}((n-1)\\mathsf{Pr})$. Thus, any $A$-Stefanich algebra $B$ determines a sequence $\\bigl((B/A)_0, (B/A)_1, \\ldots\\bigr)$.\nTo make this precise, we introduce the following adjunctions. For each $n \\geq 0$, there is an adjunction\nwhere $G_n(X) \\coloneqq \\underline{\\mathsf{Hom}}_{A_{n+1}}(\\mathbb{1}_{A_{n+1}}, X)$. By definition of a Stefanich ring, there is a canonical equivalence\n$$ A_n \\xrightarrow{\\sim} G_n(\\mathbb{1}_{A_{n+1}}) = \\mathsf{End}_{A_{n+1}}(\\mathbb{1}_{A_{n+1}}). $$ Proposition 2. Let $A$ be a Stefanich ring. There are equivalences of categories\n$$ \\begin{align*} \\mathsf{StRing}_{A/} \u0026\\simeq \\operatorname{lim}_{\\widehat{\\mathsf{Cat}}} \\Bigl( \\mathsf{CAlg}(A_1) \\xleftarrow{\\;\\mathsf{End}_{G_1(-)}(\\mathbb{1})\\;} \\mathsf{CAlg}(A_2) \\xleftarrow{\\;\\mathsf{End}_{G_2(-)}(\\mathbb{1})\\;} \\cdots \\Bigr)\\\\ \u0026\\simeq \\operatorname{colim}_{1\\mathsf{Pr}} \\Bigl( \\mathsf{CAlg}(A_1) \\xrightarrow{\\;F_1(\\mathsf{Mod}_{(-)}(A_1))\\;} \\mathsf{CAlg}(A_2) \\xrightarrow{\\;F_2(\\mathsf{Mod}_{(-)}(A_2))\\;} \\cdots \\Bigr). \\end{align*} $$Furthermore, for each $n \\geq 0$, the structure map from the $(n{+}1)$-th term of the colimit defines a fully faithful embedding\n$$ \\operatorname{Spec}_n \\colon \\mathsf{CAlg}(A_{n+1}) \\hookrightarrow \\mathsf{StRing}_{A/}, $$whose right adjoint\n$$ \\Gamma_n \\colon \\mathsf{StRing}_{A/} \\to \\mathsf{CAlg}(A_{n+1}), \\qquad B \\longmapsto (B/A)_n $$recovers the $(n{+}1)$-th component of the limit.\nProof. Step 1: Limit description.\nSince $\\mathsf{StRing}$ is defined as a limit in $\\widehat{\\mathsf{Cat}}$, the slice $\\mathsf{StRing}_{A/}$ is the limit of the corresponding slices at each level. Lurie\u0026rsquo;s identification $\\mathsf{CAlg}(n\\mathsf{Pr})_{A_n/} \\simeq \\mathsf{CAlg}(\\mathsf{Mod}_{A_n}(1\\mathsf{Pr}))$ gives\n$$ \\mathsf{StRing}_{A/} \\simeq \\operatorname{lim}_{\\widehat{\\mathsf{Cat}}} \\Bigl( \\cdots \\to \\mathsf{CAlg}(\\mathsf{Mod}_{A_{n+1}}(1\\mathsf{Pr})) \\xrightarrow{T_n} \\mathsf{CAlg}(\\mathsf{Mod}_{A_n}(1\\mathsf{Pr})) \\to \\cdots \\Bigr). $$One checks that each $T_n$ factors through $\\mathsf{CAlg}(A_{n+1})$ via\n$$ \\mathsf{CAlg}(\\mathsf{Mod}_{A_{n+1}}(1\\mathsf{Pr})) \\xrightarrow{\\mathsf{End}_{(-)}(\\mathbb{1}_{A_{n+1}})} \\mathsf{CAlg}(A_{n+1}) \\xrightarrow{G_n^*} \\mathsf{CAlg}(\\mathsf{Mod}_{A_n}(1\\mathsf{Pr})), $$so inserting the intermediate terms $\\mathsf{CAlg}(A_{n+1})$ does not change the limit.\nStep 2: Colimit description.\nThe transition functors $\\mathsf{End}_{G_n(-)}(\\mathbb{1}) \\colon \\mathsf{CAlg}(A_{n+1}) \\to \\mathsf{CAlg}(A_n)$ preserve limits and $\\aleph_1$-filtered colimits, hence lie in $\\mathsf{Pr}^R_{\\aleph_1}$. Under the equivalence $(\\mathsf{Pr}^L_{\\aleph_1})^{\\mathrm{op}} \\simeq \\mathsf{Pr}^R_{\\aleph_1}$, the limit of these right adjoints corresponds to the colimit of their left adjoints $R \\mapsto F_n(\\mathsf{Mod}_R(A_n))$ in $1\\mathsf{Pr}$.\nStep 3: Full faithfulness of $\\operatorname{Spec}_n$.\nIt suffices to show that each transition functor $R \\mapsto F_n(\\mathsf{Mod}_R(A_n))$ is fully faithful, i.e.\\ that the unit $R \\to \\mathsf{End}_{G_n(F_n(\\mathsf{Mod}_R(A_n)))}(\\mathbb{1}_{A_n})$ is an equivalence. Since $\\mathsf{Mod}_R(A_n)$ is generated by $A_n$ with $\\mathsf{End}_{\\mathsf{Mod}_R(A_n)}(A_n) \\simeq R$, and since $F_n$ is symmetric monoidal and maps $A_n$ to $\\mathbb{1}_{A_{n+1}}$, we get\n$$ \\mathsf{End}_{G_n(F_n(\\mathsf{Mod}_R(A_n)))}(\\mathbb{1}_{A_n}) \\simeq \\mathsf{End}_{F_n(\\mathsf{Mod}_R(A_n))}(\\mathbb{1}_{A_{n+1}}) \\simeq R. $$Since $\\mathsf{StRing}_{A/}$ is the colimit of these fully faithful embeddings, each structure map $\\operatorname{Spec}_n$ is fully faithful.\n$\\square$ Remark. The notation is motivated by the geometric picture. On the level of Gestalten $\\mathsf{Gest} = \\mathsf{StRing}^{\\mathrm{op}}$, the induced functor $\\mathsf{CAlg}(A_{n+1})^{\\mathrm{op}} \\to \\mathsf{Gest}_{/X}$ is a relative $\\operatorname{Spec}$ at the $n$-th categorical level, and $\\Gamma_n$ extracts the ``$n$-th level global sections\u0026rsquo;\u0026rsquo;. We write $\\operatorname{Spec}_n$ on the ring side (strictly speaking $\\operatorname{Spec}_n^{\\mathrm{op}}$) for brevity.\nThe transition functors appearing in the colimit and limit descriptions of Proposition 2 are then expressed as composites:\n$$ R \\mapsto F_n(\\mathsf{Mod}_R(A_n)) \\quad\\text{is}\\quad \\Gamma_n \\circ \\operatorname{Spec}_{n-1}, \\qquad B \\mapsto \\mathsf{End}_{G_n(B)}(\\mathbb{1}_{A_n}) \\quad\\text{is}\\quad \\Gamma_{n-1} \\circ \\operatorname{Spec}_n. $$Here $\\Gamma_n \\circ \\operatorname{Spec}_{n-1}$ takes the $(n{-}1)$-affine approximation and extracts its $n$-th level data, while $\\Gamma_{n-1} \\circ \\operatorname{Spec}_n$ forgets down one categorical level.\nLemma 3. For any Stefanich ring $A$ and $n \\geq 0$, the functor $F_n$ is fully faithful on dualizable objects of $\\mathsf{Mod}_{A_n}(1\\mathsf{Pr})$, and more generally on $\\aleph_1$-filtered colimits of dualizable objects. Proof. Since $F_n$ preserves colimits and $G_n$ preserves $\\aleph_1$-filtered colimits, it suffices to treat dualizable objects. For dualizable $X, Y$, we compute\n$$ \\operatorname{Hom}_{A_{n+1}}(F_n X, F_n Y) \\simeq \\operatorname{Hom}_{A_{n+1}}(F_n(X \\otimes Y^\\vee), \\mathbb{1}_{A_{n+1}}) \\simeq \\operatorname{Hom}_{\\mathsf{Mod}_{A_n}(1\\mathsf{Pr})} (X \\otimes Y^\\vee, G_n(\\mathbb{1}_{A_{n+1}})). $$By the defining equivalence $G_n(\\mathbb{1}_{A_{n+1}}) \\simeq \\mathbb{1}_{A_n}$, this becomes\n$$ \\operatorname{Hom}_{\\mathsf{Mod}_{A_n}(1\\mathsf{Pr})} (X \\otimes Y^\\vee, \\mathbb{1}_{A_n}) \\simeq \\operatorname{Hom}_{\\mathsf{Mod}_{A_n}(1\\mathsf{Pr})}(X, Y), $$which proves full faithfulness.\n$\\square$ Shifting For any Stefanich ring $A = (A_0, A_1, A_2, \\ldots)$, the shifted sequence $(A_1, A_2, A_3, \\ldots)$ is again a Stefanich ring: the defining equivalences $A_n \\simeq \\mathsf{End}_{A_{n+1}}(\\mathbb{1})$ simply shift their indices. This gives a shifting functor on $\\mathsf{StRing}$, which turns out to be an equivalence.\nProposition 4. The shifting functor $(A_0, A_1, A_2, \\ldots) \\mapsto (A_1, A_2, A_3, \\ldots)$ induces an equivalence\n$$ \\mathsf{StRing} \\xrightarrow{\\;\\sim\\;} \\mathsf{StRing}_{(1\\mathsf{Pr},\\, 2\\mathsf{Pr},\\, \\ldots)/}. $$More generally, shifting $n$ times yields an equivalence $\\mathsf{StRing} \\simeq \\mathsf{StRing}_{(n\\mathsf{Pr},\\, (n+1)\\mathsf{Pr},\\, \\ldots)/}$.\nProof. This follows from the identification\n$$ \\mathsf{CAlg}((n+1)\\mathsf{Pr}) = \\mathsf{CAlg}\\bigl(\\mathsf{Mod}_{n\\mathsf{Pr}}(1\\mathsf{Pr})\\bigr) = \\mathsf{CAlg}(1\\mathsf{Pr})_{n\\mathsf{Pr}/}. $$Since also $\\mathsf{CAlg}(n\\mathsf{Pr}) = \\mathsf{CAlg}(1\\mathsf{Pr})_{(n-1)\\mathsf{Pr}/}$, the slice over $n\\mathsf{Pr}$ gives\n$$ \\mathsf{CAlg}(n\\mathsf{Pr})_{n\\mathsf{Pr}/} = \\bigl(\\mathsf{CAlg}(1\\mathsf{Pr})_{(n-1)\\mathsf{Pr}/}\\bigr)_{n\\mathsf{Pr}/} = \\mathsf{CAlg}(1\\mathsf{Pr})_{n\\mathsf{Pr}/} = \\mathsf{CAlg}((n+1)\\mathsf{Pr}), $$which is exactly the transition in the colimit defining $\\mathsf{StRing}$, but based at $n\\mathsf{Pr}$ instead of $0\\mathsf{Pr}$.\n$\\square$ Remark. The shifting operation has no known geometric or intuitive meaning. Its very existence is tied to our choice of foundations: it is not a priori clear that a functor $2\\mathsf{Pr} \\to 1\\mathsf{Pr}$ should exist, and the construction is not compatible with changing the cutoff cardinal $\\kappa$ (which we fixed to be $\\aleph_1$). Had we worked stably, i.e.\\ over the base $S = (D(\\mathbb{S}), 1\\mathsf{Pr}_{\\mathrm{st}}, \\ldots)$, the shifting operation would not be available.\nNonetheless, shifting is extremely useful as a technical device: it allows us to assume without loss of generality that $n = 0$ in most proofs about properties of morphisms, thereby simplifying notation considerably.\nTensor products of Stefanich rings Limits and spectrification Limits of Stefanich rings are straightforward: they are computed levelwise. Colimits, however, are more subtle. If one forms the colimit at each level separately, the result is a sequence $D_n \\in \\mathsf{CAlg}(n\\mathsf{Pr})$ equipped with maps\n$$ D_n \\to \\mathsf{End}_{D_{n+1}}(\\mathbb{1}), $$but these usually fail to be equivalences, so $(D_n)_n$ is not yet a Stefanich ring. One must spectrify: universally enforce that these maps become equivalences. The terminology comes from the analogy with spectra in stable homotopy theory, where a Stefanich ring plays the role of an infinite loop space\n$$ X_0 \\simeq \\Omega X_1 \\simeq \\Omega^2 X_2 \\simeq \\cdots $$and one similarly needs to spectrify a sequence of spaces equipped with maps $X_n \\to \\Omega X_{n+1}$ that are not yet equivalences.\nConcretely, one replaces the sequence $(D_n)_n$ by\n$$ D'_n \\coloneqq \\mathsf{End}_{D_{n+1}}(\\mathbb{1}), $$equipped with the induced map $D_n \\to D'_n$, and iterates using transfinite induction (taking filtered colimits at limit ordinal stages). Since the functor $D \\mapsto \\mathsf{End}_D(\\mathbb{1})$ commutes with $\\aleph_1$-filtered colimits, the process stabilises after $\\aleph_1$ steps. By the adjoint functor theorem, this spectrification always exists.\nThree perspectives on pushouts In practice, we are mostly interested in pushouts (i.e.\\ relative tensor products). Consider a pushout diagram $B \\xleftarrow{f} A \\xrightarrow{g} C$ of Stefanich rings. There are three equivalent ways to describe the levelwise data before spectrification.\nPerspective 1 (absolute). Form the tensor product at each level:\n$$ (B_0 \\otimes_{A_0} C_0,\\; B_1 \\otimes_{A_1} C_1,\\; B_2 \\otimes_{A_2} C_2,\\; \\ldots), $$then spectrify.\nPerspective 2 (relative, symmetric). Under the equivalence of Proposition 2 , form the tensor product in $\\mathsf{StRing}_{A/}$:\n$$ \\bigl((B/A)_0 \\otimes_{A_1} (C/A)_0,\\; (B/A)_1 \\otimes_{A_2} (C/A)_1,\\; \\ldots\\bigr), $$then spectrify.\nPerspective 3 (relative, base change). Think of $B \\otimes_A C$ as the image of $B \\in \\mathsf{StRing}_{A/}$ under the base change functor $g^* \\colon \\mathsf{StRing}_{A/} \\to \\mathsf{StRing}_{C/}$:\n$$ \\bigl(g^*_0(B/A)_0,\\; g^*_1(B/A)_1,\\; g^*_2(B/A)_2,\\; \\ldots\\bigr), $$then spectrify.\nThe third perspective is the most useful for analysing morphisms: it expresses base change as a levelwise operation followed by spectrification. A key theme of the following sections is to isolate general classes of maps for which spectrification is unnecessary from some level onwards, making the tensor product computable in practice.\nAffine maps Motivation: a hierarchy of affineness In classical algebraic geometry, an affine morphism $f \\colon Y \\to X$ is one for which $Y$ can be completely reconstructed from its global sections: $Y \\simeq \\operatorname{Spec}(\\Gamma(Y, \\mathcal{O}_Y))$. Translated into the language of our $\\operatorname{Spec}_n \\dashv \\Gamma_n$ adjunction, this states that the counit map\n$$ \\operatorname{Spec}_0(\\Gamma_0(B)) \\to B $$is an equivalence. In other words, the single piece of algebraic data $(B/A)_0 \\in \\mathsf{CAlg}(A_1)$ is entirely sufficient to reconstruct the whole tower $B$. This is the precise formulation of 0-affineness.\nMoving one level up the categorical ladder, Gaitsgory\u0026rsquo;s notion of 1-affineness asks that a prestack $\\mathcal{Y}$ satisfies:\n$$ \\mathsf{ShvCat}(\\mathcal{Y}) \\simeq \\mathsf{QCoh}(\\mathcal{Y})\\text{-}\\mathsf{mod}, $$meaning the category of sheaves of categories on $\\mathcal{Y}$ is fully recovered just from the monoidal DG category $\\mathsf{QCoh}(\\mathcal{Y})$. In our Stefanich framework, $\\mathsf{ShvCat}(\\mathcal{Y})$ corresponds to $1\\mathsf{Pr}_\\mathcal{Y}$, and the right-hand side corresponds to $\\mathsf{Mod}_{D(\\mathcal{Y})}(1\\mathsf{Pr})$. Hence, 1-affineness is exactly the assertion that:\n$$ \\operatorname{Spec}_1(\\Gamma_1(B)) \\xrightarrow{\\sim} B. $$Here, the data $(B/A)_1 \\in \\mathsf{CAlg}(A_2)$ — which is a commutative algebra in $1\\mathsf{Pr}$-categories — is enough to determine all higher levels strictly by iteratively passing to module categories.\nThe overarching pattern is now crystal clear: $n$-affineness dictates that everything from the $n$-th categorical level onwards is rigidly generated by a single, pure piece of commutative algebra data at that level. This rigorously formalizes the mantra that ``everything becomes affine under sufficient categorification\u0026rsquo;\u0026rsquo;. In the context of our tensor product discussion, the $n$-affineness of $B$ over $A$ guarantees that spectrification is entirely unnecessary from level $n$ onwards.\nDefinition and formal properties Definition 5. Let $A$ be a Stefanich ring and $n \\geq 0$. An $A$-algebra $B \\in \\mathsf{StRing}_{A/}$ is called $n$-affine if it lies in the essential image of the fully faithful embedding\n$$ \\operatorname{Spec}_n \\colon \\mathsf{CAlg}(A_{n+1}) \\hookrightarrow \\mathsf{StRing}_{A/}, $$or equivalently, if the counit $\\operatorname{Spec}_n(\\Gamma_n(B)) \\xrightarrow{\\sim} B$ is an equivalence.\nProposition 6. Let $A$ be a Stefanich ring and $n \\geq 0$. The class of $n$-affine $B \\in \\mathsf{StRing}_{A/}$ is stable under all small colimits. Moreover:\nIf $B$ is $n$-affine over $A$, then for all $m \\geq n$ the canonical functor\n$$ \\mathsf{Mod}_{(B/A)_m}(A_{m+1}) \\to B_{m+1} $$is an equivalence. Here, the functor is the composite $\\mathsf{Mod}_{(B/A)_m}(A_{m+1}) \\hookrightarrow \\mathsf{Mod}_{A_{m+1}}(1\\mathsf{Pr}) \\xrightarrow{F_{m+1}} A_{m+2}$, which is fully faithful on ($\\aleph_1$-filtered colimits of) dualizable objects by Lemma 3 .\n(Base change.) If $g \\colon A \\to C$ is any map of Stefanich rings and $B$ is $n$-affine over $A$, then $g^*B = B \\otimes_A C$ is $n$-affine over $C$, and for all $m \\geq n$,\n$$ g^*_m(B/A)_m \\xrightarrow{\\;\\sim\\;} (B \\otimes_A C / C)_m. $$ (Composition.) For composable maps $f \\colon A \\to B$ and $g \\colon B \\to C$ with $f$ being $n$-affine, $g$ is $n$-affine if and only if $g \\circ f$ is $n$-affine.\nIn particular, $n$-affine maps are stable under composition, base change, and passage to diagonals.\nProof. Stability under small colimits is clear, as $\\operatorname{Spec}_n \\colon \\mathsf{CAlg}(A_{n+1}) \\to \\mathsf{StRing}_{A/}$ preserves small colimits.\nPart (1). If $B$ is $n$-affine, then for all $m \\geq n$, $(B/A)_{m+1}$ is by construction the image of $\\mathsf{Mod}_{(B/A)_m}(A_{m+1})$ under $F_{m+1} \\colon \\mathsf{Mod}_{A_{m+1}}(1\\mathsf{Pr}) \\to A_{m+2}$. By Lemma 3 , the functor $F_{m+1}$ is fully faithful on the subcategory containing $\\mathsf{Mod}_{(B/A)_m}(A_{m+1})$ (which is an $\\aleph_1$-filtered colimit of dualizables \u0026mdash; indeed, $\\mathsf{Mod}_R(A_{m+1})$ is dualizable whenever $R \\in \\mathsf{CAlg}(A_{m+1})$ is countably presented). Thus the underlying category is unchanged, giving $B_{m+1} \\simeq \\mathsf{Mod}_{(B/A)_m}(A_{m+1})$.\nPart (2). This follows from the commutative diagram\nwhich is assembled from two squares: (i) the assertion that module categories base change, i.e.\\ the square involving $\\mathsf{CAlg}(A_{n+1}) \\to \\mathsf{CAlg}(\\mathsf{Mod}_{A_{n+1}}(1\\mathsf{Pr}))$ and its $C$-analogue; and (ii) the square involving $\\mathsf{Mod}_{A_{n+1}}(1\\mathsf{Pr}) \\to A_{n+2}$ and its $C$-analogue, whose commutativity follows from the datum of the map $g$ of Stefanich rings.\nPart (3). If $B$ is $n$-affine over $A$, then $\\mathsf{StRing}_{B/}$ maps isomorphically to\n$$ \\mathsf{StRing}_{A/} \\times_{\\mathsf{CAlg}(A_{n+1})} \\mathsf{CAlg}(A_{n+1})_{(B/A)_n/}. $$In other words, a $B$-algebra $C$ is fully determined by the map $A \\to C$ plus a map $(B/A)_n \\to (C/A)_n$ in $\\mathsf{CAlg}(A_{n+1})$. By part (1), the category of $n$-affine $C$ over $B$ is $\\mathsf{CAlg}(B_{n+1})$, while the category of $B$-algebras $C$ that are $n$-affine over $A$ is $\\mathsf{CAlg}(A_{n+1})_{(B/A)_n/}$, and these are equivalent.\n$\\square$ Affineness as a condition on maps Warning. The converse to condition (1) is violently false. Consider the map $f \\colon * \\to B^2\\mathbb{G}_m$ over a field $k$, and let $A \\to B$ be the corresponding map of Stefanich rings. Because $f$ is a torsor for a 1-stack, it is 1-affine, so condition (1) holds perfectly for $m \\geq 1$.\nDangerously, condition (1) also appears to hold for $m = 0$. By standard pulling back, we have $A_1 = B_1 = D(k)$, and the relative global sections give $(B/A)_0 = k$. Consequently, the external equivalence holds: $\\mathsf{Mod}_k(D(k)) \\simeq D(k) = B_1$.\nHowever, $f$ is absolutely not 0-affine! Since $(B/A)_0 = k$ is merely the trivial unit object in $\\mathsf{CAlg}(A_1)$, the space freely generated by it is just the base space itself: $\\operatorname{Spec}_0(k) = A$. But obviously $A \\neq B$ ($*$ is not $B^2\\mathbb{G}_m$).\nRemark. The warning above perfectly exposes the vulnerability of lower-level truncations. It is the exact categorical analogue of the classical fact that $\\Gamma(\\mathbb{P}^1, \\mathcal{O}) = k$ but $\\mathbb{P}^1$ is undeniably not affine. The extraction functor $\\Gamma_0$ acts like a blunt \u0026ldquo;decategorification\u0026rdquo; filter: it simply cannot see the highly twisted, non-trivial 2-stacky geometry of $B^2\\mathbb{G}_m$ that lurks at higher categorical levels.\nCrucially, the equivalence $\\mathsf{Mod}_k(D(k)) \\simeq D(k) = B_1$ at $m = 0$ is genuinely induced by the map $f$ \u0026mdash; there is no mathematical error there. The catch is that checking external equivalence level-by-level is a weak test. The functor $\\Gamma_0$ loses so much hidden structural information that attempting to rebuild the space via the counit $\\operatorname{Spec}_0(\\Gamma_0(B)) \\to B$ completely fails to capture the higher stacky \u0026ldquo;ghosts\u0026rdquo;.\nThis intuitive picture also elegantly situates Gaitsgory\u0026rsquo;s 1-affineness within the broader framework. Gaitsgory\u0026rsquo;s classical condition asks for a single, isolated equivalence $\\mathsf{ShvCat}(\\mathcal{Y}) \\simeq \\mathsf{QCoh}(\\mathcal{Y})\\text{-}\\mathsf{mod}$. This amounts to merely checking that the underlying external $(\\infty,1)$-category of our reconstructed $\\operatorname{Spec}_1(\\Gamma_1(B))$ matches that of the target $B$ at level $m = 1$. Definition 5 is strictly and purposefully stronger: by demanding that the entire generated tower $\\operatorname{Spec}_1(\\Gamma_1(B)) \\to B$ is an equivalence of Stefanich $A$-algebras, it acts as a ruthless filter, guaranteeing that absolutely no higher stacky anomalies can survive in the tower. (Fortunately, the standard examples of 1-affine maps found in the literature do, in fact, satisfy this robust stronger condition.)\nProper map Étale map Reference [ACS] The Algebra of Categorical Spectra. PDF/Record. [Aok25] Ko Aoki. Higher presentable categories and limits. 2025. arXiv:2510.13503. [Lur17] Jacob Lurie. Higher Algebra. 2017. PDF. ","permalink":"https://ou-liu-red-sugar.github.io/notes/notes/gestalten/stefanich-rings/","summary":"\u003ch2 id=\"stefanich-rings\"\u003eStefanich Rings\u003c/h2\u003e\n\u003cp\u003eIn the last note, we define\n\u003c/p\u003e\n\\[\n  n\\mathsf{Pr} \\coloneqq \\mathsf{Mod}_{(n-1)\\mathsf{Pr}}(1\\mathsf{Pr}),\n  \\qquad\n  1\\mathsf{Pr} \\coloneqq \\mathsf{Pr}_{\\aleph_1}^{L}.\n\\]\u003cp\u003e\nNow, we let $0\\mathsf{Pr} \\coloneqq \\mathsf{An}$.\u003c/p\u003e\n\u003cp\u003eSince we have\n\u003c/p\u003e\n\\[\n  \\mathsf{CAlg}(\\mathcal{C}) \\hookrightarrow\n  \\mathsf{CAlg}(\\mathsf{Mod}_{\\mathcal{C}}(\\mathsf{Pr}_{\\aleph_1}^L)),\n  \\quad\n  A \\mapsto \\mathsf{Mod}_A(\\mathcal{C}),\n\\]\u003cp\u003e\nand $1\\mathsf{Pr} \\in \\mathsf{Pr}_{\\aleph_1}^L$, we obtain a sequence\n\u003c/p\u003e\n\\[\n  \\mathsf{CAlg}(0\\mathsf{Pr})\n  \\hookrightarrow\n  \\mathsf{CAlg}(1\\mathsf{Pr})\n  \\hookrightarrow\n  \\cdots\n  \\hookrightarrow\n  \\mathsf{CAlg}(n\\mathsf{Pr})\n  \\hookrightarrow\n  \\cdots .\n\\]\u003cp\u003eIn \u003ca class=\"page-ref\" href=\"/notes/notes/gestalten/presentable-categories/\"\u003ePresentable Categories\u003c/a\u003e,\nwe know that $1\\mathsf{Pr}$ admits all small colimits, which can be computed in\n$\\widehat{\\mathsf{Cat}}$ by passing to adjoint functors.\nThus, we obtain the following definition.\u003c/p\u003e","title":"Stefanich Rings"},{"content":"Six-Functor Formalisms on \\(n\\mathsf{Pr}_{(-)}\\) Let $\\mathsf{CAlg}$ be the category of derived rings, and let $\\mathsf{Aff} \\coloneqq \\mathsf{CAlg}^{\\operatorname{op}}$ be the category of affine schemes.\nQuasicoherent Sheaves of Higher Categories ","permalink":"https://ou-liu-red-sugar.github.io/notes/notes/gestalten/everything-becomes-affine-under-sufficient-categorification/","summary":"\u003ch2 id=\"six-functor-formalisms-on\"\u003eSix-Functor Formalisms on \\(n\\mathsf{Pr}_{(-)}\\)\u003c/h2\u003e\n\u003cp\u003eLet  $\\mathsf{CAlg}$ be the category of derived rings, and let $\\mathsf{Aff} \\coloneqq \\mathsf{CAlg}^{\\operatorname{op}}$ be the category of affine schemes.\u003c/p\u003e\n\u003ch2 id=\"quasicoherent-sheaves-of-higher-categories\"\u003eQuasicoherent Sheaves of Higher Categories\u003c/h2\u003e","title":"Everything Becomes Affine Under Sufficient Categorification"},{"content":"Introduction Compactly assembled categories are the right generality for modern applications where compact generation fails but the category still feels \u0026ldquo;finite-dimensional\u0026rdquo;. The motivating example: $\\mathsf{Shv}(X)$ for $X$ locally compact Hausdorff is not compactly generated, yet behaves like a compactly generated category for nearly every K-theoretic and trace-theoretic purpose.\nThe intuition: an object of a compactly assembled category is built out of compactly exhaustible atoms — sequences whose transition maps are \u0026ldquo;compact\u0026rdquo; in a precise sense. Compactness is now a property of morphisms, not just objects. In the stable world, this generality is forced on us: compactly assembled is precisely dualizable in $\\mathsf{Pr}^L_{\\mathrm{st}}$, which is what we need for K-theory, traces, and the symmetric monoidal structure to behave.\nBody A presentable category $\\mathcal{C}$ is compactly assembled if it is generated under colimits by compactly exhaustible objects — objects of the form $\\operatorname*{colim}_n X_n$ along a sequence $X_0 \\to X_1 \\to \\cdots$ in which every transition map is a compact morphism.\nA compact morphism $f\\colon X \\to Y$ is one for which, for every filtered colimit $Z \\simeq \\operatorname*{colim}_i Z_i$, the natural square\n\\[ \\begin{array}{ccc} \\operatorname*{colim}_i \\operatorname{Hom}(Y, Z_i) \u0026 \\to \u0026 \\operatorname*{colim}_i \\operatorname{Hom}(X, Z_i) \\\\ \\downarrow \u0026 \u0026 \\downarrow \\\\ \\operatorname{Hom}(Y, Z) \u0026 \\to \u0026 \\operatorname{Hom}(X, Z) \\end{array} \\]is a pullback.\nLurie–Clausen characterisation For a presentable $\\mathcal{C}$, the following are equivalent:\n$\\mathcal{C}$ is compactly assembled. There is a regular cardinal $\\kappa$ such that $\\mathcal{C}$ is $\\kappa$-presentable and the colimit functor $k\\colon \\mathsf{Ind}(\\mathcal{C}^{\\kappa}) \\to \\mathcal{C}$ has a left adjoint $\\hat y$. $\\mathcal{C}$ is a retract in $\\mathsf{Pr}^L$ of a [[Compactly generated category]]. Filtered colimits commute with all small limits in $\\mathcal{C}$. In the stable world, $\\mathcal{C} \\in \\mathsf{Pr}^L_{\\mathrm{st}}$ is compactly assembled iff it is a dualizable object of $\\mathsf{Pr}^L_{\\mathrm{st}}$.\nWhy care $\\mathsf{Shv}(X)$ for $X$ locally compact Hausdorff is compactly assembled but not in general compactly generated. Efimov K-theory extends K-theory from compactly generated to compactly assembled categories, recovering \u0026ldquo;classical\u0026rdquo; K-theory of categories like $\\mathsf{Shv}(X, \\mathsf{Sp})$. In $\\mathsf{Pr}^L_{\\mathrm{ca}}$, dualizability gives an analogue of finite-dimensional vector spaces, with internal Hom and a meaningful trace. Relation to other notions Every [[Compactly generated category]] is compactly assembled. Every compactly assembled category is a [[Presentable category]], $\\aleph_1$-presentable in fact. A [[Stable category]] is compactly assembled iff it is dualizable in $\\mathsf{Pr}^L_{\\mathrm{st}}$. Pointers Krause–Nikolaus–Pützstück, Sheaves on Manifolds, §2.3 Ramzi, Dualizable presentable ∞-categories, arXiv:2410.21537 Efimov, K-theory and localizing invariants of large categories, arXiv:2405.12169 See also: [[Presentable category]], [[Compactly generated category]], [[Stable category]].\n","permalink":"https://ou-liu-red-sugar.github.io/wiki/compactly-assembled-category/","summary":"\u003ch2 id=\"introduction\"\u003eIntroduction\u003c/h2\u003e\n\u003cp\u003eCompactly assembled categories are the \u003cstrong\u003eright generality for modern\napplications\u003c/strong\u003e where compact generation fails but the category still feels\n\u0026ldquo;finite-dimensional\u0026rdquo;. The motivating example: $\\mathsf{Shv}(X)$ for $X$\nlocally compact Hausdorff is not compactly generated, yet behaves like a\ncompactly generated category for nearly every K-theoretic and trace-theoretic\npurpose.\u003c/p\u003e\n\u003cp\u003eThe intuition: an object of a compactly assembled category is built out of\n\u003cem\u003ecompactly exhaustible\u003c/em\u003e atoms — sequences whose transition maps are\n\u0026ldquo;compact\u0026rdquo; in a precise sense. Compactness is now a property of \u003cem\u003emorphisms\u003c/em\u003e,\nnot just objects. In the stable world, this generality is forced on us:\ncompactly assembled is precisely \u003cem\u003edualizable\u003c/em\u003e in $\\mathsf{Pr}^L_{\\mathrm{st}}$,\nwhich is what we need for K-theory, traces, and the symmetric monoidal\nstructure to behave.\u003c/p\u003e","title":"Compactly assembled category"},{"content":"Introduction Compactly generated categories are the tame end of the presentable spectrum. Every object is a filtered colimit of compact ones, and most homological invariants — K-theory, Hochschild homology, cyclic homology — are determined by the small subcategory of compact objects. The idea: a \u0026ldquo;big\u0026rdquo; category is recovered, up to filtered colimits, from a small skeleton of finite/perfect/dualizable atoms.\nExamples are everywhere in algebra and topology: spectra (compact = finite spectra), modules over a ring spectrum (compact = perfect), the derived category of a ring (compact = perfect complexes). The fact that all of homotopy theory\u0026rsquo;s \u0026ldquo;large\u0026rdquo; categories of interest are compactly generated is why the theory of K-theory, traces and dimensions is as workable as it is.\nBody A presentable category $\\mathcal{C}$ is compactly generated (or $\\aleph_0$-presentable) if it is generated under filtered colimits by its compact objects $\\mathcal{C}^{\\omega} \\subset \\mathcal{C}$.\nEquivalently:\n\\[ \\mathcal{C} \\;\\simeq\\; \\mathsf{Ind}(\\mathcal{C}^{\\omega}), \\]so a compactly generated $\\mathcal{C}$ is recovered as the Ind-completion of its small subcategory of compact objects. In the classical case, this recovers the notion of a locally finitely presentable category in the sense of Adámek–Rosický.\nExamples include:\nThe derived category $\\mathsf{D}(R)$ of a ring $R$ (compact = perfect complexes). The stable category of spectra $\\mathsf{Sp}$ (compact = finite spectra). $\\mathsf{Mod}_R$ over a ring spectrum (compact = perfect modules). Many naturally occurring categories — sheaves on a locally compact Hausdorff space, for instance — are not compactly generated, but the slightly weaker notion of [[Compactly assembled category]] still applies.\nRelation to other notions A compactly generated category is automatically a [[Presentable category]] (with $\\kappa = \\aleph_0$). A [[Compactly assembled category]] is, by Lurie–Clausen, a retract in $\\mathsf{Pr}^L$ of a compactly generated one. In the stable setting, a [[Stable category]] that is compactly generated is in particular dualizable in $\\mathsf{Pr}^L_{\\mathrm{st}}$, with dual $\\mathsf{Ind}((\\mathcal{C}^{\\omega})^{\\mathrm{op}})$. Pointers Lurie, Higher Topos Theory, §5.5.7 Krause–Nikolaus–Pützstück, Sheaves on Manifolds, §2 See also: [[Presentable category]], [[Compactly assembled category]], [[Stable category]].\n","permalink":"https://ou-liu-red-sugar.github.io/wiki/compactly-generated-category/","summary":"\u003ch2 id=\"introduction\"\u003eIntroduction\u003c/h2\u003e\n\u003cp\u003eCompactly generated categories are the \u003cstrong\u003etame end of the presentable\nspectrum\u003c/strong\u003e. Every object is a filtered colimit of compact ones, and most\nhomological invariants — K-theory, Hochschild homology, cyclic homology —\nare determined by the small subcategory of compact objects. The idea: a\n\u0026ldquo;big\u0026rdquo; category is recovered, up to filtered colimits, from a small skeleton\nof finite/perfect/dualizable atoms.\u003c/p\u003e\n\u003cp\u003eExamples are everywhere in algebra and topology: spectra (compact = finite\nspectra), modules over a ring spectrum (compact = perfect), the derived\ncategory of a ring (compact = perfect complexes). The fact that all of\nhomotopy theory\u0026rsquo;s \u0026ldquo;large\u0026rdquo; categories of interest are compactly generated is\nwhy the theory of K-theory, traces and dimensions is as workable as it is.\u003c/p\u003e","title":"Compactly generated category"},{"content":"Introduction This wiki is written in the modern, homotopy-coherent dialect. The intention is to keep statements short by not repeating the qualifier \u0026ldquo;(∞,1)\u0026rdquo; on every other word. The terminology below is fixed once for the whole site, and individual entries should not redefine it.\nIf you arrive from a 1-categorical background, just read every \u0026ldquo;category\u0026rdquo; as \u0026ldquo;(∞,1)-category\u0026rdquo; and add the strictness adjective when you want it.\nBody Categorical conventions category = 1-category = (∞,1)-category. All categories on this wiki are $(\\infty,1)$-categories by default. Every notion (limits, colimits, adjunctions, presheaves, …) is the $(\\infty,1)$-categorical one.\n$n$-category = $(\\infty, n)$-category. The integer counts the depth of non-invertible morphisms; everything above level $n$ is invertible. So a $2$-category on this wiki means an $(\\infty,2)$-category, not a strict 2-category.\nclassical category is the explicit name for the 1-categorical ($\\mathsf{Set}$-enriched) notion. Whenever the 1-categorical sense is needed — to contrast with the homotopy-coherent one, or to discuss classical theorems — the qualifier classical is mandatory.\nanima denotes the $(\\infty,1)$-category of $\\infty$-groupoids (a.k.a. spaces, homotopy types, $\\mathsf{An}$). Reserved word.\nA consequence of these conventions: phrases like \u0026ldquo;the category of presentable ∞-categories\u0026rdquo; become \u0026ldquo;the category of presentable categories\u0026rdquo;, and \u0026ldquo;a 1-category in the classical sense\u0026rdquo; becomes \u0026ldquo;a classical category\u0026rdquo;.\nSuggested entry structure Every entry should have three blocks. The first two are written by hand; the third is rendered automatically by the layout.\nIntroduction — motivation and intuition. What is the entry about, and why does it deserve a name? What does it generalise or specialise? What goes wrong without it? Body — formal definitions, characterisations, the main statements. Subheadings encouraged when there is more than one route in. Neighborhood — auto-rendered. Lists outgoing $[[\\text{wikilinks}]]$, incoming references (backlinks), and a 1-hop force-directed graph centred on the current entry. Entries that need a long bibliography should add a Pointers section just before the auto-rendered Neighborhood.\nLinking Use double-bracket wikilinks for cross-references:\nA [[Compactly generated category]] is [[Presentable category]] whose generators are $\\aleph_0$-compact. The bracketed text is matched against entry titles and aliases: in the frontmatter, case-insensitive. Unresolved links render as a greyed dashed underline so they show up while drafting.\n","permalink":"https://ou-liu-red-sugar.github.io/wiki/conventions/","summary":"\u003ch2 id=\"introduction\"\u003eIntroduction\u003c/h2\u003e\n\u003cp\u003eThis wiki is written in the modern, homotopy-coherent dialect. The intention\nis to keep statements short by \u003cem\u003enot\u003c/em\u003e repeating the qualifier \u0026ldquo;(∞,1)\u0026rdquo; on\nevery other word. The terminology below is fixed once for the whole site,\nand individual entries should not redefine it.\u003c/p\u003e\n\u003cp\u003eIf you arrive from a 1-categorical background, just read every \u0026ldquo;category\u0026rdquo;\nas \u0026ldquo;(∞,1)-category\u0026rdquo; and add the strictness adjective when you want it.\u003c/p\u003e\n\u003ch2 id=\"body\"\u003eBody\u003c/h2\u003e\n\u003ch3 id=\"categorical-conventions\"\u003eCategorical conventions\u003c/h3\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003ecategory\u003c/strong\u003e = \u003cstrong\u003e1-category\u003c/strong\u003e = \u003cstrong\u003e(∞,1)-category\u003c/strong\u003e.\nAll categories on this wiki are $(\\infty,1)$-categories by default. Every\nnotion (limits, colimits, adjunctions, presheaves, …) is the\n$(\\infty,1)$-categorical one.\u003c/p\u003e","title":"Conventions"},{"content":"Introduction Presentability is the right size condition for category theory: it is large enough to contain \u0026ldquo;all the categories that come up in nature\u0026rdquo; — modules over a ring spectrum, sheaves on a site, parametrised spectra, anima, etc. — yet small enough that the adjoint functor theorem applies and that $\\mathrm{Hom}$-functors are controlled by an essentially small skeleton.\nThe intuition is that a presentable category is generated by a small piece (its $\\kappa$-compact objects) under filtered colimits. Everything else is recovered as a limit of these atoms. This is what makes the category \u0026ldquo;tame\u0026rdquo;: universal constructions exist, adjoint functors exist as soon as one side preserves the right kind of (co)limit, and almost all the categories of homotopy theory live here.\nBody A category $\\mathcal{C}$ is presentable iff:\n$\\mathcal{C}$ admits all small colimits. There is a regular cardinal $\\kappa$ and an essentially small set of $\\kappa$-compact objects that generate $\\mathcal{C}$ under $\\kappa$-filtered colimits. Equivalently (Lurie / Adámek–Rosický in the classical case), $\\mathcal{C}$ is presentable iff it is an accessible localization of a presheaf category $\\mathsf{Fun}(\\mathcal{D}^{\\mathrm{op}}, \\mathsf{An})$ for some small $\\mathcal{D}$.\nThe category of presentable categories with colimit-preserving functors is $\\mathsf{Pr}^L$. It carries the Lurie tensor product, and is itself complete and cocomplete.\nTwo important refinements live inside $\\mathsf{Pr}^L$:\nA [[Compactly generated category]] is a presentable category whose $\\kappa$-compact generators can be taken with $\\kappa = \\aleph_0$. A [[Compactly assembled category]] is a presentable category that is a retract in $\\mathsf{Pr}^L$ of a compactly generated one — equivalently, a dualizable object of $\\mathsf{Pr}^L_{\\mathrm{st}}$ in the stable case. For the 1-categorical (i.e. classical) version of these notions, see Adámek–Rosický.\nPointers Lurie, Higher Topos Theory, §5.5 Adámek–Rosický, Locally Presentable and Accessible Categories See also: [[Compactly generated category]], [[Compactly assembled category]], [[Stable category]].\n","permalink":"https://ou-liu-red-sugar.github.io/wiki/presentable-category/","summary":"\u003ch2 id=\"introduction\"\u003eIntroduction\u003c/h2\u003e\n\u003cp\u003ePresentability is the right size condition for category theory: it is large\nenough to contain \u0026ldquo;all the categories that come up in nature\u0026rdquo; — modules over\na ring spectrum, sheaves on a site, parametrised spectra, anima, etc. — yet\nsmall enough that the \u003cstrong\u003eadjoint functor theorem\u003c/strong\u003e applies and that\n$\\mathrm{Hom}$-functors are controlled by an essentially small skeleton.\u003c/p\u003e\n\u003cp\u003eThe intuition is that a presentable category is \u003cem\u003egenerated\u003c/em\u003e by a small piece\n(its $\\kappa$-compact objects) under filtered colimits. Everything else is\nrecovered as a limit of these atoms. This is what makes the category \u0026ldquo;tame\u0026rdquo;:\nuniversal constructions exist, adjoint functors exist as soon as one side\npreserves the right kind of (co)limit, and almost all the categories of\nhomotopy theory live here.\u003c/p\u003e","title":"Presentable category"},{"content":"Introduction The point of stability is that fibre and cofibre sequences become the same data. In a stable category one can build a long exact sequence in either direction and the choice is forced — there is no \u0026ldquo;missing\u0026rdquo; octahedral axiom to postulate, no shift functor whose square has to be remembered. Cones are functorial; mapping fibres glue. The triangulated structure on the homotopy category is a theorem, not a definition.\nThe motivating example is spectra. Spectra are the natural home for stable phenomena precisely because the suspension equivalence $\\Sigma$ is built in. Stable categories generalise this: any pointed category in which finite pushouts and pullbacks agree behaves like a category of spectra.\nA consequence worth absorbing: every stable category is automatically enriched in spectra, and morphism objects $\\operatorname{map}(X, Y)$ are spectra rather than anima. The triangulated homotopy category is a shadow.\nBody A stable category is a category $\\mathcal{C}$ such that:\n$\\mathcal{C}$ has a zero object, $\\mathcal{C}$ admits all finite limits and finite colimits, and a square in $\\mathcal{C}$ is a pullback iff it is a pushout. Equivalently, $\\mathcal{C}$ is pointed, has all pushouts, and the suspension functor $\\Sigma\\colon \\mathcal{C} \\to \\mathcal{C}$ is an equivalence.\nExamples The category of spectra $\\mathsf{Sp}$. $\\mathsf{Mod}_R$ for $R$ a ring spectrum. The derived category $\\mathsf{D}(\\mathcal{A})$ of an abelian category, viewed as a stable category. Quasi-coherent sheaves $\\mathsf{QCoh}(X)$ on a scheme. Inside $\\mathsf{Pr}^L$ A [[Presentable category]] that is also stable is called a presentable stable category; the resulting full subcategory $\\mathsf{Pr}^L_{\\mathrm{st}} \\subset \\mathsf{Pr}^L$ is itself presentable and inherits a symmetric monoidal structure (the Lurie tensor product). $\\mathsf{Pr}^L_{\\mathrm{st}}$ is canonically equivalent to $\\mathsf{Mod}_{\\mathsf{Sp}}(\\mathsf{Pr}^L)$.\nWithin $\\mathsf{Pr}^L_{\\mathrm{st}}$:\nA [[Compactly generated category]] that is stable corresponds, via Ind-completion, to a small idempotent-complete stable category. A [[Compactly assembled category]] that is stable is exactly a dualizable object of $\\mathsf{Pr}^L_{\\mathrm{st}}$. Stable vs triangulated The homotopy category $h\\mathcal{C}$ of a stable category is canonically triangulated, but $\\mathcal{C}$ carries strictly more data: cones are functorial in the diagram (not just up to non-unique isomorphism), the octahedral axiom is automatic, and homotopy-coherent diagrams of fibre sequences glue. A classical (i.e. 1-categorical) triangulated category is a shadow of a stable one; it is not, in general, possible to recover the stable category from its homotopy category alone.\nPointers Lurie, Higher Algebra, §1.1 Lurie, Higher Topos Theory, §5.5 See also: [[Presentable category]], [[Compactly generated category]], [[Compactly assembled category]].\n","permalink":"https://ou-liu-red-sugar.github.io/wiki/stable-category/","summary":"\u003ch2 id=\"introduction\"\u003eIntroduction\u003c/h2\u003e\n\u003cp\u003eThe point of stability is that \u003cstrong\u003efibre and cofibre sequences become the same\ndata\u003c/strong\u003e. In a stable category one can build a long exact sequence in either\ndirection and the choice is forced — there is no \u0026ldquo;missing\u0026rdquo; octahedral axiom\nto postulate, no shift functor whose square has to be remembered. Cones are\nfunctorial; mapping fibres glue. The triangulated structure on the homotopy\ncategory is a \u003cem\u003etheorem\u003c/em\u003e, not a definition.\u003c/p\u003e","title":"Stable category"},{"content":" Conventions.\nCategory means $(\\infty,1)$-category; $\\mathsf{Cat}$ is the category of all categories. $2$-category means $(\\infty,2)$-category; $\\mathsf{Cat}_2$ is the $2$-category of all categories. $\\mathsf{Pr}^L$ denotes presentable categories with left adjoints; $\\mathsf{Pr}^L_{\\mathrm{st}}$ is the stable version. $\\mathsf{CAlg}(\\mathcal M)$ denotes commutative algebra objects in a symmetric monoidal category $\\mathcal M$. Given a geometric setup $(\\mathcal C, E)$ — a category $\\mathcal C$ with finite limits and a class of morphisms $E$ closed under base change, composition and diagonals — we write $\\mathsf{Corr}(\\mathcal C, E)$ for the span $(\\infty, 2)$-category and $\\mathsf{Span}_2(\\mathcal C, E)_{I, P}$ for the CLL upgrade with biadjointability data. $\\underline{\\mathrm{Hom}}$ is internal hom; $\\mathbf 1_{X}$ is the monoidal unit of $\\mathsf D(X)$; $\\mathbb 1$ is the terminal object of a slice $\\mathcal C_{/Y}$. Part I · Six-functor formalisms 1. From cohomology to six functors A cohomology theory associates with every \u0026ldquo;space\u0026rdquo; $X$ a complex $\\Gamma(X; \\Lambda) \\in \\mathsf D(\\Lambda)$ of $\\Lambda$-modules — singular cohomology, étale cohomology, coherent cohomology, and many more. We then expect $\\Gamma(X; \\Lambda)$ to satisfy structural identities like the Künneth formula, Poincaré duality, proper base change, excision. Each is classically proved by hand, with most of the work intertwined with whatever specific sheaf-theoretic gadget produced $\\Gamma(X; \\Lambda)$.\nThe six-functor formalism flips the order: we fix a much richer object — a category $\\mathsf D(X)$ of \u0026ldquo;sheaves on $X$\u0026rdquo; with six interrelated functors — and demand a small list of compatibilities. Cohomology becomes a derived quantity, and the structural identities fall out as formal consequences.\nThe data Fix an $(\\infty,1)$-category $\\mathcal C$ of geometric objects with finite limits and terminal object $\\ast$. A six-functor formalism $\\mathsf D$ on $\\mathcal C$ assigns:\nto each $X \\in \\mathcal C$, a closed symmetric monoidal category $\\mathsf D(X)$ — sheaves on $X$ — with $\\otimes$, $\\underline{\\mathrm{Hom}}$, and unit $\\mathbf 1_{X}$; to each map $f\\colon Y \\to X$ in $\\mathcal C$, an adjunction $f^{\\ast} \\dashv f_{\\ast}$; to each map $f\\colon Y \\to X$ in a distinguished class $E$ (the exceptional morphisms — typically maps admitting a compactification), an adjunction $f_{!} \\dashv f^{!}$. For $\\mathsf D \\coloneqq \\mathsf D(\\ast)$ — usually $\\mathsf D(\\Lambda)$ — and $X$ with structure map $p\\colon X \\to \\ast$, four (co)homology theories are immediate:\nCohomology $\\Gamma(X; \\mathbf 1) \\coloneqq p_{\\ast} \\mathbf 1_{X}$; Compactly-supported cohomology $\\Gamma_{c}(X; \\mathbf 1) \\coloneqq p_{!} \\mathbf 1_{X}$; Borel–Moore homology $\\Gamma^{\\mathrm{BM}}(X; \\mathbf 1) \\coloneqq p_{\\ast} p^{!} \\mathbf 1$; Homology $\\Gamma^{\\vee}(X; \\mathbf 1) \\coloneqq p_{!} p^{!} \\mathbf 1$. The compatibilities For the formalism to do work, the functors are coupled by three axioms:\nPullback is monoidal. $f^{\\ast}$ is symmetric monoidal — in particular $f^{\\ast}$ commutes with $\\otimes$. Proper base change. For a cartesian square with $f \\in E$, the natural map $g^{\\ast} f_{!} \\xrightarrow{\\sim} f'_{!} g'^{\\ast}$ is an equivalence. Projection formula. For $f \\in E$, $M \\otimes f_{!} N \\xrightarrow{\\sim} f_{!}(f^{\\ast} M \\otimes N)$ is an equivalence. With just these, the classical theorems are one-line consequences.\nProposition 1 (Künneth formula). For $X, Y \\in \\mathcal C$ with exceptional structure maps,\n$$ \\Gamma_{c}(X \\times Y;\\, \\mathbf 1) \\;\\simeq\\; \\Gamma_{c}(X;\\, \\mathbf 1) \\;\\otimes\\; \\Gamma_{c}(Y;\\, \\mathbf 1). $$ Proof. Apply base change and projection to $X \\xleftarrow{p_{X}} X \\times Y \\xrightarrow{p_{Y}} Y$ over $\\ast$:\n$$ \\begin{aligned} \\Gamma_{c}(X \\times Y;\\, \\mathbf 1) \u0026\\;\\simeq\\; (p_{Y})_{!}\\bigl( (p_{Y})^{\\ast} (p_{X})_{!} \\mathbf 1 \\bigr) \\quad \\text{(base change)} \\\\ \u0026\\;\\simeq\\; (p_{X})_{!}\\, \\mathbf 1 \\;\\otimes\\; (p_{Y})_{!}\\, \\mathbf 1 \\quad \\text{(projection)}. \\end{aligned} $$ $\\square$ Proposition 2 (Poincaré duality). For $X \\in \\mathcal C$ \u0026ldquo;smooth\u0026rdquo; (precise notion in §6) with $\\omega_{X} \\coloneqq p^{!} \\mathbf 1$,\n$$ \\Gamma(X;\\, \\omega_{X}) \\;\\simeq\\; \\underline{\\mathrm{Hom}}\\bigl( \\Gamma_{c}(X;\\, \\mathbf 1),\\, \\mathbf 1 \\bigr). $$ Proof. For any $M \\in \\mathsf D$, two adjunctions plus projection give\n$$ \\mathrm{Hom}(M, p_{\\ast} p^{!} \\mathbf 1) \\simeq \\mathrm{Hom}(p_{!} p^{\\ast} M, \\mathbf 1) \\simeq \\mathrm{Hom}(M \\otimes \\Gamma_{c}(X; \\mathbf 1), \\mathbf 1). $$Yoneda. For an oriented $n$-manifold $\\omega_{X} \\simeq \\mathbf 1[n]$, and this becomes the classical $H^{i}(X; \\Lambda) \\simeq H_{c}^{n-i}(X; \\Lambda)^{\\vee}$.\n$\\square$ Two short proofs, no bespoke analysis. The work has been shifted out of the cohomology theory and into the axioms — and that is what we now want to package efficiently.\n2. The span picture Three compatibilities is a small list, but each drags an $\\infty$-dimensional web of higher coherences (associativity of $\\otimes$, naturality of base change in two variables, projection formula compatible with the symmetric monoidal structure, …). Writing this down by hand is hopeless. The trick — initiated by Liu–Zheng [liu-zheng] and brought to its modern form by Mann [mann-rigid] and Scholze [scholze-six] — packages everything as a single lax symmetric monoidal functor out of a span category.\nFix a geometric setup $(\\mathcal C, E)$. The span category $\\mathsf{Corr}(\\mathcal C, E)$ has:\nobjects: those of $\\mathcal C$; morphisms $X \\to Y$: spans $X \\xleftarrow{} W \\xrightarrow{g} Y$ with $g \\in E$; composition: pullback of spans. When $\\mathcal C$ has finite products, $\\mathsf{Corr}(\\mathcal C, E)$ is symmetric monoidal under $\\times$.\nDefinition 3 (Three- and six-functor formalisms). A three-functor formalism on $(\\mathcal C, E)$ is a lax symmetric monoidal functor\n$$ \\mathsf D\\colon \\mathsf{Corr}(\\mathcal C, E) \\longrightarrow \\mathsf{Cat}. $$It is a six-functor formalism if $-\\otimes A$, $f^{\\ast}$ and $f_{!}$ admit right adjoints (i.e.\\ $\\underline{\\mathrm{Hom}}$, $f_{\\ast}$, $f^{!}$ exist). The right adjoints are a property, not extra data: adjoints come with all coherences automatically. ([scholze-six, Lecture II] ; [heyer-mann] .)\nWhat a 3-functor formalism encodes:\nPullback. $g\\colon Y \\to X$ gives the span $X \\xleftarrow{g} Y \\xrightarrow{\\mathrm{id}} Y$ → $g^{\\ast}$. Exceptional pushforward. $f\\colon Y \\to X$ in $E$ gives $Y \\xleftarrow{\\mathrm{id}} Y \\xrightarrow{f} X$ → $f_{!}$. Tensor product. Lax symmetric monoidality gives $\\mathsf D(X) \\times \\mathsf D(X) \\to \\mathsf D(X \\times X)$; precomposing with $\\Delta_{X}^{\\ast}$ gives $\\otimes$. General span. $X \\xleftarrow{g} Z \\xrightarrow{f} Y$ goes to $f_{!} g^{\\ast}$. Most of the magic: the two non-trivial compatibilities of §1 are consequences of the single statement \u0026ldquo;$\\mathsf D$ is a lax symmetric monoidal functor\u0026rdquo;:\nproper base change = functoriality on composable spans; the projection formula = the lax-monoidal coherence on $X \\times X$. So three axioms plus one adjoint-existence property = six functors with all their coherences.\n3. Construction via CLL universality Producing a lax symmetric monoidal functor out of $\\mathsf{Corr}(\\mathcal C, E)$ from scratch is just as hopeless as spelling out the hexagonal compatibility web by hand. The standard idea: split exceptional morphisms into two simpler classes and build the formalism on each.\nA suitable decomposition of $E$ is a pair $I, P \\subseteq E$ (\u0026ldquo;open immersions\u0026rdquo; and \u0026ldquo;proper morphisms\u0026rdquo;) such that every $f \\in E$ factors as $f = p \\circ i$ with $i \\in I$, $p \\in P$, subject to mild cancellation/truncation axioms. Geometrically: $i_{!}$ is left adjoint to $i^{\\ast}$ (extension by zero); $p_{!}$ is right adjoint to $p^{\\ast}$ (direct image of a proper map); $f_{!} = p_{!} i_{!}$. The input data we want to carry is captured by two base-change conditions.\nDefinition 4 (Left and right base change). Let $\\mathsf D_{0}\\colon \\mathcal C^{\\mathrm{op}} \\to \\mathsf{CAlg}(\\mathsf{Cat})$ encode pullbacks and tensor.\n$g\\colon C \\to D$ satisfies left base change if, for every cartesian square\n$k^{\\ast}$ has a left adjoint $k_{\\sharp}$ and the canonical $k_{\\sharp} h^{\\ast} \\xrightarrow{\\sim} f^{\\ast} g_{\\sharp}$ is an equivalence.\n$f$ satisfies right base change if dually $h^{\\ast}$ has a right adjoint $h_{\\ast}$ and $g^{\\ast} f_{\\ast} \\xrightarrow{\\sim} h_{\\ast} k^{\\ast}$ is an equivalence.\nEither condition packages the corresponding adjoint plus its compatibility with base change as a single property.\nThe cleanest path now goes via the universal property of Cnossen–Lenz–Linskens [cll-universal] : enhance $\\mathsf{Corr}(\\mathcal C, E)$ to a $2$-category whose $2$-morphisms encode the adjunction data of $i_{\\sharp}$ and $p_{\\ast}$ coherently, and produce a universal $(I, P)$-biadjointable functor.\nConstruction 5 (The CLL 2-categorical span category). $\\mathsf{Span}_2(\\mathcal C, E)_{I, P}$ has:\nObjects. Objects of $\\mathcal C$.\n$1$-morphisms $X \\to Z$: spans\nwith $g \\in E$.\n$2$-morphisms between two such spans: diagrams\nwith $a \\in P$ and $b \\in I$.\nTheorem 6 (CLL universality). Assume $I, P \\subseteq E$ are wide and closed under base change, every $e \\in E$ factors as $p \\circ i$ with $p \\in P$ and $i \\in I$, and maps in $I \\cap P$ are truncated. Then\n$$ \\mathcal C^{\\mathrm{op}} \\hookrightarrow \\mathsf{Span}_2(\\mathcal C, E)_{I, P} $$is the universal $(I, P)$-biadjointable functor — i.e.\\ the initial functor under which morphisms in $I$ satisfy left base change and morphisms in $P$ satisfy right base change. ([cll-universal, Thm. 5.17] .)\nSo a six-functor formalism on $(\\mathcal C, E)$ is, by definition, a lax symmetric monoidal functor out of $\\mathsf{Span}_{2}(\\mathcal C, E)_{I, P}$; constructing one reduces to verifying left base change for $I$, right base change for $P$, and the mixed Beck–Chevalley equivalence on cartesian squares with one $I$-edge and one $P$-edge. Everything else is formal.\nCorollary 7 (Mann\u0026#39;s extension theorem). Suppose $\\mathsf D_{0}\\colon \\mathcal C^{\\mathrm{op}} \\to \\mathsf{CAlg}(\\mathsf{Cat})$ satisfies left base change for every $i \\in I$, right base change for every $p \\in P$, and the mixed Beck–Chevalley condition: for every cartesian square with $i \\in I$, $p \\in P$, the natural map $i_{\\sharp} p'_{\\ast} \\xrightarrow{\\sim} p_{\\ast} i'_{\\sharp}$ is an equivalence. Then $\\mathsf D_{0}$ extends uniquely to a 3-functor formalism with $f_{!} = p_{\\ast} i_{\\sharp}$. ([mann-rigid, Prop. A.5.10] ; [liu-zheng] .) Heyer–Mann observed that mixed Beck–Chevalley is often automatic:\nProposition 8 (Auto-Beck–Chevalley for monomorphic open immersions). If all $i \\in I$ are monomorphisms, the mixed Beck–Chevalley condition holds automatically. ([heyer-mann] .) The slogan: in any setup where \u0026ldquo;open immersion\u0026rdquo; really means open immersion (not some derived enlargement of one), the construction reduces to two independent base-change conditions, no mixing required.\nPart II · From localization to $\\mathsf{SH}$ We now make the CLL/Mann input concrete in the motivic setting: localization plus base change, two further geometric properties (excision + $\\mathbb A^{1}$-invariance), produces $\\mathsf{SH}$ — the initial six-functor formalism with these properties.\n4. Localization, base change, and cdh descent The CLL construction asks for left base change on $I$, right base change on $P$, and mixed Beck–Chevalley. To make these concrete in the motivic setting we need one more axiom — localization (recollement) — which together with base change forces cdh descent.\nDefinition 9 (Recollement and localization). A diagram\n$$ \\mathcal C_U \\xleftarrow{\\;j^{\\ast}\\;} \\mathcal C_X \\xrightarrow{\\;i^{\\ast}\\;} \\mathcal C_Z $$in $\\mathsf{Pr}^L_{\\mathrm{st}}$ is a recollement if $j^{\\ast}$ has a left adjoint $j_{!}$ with $j_{!} j^{\\ast} \\to \\mathrm{id}$ an equivalence, $i^{\\ast}$ has a right adjoint $i_{\\ast}$ with $\\mathrm{id} \\to i_{\\ast} i^{\\ast}$ an equivalence, $j^{\\ast} i_{\\ast} \\simeq 0$, and the square\nis a pushout in $\\mathrm{End}(\\mathcal C_X)$.\n$\\mathsf D\\colon (\\mathsf{Sch}^{\\mathrm{qcqs}})^{\\mathrm{op}} \\to \\mathsf{Pr}^L_{\\mathrm{st}}$ satisfies the localization axiom if every closed immersion $i\\colon Z \\hookrightarrow X$ with open complement $j\\colon U \\hookrightarrow X$ gives a recollement of $\\mathsf D(U), \\mathsf D(X), \\mathsf D(Z)$.\nLocalization plus base change feeds directly into descent.\nProposition 10 (Localization implies cdh descent). Let $\\mathsf D$ satisfy the localization axiom.\nIf every étale morphism satisfies left base change, then $\\mathsf D$ satisfies Nisnevich descent. If every finitely presented proper morphism satisfies right base change, then $\\mathsf D$ satisfies abstract blowup descent. Under both, $\\mathsf D$ is a cdh sheaf.\nProof. We do abstract blowup; Nisnevich is dual. For\nwrite $j\\colon U = X \\setminus Z \\hookrightarrow X$ and $h\\colon U \\hookrightarrow \\tilde X$. We must show\n$$ (p^{\\ast}, i^{\\ast})\\colon \\mathsf D(X) \\to \\mathsf D(\\tilde X) \\times_{\\mathsf D(E)} \\mathsf D(Z) $$is an equivalence. Right adjoint: $G(A, B) = p_{\\ast} A \\times_{(iq)_{\\ast} C} i_{\\ast} B$.\nConservativity. Localization says $(i^{\\ast}, j^{\\ast})$ is jointly conservative; $j^{\\ast} \\simeq h^{\\ast} p^{\\ast}$ since $p$ is iso over $U$, so $(p^{\\ast}, i^{\\ast})$ is too.\n$i^{\\ast}$-component. Right base change for $p$ along $i$ gives $i^{\\ast} p_{\\ast} A \\simeq q_{\\ast} C$, and full faithfulness of $i_{\\ast}$ collapses the pullback to $B$.\n$p^{\\ast}$-component. Apply $k^{\\ast}$ and $h^{\\ast}$ separately: $k^{\\ast}$ is the previous case, $h^{\\ast}$ kills the $Z$-supported terms and is left with $j^{\\ast} p_{\\ast} A$, which by right base change for $i$ along $p$ and the localization triangle equals $h^{\\ast} A$.\n$\\square$ 5. SH and Drew–Gallauer initiality Theorem 11 (Six-functor formalism on SH). $\\mathsf{SH}\\colon (\\mathsf{Sch}^{\\mathrm{qcqs}})^{\\mathrm{op}} \\to \\mathsf{Pr}^L_{\\mathrm{st}}$ satisfies the localization axiom ([morel-voevodsky] ). The data of $\\mathsf{SH}$ together with smooth $i_{\\sharp}$ and proper $p_{\\ast}$ extends essentially uniquely to a lax symmetric monoidal\n$$ \\mathsf{SH}\\colon \\mathsf{Span}_{2}(\\mathsf{Sch}^{\\mathrm{qcqs}}, E)_{I, P}^{\\otimes} \\longrightarrow \\mathsf{CAlg}(\\mathsf{Pr}^L_{\\mathrm{st}}) $$with $E$ = locally finitely presented, $I$ = open immersions, $P$ = proper morphisms, and $f_{!} = p_{\\ast} i_{\\sharp}$ for $f = p \\circ i$. ([ayoub] ; [cisinski-deglise] ; [cll-universal] .)\nThe proof has a formal half (invoke Theorem 6 ) and a geometric half (smooth/étale base change for $i_{\\sharp}$, localization, proper base change for $p_{\\ast}$ — Ayoub for projective, Cisinski–Déglise for arbitrary proper via Chow\u0026rsquo;s lemma). The mixed Beck–Chevalley is not additional input: it follows from proper base change across a square with one étale edge.\nCombining Proposition 10 with proper base change: $\\mathsf{SH}$ is a cdh sheaf.\nIn what sense is $\\mathsf{SH}$ universal?\nTheorem 12 (Drew–Gallauer initiality). Over a noetherian base $k$ of finite Krull dimension, with $\\mathcal C$ = separated finite-type $k$-schemes, $I$ open immersions, $P$ proper morphisms: the initial presentable six-functor formalism\n$$ \\mathsf D\\colon \\mathsf{Corr}(\\mathcal C) \\to \\mathsf{Pr}^{L}_{\\mathrm{st}} $$with open immersions cohomologically étale, proper maps cohomologically proper, every smooth morphism cohomologically smooth, satisfying excision ($j_{!} \\mathbf 1 \\to \\mathbf 1 \\to i_{\\ast} \\mathbf 1$ a fibre sequence for $i \\hookrightarrow X$ closed) and $\\mathbb A^{1}$-invariance ($\\pi^{\\ast}\\colon \\mathsf D(X) \\to \\mathsf D(\\mathbb A^{1}_{X})$ fully faithful) is $\\mathsf{SH}$. ([drew-gallauer] ; cf.\\ [scholze-six, Lecture XI] .)\nThe proof is constructive and recovers Morel–Voevodsky: smooth $f_{\\sharp}$ forces $\\mathsf{PShv}(\\mathsf{Sm}_{X}; \\mathsf{Sp})$; excision forces Nisnevich localization; $\\mathbb A^{1}$-invariance forces $\\mathbb A^{1}$-localization; $\\otimes$-inverting the Tate object yields $\\mathsf{SH}$.\nPart III · Cohomological smoothness and the 2-category of kernels We now zoom into a fixed six-functor formalism $\\mathsf D$ and ask which morphisms satisfy Poincaré duality. This is the question that motivates the $2$-category of kernels.\n6. Cohomological smoothness Classical Poincaré duality says: for a proper manifold bundle $f\\colon X \\to Y$ of relative dimension $d$, $f_{!}$ has a right adjoint $f^{!}$ of the form $f^{!} \\simeq f^{\\ast}(-) \\otimes \\omega_{f}$, where $\\omega_{f} = f^{!}(\\mathbf 1_{Y})$ is the dualizing complex of $f$, locally isomorphic to $\\mathbf 1[d]$.\nFor an abstract $\\mathsf D$, this motivates axiomatising the class of morphisms with this behaviour:\nDefinition 13 (Cohomologically smooth morphism). A morphism $f\\colon X \\to Y$ in $E$ is $\\mathsf D$-cohomologically smooth (or $\\mathsf D$-smooth) if:\n$f^{!}$ exists, and $f^{!}(\\mathbf 1_{Y}) \\otimes f^{\\ast}(-) \\to f^{!}(-)$ is an equivalence of functors $\\mathsf D(Y) \\to \\mathsf D(X)$. $\\omega_{f} \\coloneqq f^{!}(\\mathbf 1_{Y})$ is $\\otimes$-invertible. (1) and (2) hold for every base change $f'$ of $f$, and the natural map $g'^{\\ast}\\omega_{f} \\to \\omega_{f'}$ is an equivalence. ([scholze-six, Lecture V] .)\nChecking (1)–(3) looks impossibly hard: each piece involves base changes, and $f^{!}$ is abstractly an adjoint. The point of the next two sections is Scholze\u0026rsquo;s organising observation: all three conditions together follow for free from a single, minimal piece of $2$-categorical data on $X \\times_Y X$ — once we have built the right $2$-category.\n7. Integral transforms and the 2-category of kernels The classical integral transform $T_{K}(F)(x) = \\int_{Y} K(x, y)\\, F(y)\\, dy$ encodes a function $L^{2}(Y) \\to L^{2}(X)$ as a function on $X \\times Y$. Its categorification — the Fourier–Mukai transform — replaces integration by $\\pi_{!}$ and tensor: for $K \\in \\mathsf D(X \\times_{Y} X')$,\n$$ \\Phi_{K}\\colon \\mathsf D(X) \\to \\mathsf D(X'), \\qquad A \\mapsto (\\pi_{X'})_{!}\\bigl(\\pi_{X}^{\\ast} A \\otimes K\\bigr). $$Two familiar functors are themselves Fourier–Mukai transforms with simple kernels. Take $K = \\mathbf 1_{X}$ regarded as an element of $\\mathsf D(X \\times_{Y} Y) = \\mathsf D(X)$ (the \u0026ldquo;$X \\to Y$\u0026rdquo; slot): then $\\Phi_{K} = f_{!}$. Take instead $K = \\mathbf 1_{X}$ in $\\mathsf D(Y \\times_{Y} X) = \\mathsf D(X)$ (the \u0026ldquo;$Y \\to X$\u0026rdquo; slot): then $\\Phi_{K} = f^{\\ast}$.\nSo \u0026ldquo;$f_{!}$ is left adjoint to $f^{\\ast}$\u0026rdquo; translates to \u0026ldquo;$\\mathbf 1_{X}$ (in the $X \\to Y$ slot) is left adjoint to $\\mathbf 1_{X}$ (in the $Y \\to X$ slot) in a suitable $2$-category whose $1$-morphisms are kernels\u0026rdquo;. Making this precise:\nDefinition 14 (The 2-category of kernels). For $\\mathsf D\\colon \\mathsf{Corr}(\\mathcal C, E) \\to \\mathsf{Cat}$ and $Y \\in \\mathcal C$, the $2$-category of kernels $\\mathbb K_{\\mathsf D, Y}$ has:\nObjects. Objects of $(\\mathcal C_{E})_{/Y}$ — i.e.\\ maps $X \\to Y$ in $E$. Morphism categories. $\\mathsf{Hom}_{Y}(X_{1}, X_{2}) \\coloneqq \\mathsf D(X_{1} \\times_{Y} X_{2})$. Composition. $M \\circ N \\coloneqq \\pi_{13!}(\\pi_{12}^{\\ast} M \\otimes \\pi_{23}^{\\ast} N)$. Identity. $\\mathrm{id}_{X} = (\\Delta_{X/Y})_{!}\\mathbf 1_{X}$. ([lu-zheng] ; [fargues-scholze, §2.3] ; [scholze-six, Lecture V] ; [heyer-mann] .)\nThree structural features fall out of the definition.\nRealisation. There is a \u0026ldquo;realisation\u0026rdquo; $2$-functor $\\Psi_{\\mathsf D, Y} \\coloneqq \\mathsf{Hom}_{Y}(Y, -)\\colon \\mathbb K_{\\mathsf D, Y} \\to \\mathsf{Cat}_{2}$ sending $X \\mapsto \\mathsf D(X)$ and a kernel $M$ to its Fourier–Mukai functor. Working in $\\mathbb K_{\\mathsf D, Y}$ amounts to working with kernels of functors directly, instead of the induced functors.\nFactorization ([heyer-mann] ). $\\mathsf D$ factors:\n$$ \\mathsf{Corr}((\\mathcal C_{E})_{/Y}) \\xrightarrow{\\;\\Phi_{\\mathsf D, Y}\\;} \\mathbb K_{\\mathsf D, Y} \\xrightarrow{\\;\\Psi_{\\mathsf D, Y}\\;} \\mathsf{Cat}, $$with $\\Psi \\circ \\Phi = \\mathsf D$. The first $2$-functor is identity on objects and sends a span $X \\xleftarrow{g} Z \\xrightarrow{f} Y$ to $(f, g)_{!} \\mathbf 1_{Z}$.\nFunctoriality. As $Y$ varies, $\\mathbb K_{\\mathsf D, (-)}$ is itself a 3-functor formalism, one categorical level higher:\nTheorem 15 (Functoriality of the kernel 2-category). $\\mathbb K_{\\mathsf D, (-)}\\colon \\mathsf{Corr}(\\mathcal C, E) \\to \\mathsf{Cat}_{2}$ is a lax symmetric monoidal $2$-functor. ([heyer-mann] .) That is: just as $\\mathsf D$ is a categorified cohomology theory valued in $\\mathsf{Cat}$, $\\mathbb K_{\\mathsf D}$ is a $2$-categorified one valued in $\\mathsf{Cat}_{2}$. This both promotes the classical \u0026ldquo;base change of suave/prim\u0026rdquo; lemmas to formal $2$-functoriality, and is the gateway to iteration in §10.\n8. Cohomological smoothness as dualizability in $\\mathbb K_{\\mathsf D}$ Now the punchline. After sliding $\\mathcal C$ down to $\\mathcal C_{/Y}$, assume $Y$ is final. Consider $\\mathbf 1_{X}$ as a $1$-morphism $X \\to Y$ in $\\mathbb K_{\\mathsf D}$; its realisation under $\\Psi$ is $f_{!}$.\nTheorem 16 (Poincaré duality as dualizability). $f$ is $\\mathsf D$-cohomologically smooth iff:\n$\\mathbf 1_{X} \\in \\mathsf{Hom}_{\\mathbb K_{\\mathsf D}}(X, Y) = \\mathsf D(X)$ is a left adjoint in $\\mathbb K_{\\mathsf D}$ — i.e.\\ admits a right adjoint $\\omega_{f} \\in \\mathsf D(X)$; $\\omega_{f}$ is $\\otimes$-invertible in $\\mathsf D(X)$. In this case $\\omega_{f} \\simeq f^{!}(\\mathbf 1_{Y})$. ([scholze-six, Lecture V] .)\nTwo consequences:\nDropping invertibility in (2) gives a strictly weaker, far more robust notion — suaveness, the subject of §9. Base-change compatibility of $\\omega_{f}$ (axiom (3) of Definition 13 ) is automatic, because $\\mathbb K_{\\mathsf D, Y}$ is functorial in $Y$ (Theorem 15 ) and adjunctions are preserved by $2$-functors. Even better, the $2$-categorical adjointness of (1) can be verified from a surprisingly small amount of data on $X$, $Y$, $X \\times_{Y} X$:\nTheorem 17 (Small-data criterion). $f\\colon X \\to Y$ (with $Y$ final) is cohomologically smooth iff there exist:\na $\\otimes$-invertible $L \\in \\mathsf D(X)$, $\\alpha\\colon \\Delta_{!}\\mathbf 1_{X} \\to \\pi_{2}^{\\ast} L$ in $\\mathsf D(X \\times_{Y} X)$, $\\beta\\colon f_{!} L \\to \\mathbf 1_{Y}$ in $\\mathsf D(Y)$, such that the two composites\n$$ \\mathbf 1_{X} \\cong \\pi_{1!}\\Delta_{!}\\mathbf 1_{X} \\xrightarrow{\\pi_{1!}\\alpha} f^{\\ast} f_{!} L \\xrightarrow{f^{\\ast}\\beta} \\mathbf 1_{X}, $$$$ L \\cong \\pi_{2!}(\\pi_{1}^{\\ast} L \\otimes \\Delta_{!}\\mathbf 1_{X}) \\xrightarrow{\\alpha} f^{\\ast} f_{!} L \\otimes L \\xrightarrow{f^{\\ast}\\beta \\otimes L} L $$are the identity. Then $\\omega_{f} \\simeq L$. ([scholze-six, Lecture V] .)\nThe proof in $(\\Leftarrow)$ is the heart of the picture: $(\\alpha, \\beta)$ translate literally into unit/counit $2$-morphisms making $\\mathbf 1_{X}$ (in the $X \\to Y$ direction) and $L$ (in the $Y \\to X$ direction) an adjoint pair in $\\mathbb K_{\\mathsf D}$. Applying $\\Psi$ gives the adjunction $f_{!} \\dashv (f^{\\ast}(-) \\otimes L)$ in $\\mathsf{Cat}$, hence $f^{!} \\simeq f^{\\ast}(-) \\otimes L$ and $L \\simeq f^{!}(\\mathbf 1_{Y})$.\nExample 18 (ℝ → ∗ in topology). For $f\\colon \\mathbb R \\to \\ast$ in $\\mathsf D = \\mathsf D(\\mathsf{Ab})$, take $L = \\mathbb Z[1]$. Here $\\alpha$ is a generator of $R\\Gamma_{c}(\\mathbb R, \\mathbb Z[1]) \\simeq \\mathbb Z$ and $\\beta$ is built from the triangle $0 \\to \\Delta_{!}\\mathbb Z \\to \\mathbb Z \\to j_{!}\\mathbb Z \\to 0$ on $\\mathbb R^{2}$. Compatibility of signs is automatic. Base-changing, every manifold bundle is $\\mathsf D$-smooth. Part IV · Suave, prim, and transmutation to Gestalten 9. Suave, prim, étale, proper Dropping invertibility from Theorem 16 produces a strictly larger, far more robust class of morphisms. This was the observation of Heyer–Mann.\nDefinition 19 (Suave and prim). Let $f\\colon X \\to Y$ in $E$. Note $\\mathsf{Hom}_{\\mathbb K_{\\mathsf D, Y}}(X, Y) = \\mathsf D(X)$, so the unit $\\mathbf 1_{X}$ is a $1$-morphism $X \\to Y$.\n$f$ is $\\mathsf D$-suave if $\\mathbf 1_{X}$ is a left adjoint in $\\mathbb K_{\\mathsf D, Y}$. The right adjoint $\\omega_{f} \\in \\mathsf D(X)$ is the dualizing complex. $f$ is $\\mathsf D$-prim if $\\mathbf 1_{X}$ is a right adjoint in $\\mathbb K_{\\mathsf D, Y}$. The left adjoint $\\delta_{f} \\in \\mathsf D(X)$ is the codualizing complex. The names: suave (Scholze, \u0026ldquo;close to smooth\u0026rdquo;); prim (Hansen, \u0026ldquo;close to but not proper\u0026rdquo;). $\\mathsf D$-cohomological smoothness = $\\mathsf D$-suave + $\\omega_{f}$ invertible. ([scholze-six, Lecture VI] ; [heyer-mann] .)\nRealising through $\\Psi$, $f$-suaveness gives the twist formula\n$$ f^{!} \\simeq \\omega_{f} \\otimes f^{\\ast}, \\qquad f^{\\ast} \\simeq \\underline{\\mathrm{Hom}}(\\omega_{f}, f^{!}), $$so suaveness is the structural half of cohomological smoothness — the identity $f^{!} = \\omega_{f} \\otimes f^{\\ast}$ — without requiring $\\omega_{f}$ invertible. Dually for prim and $\\delta_{f}$.\nConcrete criterion Proposition 20 (Pointwise criterion for suaveness). $f\\colon X \\to Y$ is $\\mathsf D$-suave iff\n$$ \\pi_{1}^{\\ast} \\underline{\\mathrm{Hom}}(\\mathbf 1_{X}, f^{!} \\mathbf 1_{Y}) \\otimes \\pi_{2}^{\\ast} \\mathbf 1_{X} \\;\\xrightarrow{\\;\\sim\\;}\\; \\underline{\\mathrm{Hom}}(\\pi_{1}^{\\ast} \\mathbf 1_{X}, \\pi_{2}^{!} \\mathbf 1_{X}) $$is an isomorphism in $\\mathsf D(X \\times_{Y} X)$. Then $\\omega_{f} = f^{!}\\mathbf 1_{Y}$. ([heyer-mann] ; [scholze-six, Lecture VI] .)\nStability and geometric meaning All from formal $2$-categorical adjointness (using Theorem 15 ):\nLocality on the target: both notions descend along $\\mathsf D^{\\ast}$-covers. Stability: suave morphisms compose, base-change, and are stable under suave pullback. Self-duality: $\\omega_{f}$ for $f$ suave is itself dualizable, and $\\omega_{f}^{\\vee} \\simeq \\omega_{f}$ by uniqueness of adjoints. ([heyer-mann] .) Geometric content:\nÉtale sheaves on schemes: $f$-suave = ULA — the original motivation of the kernel category in [fargues-scholze] . Topology: topological manifolds are $\\mathsf D$-suave with $\\omega_{f} \\simeq \\mathbf 1[\\dim f]$. Representation theory of locally profinite groups: on classifying stacks, $f$-suave = admissible representations, $f$-prim = compact representations ([heyer-mann] , the main application of HM\u0026rsquo;s paper). Étale and proper, via the diagonal When the dualizing/codualizing complex is trivial — $\\omega_{f} \\simeq \\mathbf 1$ or $\\delta_{f} \\simeq \\mathbf 1$ — we recover the étale/proper hierarchy. The natural definition is recursive on the diagonal.\nFor $f$ truncated, using the diagonal factorisation, one builds a natural map $f^{!} \\to f^{\\ast}$ provided $\\Delta_{f}^{!} \\simeq \\Delta_{f}^{\\ast}$:\n$$ f^{!} \\simeq \\Delta_{f}^{\\ast} \\pi_{2}^{\\ast} f^{!} \\xrightarrow{\\sim} \\Delta_{f}^{!} \\pi_{2}^{\\ast} f^{!} \\to \\Delta_{f}^{!} \\pi_{1}^{!} f^{\\ast} \\simeq f^{\\ast}. $$The recursion terminates because the diagonal of an $n$-truncated map is $(n-1)$-truncated.\nDefinition 21 (Cohomologically étale and proper, inductively). Let $f\\colon X \\to Y$ be $n$-truncated in $E$.\n$f$ is cohomologically étale if $f$ is $\\mathsf D$-suave and $\\Delta_{f}$ is either an isomorphism or cohomologically étale. $f$ is cohomologically proper if $f$ is $\\mathsf D$-prim and $\\Delta_{f}$ is either an isomorphism or cohomologically proper. ([scholze-six, Lecture VI] ; [heyer-mann] .)\nThe étale-proper dichotomy materialises:\nProposition 22 (Étale ⇔ f^! = f^*; proper ⇔ f_! = f_*). For $f$ truncated:\nIf $\\Delta_{f}$ is cohomologically étale, the following are equivalent: $f$ is cohomologically étale; $f^{!}\\mathbf 1_{Y} \\simeq f^{\\ast}\\mathbf 1_{Y}$; $f^{!} \\xrightarrow{\\sim} f^{\\ast}$. If $\\Delta_{f}$ is cohomologically proper, the following are equivalent: $f$ is cohomologically proper; $f_{!}\\mathbf 1_{X} \\simeq f_{\\ast}\\mathbf 1_{X}$; $f_{!} \\xrightarrow{\\sim} f_{\\ast}$. ([scholze-six, Lecture VI] ; [heyer-mann] .)\nAoki\u0026rsquo;s one-step reformulation The inductive definition is clean but unfolds truncations. Aoki observed that the same content packages in one step once one passes through monomorphisms.\nDefinition 23 (Open and closed immersions). A morphism $f\\colon X \\to Y$ of stacks is an open immersion if it is a $\\mathsf D$-suave monomorphism, a closed immersion if it is a $\\mathsf D$-prim monomorphism. ([aoki-motives] .) Equivalently ([aoki-motives] ): $[X] \\to [Y]$ admits a fully faithful $[Y]$-linear left (resp. right) adjoint.\nDefinition 24 (Unramified, étale, separated, proper (Aoki)). A static (i.e.\\ $0$-truncated) morphism $f$ of stacks is:\nunramified if $\\Delta_{f}$ is an open immersion; étale if $f$ is suave and unramified; separated if $\\Delta_{f}$ is a closed immersion; proper if $f$ is prim and separated. ([aoki-motives] .)\nRemark(Why static?). For a static morphism, both source and target are $0$-truncated, so $X \\times_{Y} X$ is $0$-truncated, hence $\\Delta_{f}\\colon X \\to X \\times_{Y} X$ is automatically a monomorphism. Without staticity, requiring $\\Delta_{f}$ to be open/closed immersion would impose an extra mono condition. The recursion of Definition 21 is now implicit: open/closed immersions encode all the truncated-diagonal data in one step. Remark(Coda: six operations without compactification). Aoki\u0026rsquo;s formulation has a structural dividend. With $\\mathcal C$ stacks and $E$ = exceptional morphisms (those $n$-truncated $f$ along which $[Y]$ is iteratively dualizable over $[Y]^{\\otimes_{[X]} S^{n-1}}$), the $(E, E)$-biadjointability condition of Theorem 6\nholds, so CLL produces a six-functor formalism without input compactification. This recovers the classical six operations on locally compact Hausdorff spaces of countable weight built from open subsets of $\\mathbb R^{n}$. ([aoki-motives] .)\n10. Transmutation to Gestalten The functoriality theorem (Theorem 15 ) makes $\\mathbb K_{\\mathsf D, (-)}$ a 3-functor formalism one categorical level higher. The natural next step is to iterate. The output is a tower of $n$-categories controlled by a single algebraic object — a Stefanich ring — and the geometric content of $\\mathsf D$ is captured by a functor into $\\mathsf{Gest}$.\nFunctorial form of $\\mathbb K_{\\mathsf D, Y}$ For iteration we want a description of $\\mathbb K_{\\mathsf D, Y}$ in which $\\mathsf D$-linearity is built in from the start. Set $\\mathsf{Corr}_{Y} \\coloneqq \\mathsf{Corr}((\\mathcal C_{E})_{/Y})$. Under Day convolution, $\\mathsf D$ (restricted to $\\mathsf{Corr}_{Y}$) is a commutative algebra in $\\mathsf{Fun}(\\mathsf{Corr}_{Y}, 1\\mathsf{Pr})$.\nProposition 25 (Functorial description of $\\mathbb K_{\\mathsf D, Y}$). $\\mathbb K_{\\mathsf D, Y}$ is identified with the full sub-$2$-category of\n$$ \\mathsf{Mod}_{\\mathsf D}\\!\\bigl(\\mathsf{Fun}(\\mathsf{Corr}_{Y}, 1\\mathsf{Pr})\\bigr) $$spanned by representable $\\mathsf D$-modules $[X] \\coloneqq \\mathsf D(- \\times_{Y} X)$. ([scholze-six, Lecture V, appendix] ; [aoki-motives] .)\nIn this form, the $\\mathsf D$-module structure is intrinsic: a kernel $K \\in \\mathsf D(X_{1} \\times_{Y} X_{2})$ becomes a $\\mathsf D$-module morphism $\\varphi_{K}\\colon [X_{1}] \\to [X_{2}]$, and asking for $\\varphi_{K}$ to admit a linear adjoint is a direct condition inside $\\mathsf{Mod}_{\\mathsf D}$. Statements like \u0026ldquo;$[Y]$ is self-dual over $[X]$\u0026rdquo; or \u0026ldquo;the trace of a kernel lives in $\\mathsf D(Y)$\u0026rdquo; become naked symmetric-monoidal statements.\nIteration to Stefanich rings The functorial form iterates. From $\\mathsf D\\colon \\mathsf{Corr}(\\mathcal C) \\to 1\\mathsf{Pr}$, the functoriality theorem gives a kernel $2$-category $\\mathbb K_{\\mathsf D, X}$ for each $X$; the same machinery applied to $\\mathbb K_{\\mathsf D, (-)}$ gives a $3$-category $\\mathbb K_{\\mathsf D, X}^{(2)}$, and so on. Assembling everything yields a Stefanich ring\n$$ A_{\\mathsf D, X} \\;\\coloneqq\\; \\bigl( \\mathsf D(X),\\; \\mathbb K_{\\mathsf D, X},\\; \\mathbb K_{\\mathsf D, X}^{(2)},\\; \\ldots \\bigr) \\;\\in\\; \\mathsf{StRing}. $$([scholze-gestalten, §3] , building on [stefanich-thesis] .) Functorially: $\\mathcal C^{\\mathrm{op}} \\to \\mathsf{StRing}$.\nThe transmutation theorem The Gestalt category is $\\mathsf{Gest} \\coloneqq \\mathsf{StRing}^{\\mathrm{op}}$. Composing with $\\mathsf{Spec}_{\\infty}\\colon \\mathsf{StRing} \\to \\mathsf{Gest}^{\\mathrm{op}}$ gives the transmutation:\nTheorem 26 (Transmutation). $X \\mapsto [X]_{\\mathsf D} \\coloneqq \\mathsf{Spec}_{\\infty}(A_{\\mathsf D, X})$ extends to a finite-limit-preserving functor $[-]_{\\mathsf D}\\colon \\mathcal C \\to \\mathsf{Gest}$, and for every $f\\colon X \\to Y$ in $\\mathcal C$:\n$[f]_{\\mathsf D}$ is $1$-étale and $1$-proper automatically; $f$ is $\\mathsf D$-suave (resp.\\ $\\mathsf D$-prim) iff $[f]_{\\mathsf D}$ is $0$-suave (resp.\\ $0$-prim); $f$ is truncated and $\\mathsf D$-cohomologically étale (resp.\\ proper) iff $f$ is truncated and $[f]_{\\mathsf D}$ is $0$-étale (resp.\n$0$-proper). ([scholze-gestalten, Prop. 9.5, Rem. 9.6] .)\nTwo things at once:\nLevel $1$ free. Whatever $f$ is geometrically, $[f]_{\\mathsf D}$ is \u0026ldquo;doubly nice\u0026rdquo; at the next level — ambidexterity in the Stefanich-ring tower. The 6FF sees only level $0$; $\\mathsf{Gest}$ remembers all. Level $0$ faithful. Suave, prim, étale, proper for $f$ correspond exactly to their $0$-versions for $[f]_{\\mathsf D}$. Unpacking the dictionary For $f\\colon X \\to Y$, write $A = A_{\\mathsf D, Y}$, $B = A_{\\mathsf D, X}$. The unit $\\mathbf 1_{X} \\in \\mathsf D(X) = \\mathrm{Fun}_{Y}(X, \\mathbb 1)$ — a morphism $X \\to \\mathbb 1$ in $\\mathbb K_{\\mathsf D, Y}$ realising as $f_{!}$ — is the structure map $\\mathbb 1 \\to (B/A)_{1}$ of the algebra $(B/A)_{1} \\in A_{2}$. Its Koszul dual is the counit $(B/A)^{!}_{1} \\to \\mathbb 1$ of the coalgebra $(B/A)^{!}_{1} = f_{1, \\sharp}(\\mathbb 1)$. Then ([scholze-gestalten, Defs. 6.9, 6.16] ):\n$[f]_{\\mathsf D}$ is $0$-prim iff for all $m \\geq 1$ the map $\\mathbb 1 \\to (B/A)_{m}$ admits a right adjoint in $A_{m+1}$. For transmuted maps, $m \\geq 2$ follow from ambidexterity, so the content is at $m = 1$: $\\mathbb 1 \\to (B/A)_{1}$ admits a right adjoint in $A_{2}$.\n$[f]_{\\mathsf D}$ is $0$-suave iff for all $m \\geq 1$ both $\\mathbb 1 \\to (B/A)_{m}$ admits a left adjoint and $(B/A)^{!}_{m} \\to \\mathbb 1$ admits a right adjoint — the second a strengthening. Reduces to $m = 1$.\nSo suave/prim conditions on $f$, abstract adjunction statements in $\\mathbb K_{\\mathsf D}$, become bare adjoint existence for the unit map $\\mathbb 1 \\to (B/A)_{1}$ in $A_{2}$.\nWhat this buys us Six-functor formalisms forget structure. Higher-categorical data hidden in $\\mathsf D$ becomes manifest in $A_{\\mathsf D}$; the level-$1$ ambidexterity that was previously a theorem is now a tautology. Geometry independent of the formalism. Different 6FFs on $\\mathcal C$ can produce the same Gestalt $[X]_{\\mathsf D}$, reducing comparison of cohomology theories to comparison of geometric objects. The site is encoded. The Morel–Voevodsky example $[\\operatorname{Spec}(\\mathbb Z)]_{\\mathsf{SH}} \\in \\mathsf{Gest}$ is $2$-affine, generated by $[\\mathbb A^{n}_{\\mathbb Z}]_{\\mathsf{SH}}$, $n \\geq 0$. ([scholze-gestalten, Prop. 9.9] .) The \u0026ldquo;site\u0026rdquo; of algebraic geometry is reconstructed from $\\mathsf D$ alone. A Gestalt is a six-functor formalism with all its hidden higher-categorical data made manifest. Cohomology, suaveness, properness, smoothness sit at level $0$; everything else encoded above.\nReferences [Aoki] Ko Aoki. Algebraic 2-motives and ring stacks. Preprint (2025).\n[Ayo07] Joseph Ayoub. Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. Astérisque 314, 315 (2007).\n[CD19] Denis-Charles Cisinski and Frédéric Déglise. Triangulated Categories of Mixed Motives. Springer Monographs in Mathematics (2019).\n[CLL25] Bastiaan Cnossen, Tobias Lenz, Sil Linskens. Universal six-functor formalisms. arXiv:2505.19192 (2025).\n[DK25] Adam Dauser and Josefien Kuijper. Uniqueness of six-functor formalisms. arXiv:2412.15780 (2025).\n[DG22] Brad Drew and Martin Gallauer. The universal six-functor formalism. Annals of K-theory 7 (2022), 599–649.\n[FS24] Laurent Fargues and Peter Scholze. Geometrization of the local Langlands correspondence. arXiv:2102.13459 (2024).\n[HM24] Claudius Heyer and Lucas Mann. 6-functor formalisms and smooth representations. Preprint (2024).\n[LZ12] Yifeng Liu and Weizhe Zheng. Enhanced six operations and base change theorem for higher Artin stacks. arXiv:1211.5948 (2012).\n[LZ22] Qing Lu and Weizhe Zheng. Categorical traces and a relative Lefschetz–Verdier formula. Forum of Mathematics, Sigma 10 (2022).\n[Man22] Lucas Mann. A $p$-adic 6-functor formalism in rigid-analytic geometry. arXiv:2206.02022 (2022).\n[MV99] Fabien Morel and Vladimir Voevodsky. $\\mathbb A^{1}$-homotopy theory of schemes. Publ. Math. IHES 90 (1999), 45–143.\n[Sch26] Peter Scholze. Geometry and higher category theory. Lecture notes (2025/26).\n[Sch22] Peter Scholze. Six-functor formalisms. Lecture notes, Bonn (WS 2022/23).\n[Ste20] Germán Stefanich. Presentable $(\\infty,n)$-categories. arXiv:2011.03035 (2020).\n","permalink":"https://ou-liu-red-sugar.github.io/notes/notes/a-brief-introduction-to-6-functor-formalisms/","summary":"\u003cblockquote\u003e\n\u003cp\u003e\u003cstrong\u003eConventions.\u003c/strong\u003e\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cem\u003eCategory\u003c/em\u003e means $(\\infty,1)$-category; $\\mathsf{Cat}$ is the category\nof all categories.\u003c/li\u003e\n\u003cli\u003e\u003cem\u003e$2$-category\u003c/em\u003e means $(\\infty,2)$-category; $\\mathsf{Cat}_2$ is the\n$2$-category of all categories.\u003c/li\u003e\n\u003cli\u003e$\\mathsf{Pr}^L$ denotes presentable categories with left adjoints;\n$\\mathsf{Pr}^L_{\\mathrm{st}}$ is the stable version.\u003c/li\u003e\n\u003cli\u003e$\\mathsf{CAlg}(\\mathcal M)$ denotes commutative algebra objects in a\nsymmetric monoidal category $\\mathcal M$.\u003c/li\u003e\n\u003cli\u003eGiven a \u003cem\u003egeometric setup\u003c/em\u003e $(\\mathcal C, E)$ — a category $\\mathcal C$\nwith finite limits and a class of morphisms $E$ closed under base\nchange, composition and diagonals — we write\n$\\mathsf{Corr}(\\mathcal C, E)$ for the span $(\\infty, 2)$-category and\n$\\mathsf{Span}_2(\\mathcal C, E)_{I, P}$ for the CLL upgrade with\nbiadjointability data.\u003c/li\u003e\n\u003cli\u003e$\\underline{\\mathrm{Hom}}$ is internal hom; $\\mathbf 1_{X}$ is the\nmonoidal unit of $\\mathsf D(X)$; $\\mathbb 1$ is the terminal object of\na slice $\\mathcal C_{/Y}$.\u003c/li\u003e\n\u003c/ul\u003e\u003c/blockquote\u003e\n\u003chr\u003e\n\u003ch1 id=\"part-i--six-functor-formalisms\"\u003ePart I · Six-functor formalisms\u003c/h1\u003e\n\u003ch2 id=\"1-from-cohomology-to-six-functors\"\u003e1. From cohomology to six functors\u003c/h2\u003e\n\u003cp\u003eA cohomology theory associates with every \u0026ldquo;space\u0026rdquo; $X$ a complex\n$\\Gamma(X; \\Lambda) \\in \\mathsf D(\\Lambda)$ of $\\Lambda$-modules — singular\ncohomology, étale cohomology, coherent cohomology, and many more. We then\nexpect $\\Gamma(X; \\Lambda)$ to satisfy structural identities like the\nKünneth formula, Poincaré duality, proper base change, excision. Each is\nclassically proved by hand, with most of the work intertwined with\nwhatever specific sheaf-theoretic gadget produced $\\Gamma(X; \\Lambda)$.\u003c/p\u003e","title":"A brief introduction to 6-functor formalisms"},{"content":" Disclaimer: This is my personal learning log and decision framework. Nothing here is financial advice.\nStyle: long-term (months to years), no leverage, low-frequency execution.\n1. Scope \u0026amp; Objectives Time horizon: months to years\nPrimary objective: sustainable long-term compounding with explainable decisions\nSecondary objective: keep investing in the background (study/life first)\n2. Non-Negotiables No leverage No high-frequency trading Do not change long-term structure due to short-term volatility Avoid complexity for the sake of being “smart” Default to low action when evidence is weak 3. Portfolio Architecture 3.1 Core ETFs = Market Tracking (Beta Engine) The core sleeve is designed to follow broad markets with low friction. This is the default compounding engine.\nTypical instruments:\nVOO / RSP (US large-cap / S\u0026amp;P 500 exposure with different weighting) VEU (ex-US equity exposure) SCHG (growth tilt) 3.2 Individual Stocks = Defense + Opportunities (Satellite Sleeve) Stocks are not for market tracking. They serve two roles:\nDefense\nResilient businesses meant to stabilize the portfolio under stress\n(defensive ≠ no drawdown; the real defense is sizing + emotional distance).\nOpportunities\nSmall, controlled bets on clear mispricing or structural themes.\nDefault filter (to avoid randomness):\nPrefer high-quality, well-covered mega-caps temporarily weak / range-bound\nover obscure small caps, unless I can articulate a real edge. 3.3 Gold + Cash = Risk Tools Gold: diversifier / hedge (can be volatile in the short run) Cash: optionality + psychological safety; allows patient deployment 4. Allocation Bands (targets, not fixed weights) I manage allocation by bands rather than exact weights.\nCore ETFs (Market): 35–60% Stocks (Defense + Opportunities): 20–50% Gold: 8–15% Cash: 5–30% Important: Bands are not meant to hit upper bounds simultaneously.\nPriority rule: keep the Core healthy; correct drift mainly via cash flows.\nNeutral baseline (informal):\nCore ~50%, Stocks ~40%, Gold ~10%, Cash 0–20% (can go higher temporarily after deposits) Band violation protocol:\nIf bands are violated through passive drift (price moves): correct via cash flows; no urgency. If bands are violated through active rotation: the rotation must meet the conditions in Section 6.5. If not, it is a process failure — review in the next monthly review. The bands are guardrails, not prison walls. A thoughtful, written exception is allowed; an impulsive breach is not. 5. Decision Protocol (before any action) Before any buy/sell/add/reduce decision:\nMirror intent: structural allocation or emotional reaction? New information check: did the thesis change, or is it just volatility/noise? Default action: if evidence is weak → no action or tiny adjustment using new cash only AI rule (tool, not leader):\nAI may help research, but it cannot define risk boundaries. If I cannot state my cushion (price + sizing + liquidity + falsification) in one sentence, default to smaller size or no action. 6. Rebalancing Rules (low action by default) 6.1 Primary method: rebalance via new cash Prefer fixing drift by directing new contributions into the underweight sleeve. Avoid forced selling unless structure is meaningfully broken. 6.2 Triggers If Stocks \u0026gt; 50%:\nstop adding to stocks; direct new buys to Core ETFs until back in range. If Core ETFs \u0026lt; 35%:\nnew buys go to the Core by default (staged entries). If Cash \u0026gt; 30%:\ndeploy gradually into Core ETFs unless a clearly defined opportunity exists. If Gold \u0026gt; 15%:\nprefer correcting via cash flows; trim only if necessary to restore structure. 6.3 Execution style Prefer 3-step entries over all-in timing: Step 1: starter position Step 2: add if thesis holds and price improves / risk improves Step 3: complete only if evidence strengthens Avoid “revenge trades” after losses. Minimum gap between steps: 5 trading days. Same-day or next-day adds count as the same step for discipline purposes. 6.4 Cooling-off rule after large deposits (hard rule) After any large deposit:\nwrite the plan first, then deploy default first buys go to Core ETFs any stock entries must be staged (3 steps) and must fit allocation bands 6.5 Valuation-based rotation (conditional, not default) Selling Core ETFs to fund stock purchases is allowed only when all five conditions are met:\nWritten justification with a specific valuation metric (e.g., \u0026ldquo;SPX PE ~29x, well above 20-year avg of ~16x\u0026rdquo;). Named destination with a clear risk/reward case at current prices. Minimum Core floor: at least one core ETF must remain ≥10% weight. Gradual execution: spread over at least 2–3 weeks. Sector check: destination must not push any single sector above 25%. Without all five conditions, the default remains: use new cash only.\nRelated lesson: Rotation is valid — magnitude is the risk\n6.6 Sector concentration cap No more than 25% of the portfolio in a single GICS sector. Before adding to any position, check total sector exposure:\nIf a sector exceeds 25% through passive drift, correct via cash flows. If a sector exceeds 25% through active buying, stop immediately. 7. Exception Clause (allowed, but controlled) I may break a default rule only if:\nthe action is small (does not alter overall structure), the reason is written in 1–2 sentences: “What hypothesis am I betting on?” “If I’m wrong, how will I exit (time/price/size)?” exceptions are reviewed in the next monthly review. 8. Review Cadence (and attention budget) Log: short entries written close to the decision (3–10 minutes) Monthly review: once per month, focusing on: structure drift (Core / Stocks / Gold / Cash) whether any thesis actually changed whether investing is consuming too much attention Lessons: if a mistake repeats twice, convert it into an explicit rule Attention budget (hard constraint):\nDaily: 15–25 minutes max (check + log + orders) Weekly: one deep block (60–90 minutes) for reading + thesis maintenance If math/study suffers → reduce investing frequency, not “try harder”. Snapshots live in the Log, not the Playbook.\nSee: /invest/log/ 9. Change Log (Playbook edits) 2026-01-30: initial version; defined Core ETFs as market tracking, stocks as defense/opportunities; established allocation bands and rebalancing rules. 2026-01-30 (v1.1): removed baseline snapshots from Playbook (moved to Log); added deposit cooling-off rule, AI tool constraint, and attention budget. 2026-02-28 (v1.2): Added valuation-based rotation clause (6.5), time-staged minimum gap (6.3), sector concentration cap (6.6), and band violation protocol (4). Driven by February valuation rotation and SPGI entry speed. ","permalink":"https://ou-liu-red-sugar.github.io/invest/playbook/","summary":"\u003cblockquote\u003e\n\u003cp\u003e\u003cstrong\u003eDisclaimer:\u003c/strong\u003e This is my personal learning log and decision framework. Nothing here is financial advice.\u003cbr\u003e\n\u003cstrong\u003eStyle:\u003c/strong\u003e long-term (months to years), no leverage, low-frequency execution.\u003c/p\u003e\u003c/blockquote\u003e\n\u003chr\u003e\n\u003ch2 id=\"1-scope--objectives\"\u003e1. Scope \u0026amp; Objectives\u003c/h2\u003e\n\u003cp\u003e\u003cstrong\u003eTime horizon:\u003c/strong\u003e months to years\u003cbr\u003e\n\u003cstrong\u003ePrimary objective:\u003c/strong\u003e sustainable long-term compounding with explainable decisions\u003cbr\u003e\n\u003cstrong\u003eSecondary objective:\u003c/strong\u003e keep investing in the background (study/life first)\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"2-non-negotiables\"\u003e2. Non-Negotiables\u003c/h2\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eNo leverage\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eNo high-frequency trading\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003eDo not change long-term structure due to short-term volatility\u003c/li\u003e\n\u003cli\u003eAvoid complexity for the sake of being “smart”\u003c/li\u003e\n\u003cli\u003eDefault to \u003cstrong\u003elow action\u003c/strong\u003e when evidence is weak\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"3-portfolio-architecture\"\u003e3. Portfolio Architecture\u003c/h2\u003e\n\u003ch3 id=\"31-core-etfs--market-tracking-beta-engine\"\u003e3.1 Core ETFs = Market Tracking (Beta Engine)\u003c/h3\u003e\n\u003cp\u003eThe core sleeve is designed to \u003cstrong\u003efollow broad markets\u003c/strong\u003e with low friction.\nThis is the default compounding engine.\u003c/p\u003e","title":"Playbook"},{"content":"This note is not part of the original lecture course; it grew out of discussions about understanding sheaf cohomology from a derived / animated perspective. The treatment follows Lurie\u0026rsquo;s Spectral Algebraic Geometry ([lurie-sag] ) and Mathew\u0026rsquo;s work on Galois groups in stable homotopy theory ([mathew-galois] ).\nConventions Throughout we work in the derived setting and drop the $\\mathrm{R}$-prefix on all functors. Concretely:\nAll limits, colimits, and tensor products are derived. The symbol $\\otimes$ denotes derived tensor product; the classical tensor product is recovered as $\\pi_0(- \\otimes -)$. $\\Gamma(X, \\mathcal{F})$ denotes derived global sections; the classical $\\Gamma$ is $H^0(X, \\mathcal{F}) \\coloneqq \\pi_0\\,\\Gamma(X, \\mathcal{F})$. $\\Hom_R(M, N)$ denotes derived $\\Hom$, so $\\operatorname{Ext}^i_R(M, N) = \\pi_{-i}\\Hom_R(M, N)$. $\\mathrm{Mod}_R$ denotes the $\\infty$-category of (left) module spectra over a connective $\\mathbb{E}_\\infty$-ring $R$ (equivalently, $\\mathsf{D}(R)$ when $R$ is discrete). Presheaves take values in a stable presentable category $\\mathcal{D}$ — typically $\\mathsf{D}(\\mathbb{Z})$, $\\mathsf{Sp}$, or $\\mathrm{Mod}_R$ for a base ring. The classical $1$-categorical theory is recovered by passing to $\\pi_0$ at the end.\nThe thesis: cohomology = sheafification + global sections The core observation is captured in a single line. For a scheme $X$ and a presheaf $\\mathcal{F} \\in \\mathrm{Fun}(\\mathsf{Sch}^{\\mathrm{op}}, \\mathcal{D})$, set \\[ \\boxed{\\,\\Gamma(X, \\mathcal{F}) \\coloneqq (L\\mathcal{F})(X),\\,} \\] where $L\\colon \\mathrm{PShv}(\\mathsf{Sch}, \\mathcal{D}) \\to \\mathrm{Shv}_\\tau(\\mathsf{Sch}, \\mathcal{D})$ is sheafification with respect to a Grothendieck topology $\\tau$ (we focus on the Zariski topology). All cohomological information about $\\mathcal{F}$ is packed into this single derived object. Three immediate consequences illustrate the formalism.\nMayer–Vietoris. For an open cover $X = U \\cup V$, the sheaf condition is exactly the fibre square Čech cohomology. For an open cover $\\mathfrak{U} = (U_i \\to X)_{i \\in I}$, write \\[ U_n \\coloneqq \\coprod_{(i_0, \\dots, i_n) \\in I^{n+1}} U_{i_0} \\cap \\cdots \\cap U_{i_n}. \\] The sheaf condition along $\\mathfrak{U}$ unfolds into the equivalence \\[ \\Gamma(X, \\mathcal{F}) \\xrightarrow{\\;\\sim\\;} \\lim_{[n] \\in \\mathbf{\\Delta}} \\Gamma(U_n, \\mathcal{F}). \\] We call the right-hand side the derived Čech complex of $\\mathfrak{U}$.\nPushforward and base change. For $f\\colon Y \\to X$, the pushforward $f_*\\colon \\mathrm{Shv}_\\tau(Y, \\mathcal{D}) \\to \\mathrm{Shv}_\\tau(X, \\mathcal{D})$ is determined by $\\Gamma(X, f_* \\mathcal{F}) \\simeq \\Gamma(Y, \\mathcal{F})$. Higher direct images are $H^i(X, f_*\\mathcal{F}) = \\pi_{-i} f_*\\mathcal{F}$ in the appropriate $t$-structure.\nThese are the standard cohomological tools, derived without spectral sequences, injective resolutions, or flabby/soft sheaves: everything is encoded in the sheaf condition together with the universal property of sheafification.\nRemark(Where classical Čech can fail). The classical naïve Čech complex of a cover is the cosimplicial abelian group $[n] \\mapsto H^0(U_n, \\mathcal{F})$ — the degree-zero truncation of $\\Gamma(U_n, \\mathcal{F})$. When some $\\Gamma(U_n, \\mathcal{F})$ has non-zero higher cohomology, this truncation drops information and naïve Čech cohomology disagrees with sheaf cohomology. We will see in the final section that the two agree precisely when each $\\Gamma(U_n, \\mathcal{F})$ is concentrated in degree zero. Quasi-coherent sheaves are sheaves For a scheme $X$, recall that the category of quasi-coherent modules is the limit \\[ \\mathrm{Mod}_X = \\lim_{(R,\\, x\\colon \\Spec(R) \\to X)} \\mathrm{Mod}_R \\] with derived base change as transition functors. For an $R$-module $M$, the associated presheaf on $\\mathrm{CAlg}_R$ is \\[ \\mathcal{F}_M\\colon \\mathrm{CAlg}_R \\to \\mathcal{D}, \\qquad A \\mapsto M \\otimes_R A. \\]The main theorem of this note is:\nTheorem 1 (Quasi-coherent modules are Zariski sheaves). For any ring $R$ and any $R$-module $M$, the presheaf $\\mathcal{F}_M$ is a Zariski sheaf on $\\mathrm{CAlg}_R$. Equivalently, $L\\mathcal{F}_M \\simeq \\mathcal{F}_M$, and consequently \\[ \\Gamma(\\Spec(R), \\mathcal{F}_M) \\simeq \\mathcal{F}_M(R) = M. \\] Since $M$ is concentrated in degree zero (for $M \\in \\mathrm{Mod}_R^{\\heartsuit}$), the higher cohomology vanishes:\nCorollary 2 (Affine cohomology vanishing). For an affine scheme $X = \\Spec(R)$ and a discrete quasi-coherent module $M$, \\[ H^i(X, \\mathcal{F}_M) = 0 \\quad \\text{for all } i \u003e 0. \\] The strategy for Theorem 1 is to establish the stronger statement that $\\mathcal{F}_M$ satisfies faithfully flat descent. This is the content of the next section, via Mathew\u0026rsquo;s framework of descendable algebras.\nFaithfully flat descent Throughout this section $\\mathcal{C}$ is a stable presentable symmetric monoidal category. In our application $\\mathcal{C} = \\mathrm{Mod}_R$.\nDescendable algebras Definition 3 (Thick tensor ideal). For $A \\in \\mathrm{CAlg}(\\mathcal{C})$, the thick tensor ideal generated by $A$, denoted $\\langle A \\rangle \\subset \\mathcal{C}$, is the smallest full subcategory containing $A$ and closed under finite (co)limits, retracts, and tensor products $- \\otimes X$ for arbitrary $X \\in \\mathcal{C}$. Definition 4 (Descendable algebra). $A \\in \\mathrm{CAlg}(\\mathcal{C})$ is descendable if $\\mathbf{1}_{\\mathcal{C}} \\in \\langle A \\rangle$.\n([mathew-galois, Def. 3.18] .)\nThe cobar construction associated to $A$ is the cosimplicial algebra \\[ A^{\\otimes \\bullet}\\colon \\mathbf{\\Delta} \\to \\mathrm{CAlg}(\\mathcal{C}), \\qquad [n] \\mapsto A^{\\otimes(n+1)}. \\] Definition 5 (Pro-constant cosimplicial object). A cosimplicial object $M^{\\bullet}\\colon \\mathbf{\\Delta} \\to \\mathcal{C}$ is pro-constant if its filtered diagram of partial totalisations \\[ n \\mapsto \\mathrm{Tot}_{\\le n}(M^{\\bullet}) \\coloneqq \\lim_{m \\in \\mathbf{\\Delta}_{\\le n}} M^m \\] is pro-equivalent (in $\\mathsf{Pro}(\\mathcal{C})$) to a constant pro-system.\nExample 6 (Split cosimplicial objects). If $M^{\\bullet}$ is split (admits a coaugmentation that is a section in homotopy at each level), then $\\mathrm{Tot}_{\\le n}(M^{\\bullet})$ stabilises for $n \\gg 0$, so $M^{\\bullet}$ is automatically pro-constant. Example 7 (Tensoring with pro-constants). In any stable symmetric monoidal category where $\\otimes$ preserves limits in each variable, tensoring with a pro-constant cosimplicial object remains pro-constant, and the limit commutes with the tensor: \\[ \\Big(\\lim_{\\mathbf{\\Delta}} M^{\\bullet}\\Big) \\otimes X \\simeq \\lim_{\\mathbf{\\Delta}} (M^{\\bullet} \\otimes X). \\] This is automatic for $\\mathcal{C} = \\mathrm{Mod}_R$, since $\\otimes_R$ is exact.\nTwo theorems of Mathew connect descendability with pro-constancy.\nTheorem 8 (Mathew). $A \\in \\mathrm{CAlg}(\\mathcal{C})$ is descendable iff the cobar $A^{\\otimes \\bullet}$ is pro-constant with limit $\\mathbf{1}_{\\mathcal{C}}$.\n([mathew-galois, Prop. 3.20] .)\nProof. ($\\Rightarrow$). Let $\\mathcal{X} \\subset \\mathcal{C}$ be the full subcategory of those $X$ such that $X \\otimes A^{\\otimes \\bullet}$ is pro-constant with limit $X$. Example 7 shows $\\mathcal{X}$ is closed under finite (co)limits, retracts, and tensor products. By Example 6 , $A \\in \\mathcal{X}$ — the cosimplicial object $A \\otimes A^{\\otimes \\bullet}$ is split via the obvious extra degeneracy. Hence $\\langle A \\rangle \\subset \\mathcal{X}$; in particular $\\mathbf{1} \\in \\mathcal{X}$.\n($\\Leftarrow$). If $A^{\\otimes \\bullet}$ is pro-constant with limit $\\mathbf{1}$, some $\\mathrm{Tot}_{\\le n}(A^{\\otimes \\bullet})$ admits $\\mathbf{1}$ as a retract. But $\\mathrm{Tot}_{\\le n}(A^{\\otimes \\bullet})$ is a finite limit of $A, A^{\\otimes 2}, \\dots$, hence lies in $\\langle A \\rangle$.\n$\\square$ Theorem 9 (Mathew). If $A \\in \\mathrm{CAlg}(\\mathcal{C})$ is descendable, then the canonical functor \\[ \\mathcal{C} \\xrightarrow{\\;\\sim\\;} \\lim_{\\mathbf{\\Delta}} \\mathrm{Mod}_{A^{\\otimes(\\bullet+1)}}(\\mathcal{C}) \\] is an equivalence.\n([mathew-galois, Thm. 3.26] .)\nProof. The adjunction $- \\otimes A\\colon \\mathcal{C} \\rightleftarrows \\mathrm{Mod}_A(\\mathcal{C}) \\colon F$ satisfies the Barr–Beck–Lurie criterion.\nConservativity of $- \\otimes A$. If $X \\otimes A \\simeq 0$, the subcategory $\\mathcal{Y} = \\{Y : X \\otimes Y \\simeq 0\\}$ contains $A$ and is closed under finite (co)limits, retracts, and tensor products, so $\\mathbf{1} \\in \\langle A \\rangle \\subset \\mathcal{Y}$, giving $X \\simeq 0$.\nLimit-exchange for split cosimplicial objects. Similar, via Example 7 and Theorem 8 .\n$\\square$ Faithfully flat maps are descendable We specialise to $\\mathcal{C} = \\mathrm{Mod}_R$ with $A = S$ for a connective ring map $R \\to S$.\nDefinition 10 (Flat / faithfully flat). A connective $R$-module $M$ is flat if $- \\otimes_R M$ is $t$-exact; equivalently, $\\pi_0 M$ is a flat $\\pi_0 R$-module and the canonical map $\\pi_n R \\otimes_{\\pi_0 R} \\pi_0 M \\to \\pi_n M$ is an isomorphism for every $n$.\nA connective ring map $R \\to S$ is faithfully flat if $S$ is a flat $R$-module and $\\pi_0 R \\to \\pi_0 S$ is faithfully flat in the classical sense (equivalently, $\\Spec \\pi_0 S \\to \\Spec \\pi_0 R$ is surjective).\nThe technical engine is a flat-Ext vanishing lemma due to Lurie:\nLemma 11 (Flat Ext vanishing). Let $R$ be a connective ring and $M$ a flat $R$-module such that $\\pi_0 M$ is $\\aleph_n$-presentable as a $\\pi_0 R$-module. Then for any connective $N \\in \\mathrm{Mod}_R$ and any $k \u003e n$, \\[ \\operatorname{Ext}^k_R(M, N) = 0. \\]([lurie-sag, Lem. D.3.3.6] .)\nTheorem 12 (Faithfully flat descent). Let $R \\to S$ be a faithfully flat connective ring map, with the cardinality bound that $\\pi_0 S$ is $\\aleph_n$-presentable over $\\pi_0 R$ for some $n$ (e.g. $\\aleph_{\\omega}$ — automatic for finitely or countably presented algebras). Then $S$ is descendable as an $R$-algebra, and consequently \\[ \\mathrm{Mod}_R \\xrightarrow{\\;\\sim\\;} \\lim_{\\mathbf{\\Delta}} \\mathrm{Mod}_{S^{\\otimes(\\bullet+1)}}. \\] Proof. By Theorem 9 , descendability of $S$ implies the equivalence; we show $S$ is descendable.\nSet $K \\coloneqq \\mathrm{fib}(R \\to S)$ and $C \\coloneqq \\mathrm{cofib}(R \\to S)$, related by $K \\simeq C[-1]$. The structure map $\\rho\\colon K \\to R$ assembles, for each $m \\ge 1$, into the $m$-fold tensor power \\[ \\rho^{(m)}\\colon K^{\\otimes m} \\xrightarrow{\\;\\rho \\otimes \\cdots \\otimes \\rho\\;} R. \\]Claim. $\\rho^{(m)}$ is null-homotopic for $m$ large enough.\nBy the shift $K \\simeq C[-1]$, \\[ \\rho^{(m)} \\in [K^{\\otimes m}, R]_{\\mathrm{Mod}_R} = [C^{\\otimes m}[-m], R]_{\\mathrm{Mod}_R} = \\operatorname{Ext}^m_R(C^{\\otimes m}, R). \\] Flatness of $S$ makes $C = \\mathrm{cofib}(R \\to S)$ flat too (long exact sequence on $\\pi_*$), so $C^{\\otimes m}$ is flat. The cardinality assumption makes $\\pi_0(C^{\\otimes m})$ $\\aleph_n$-presentable, and Lemma 11 gives $\\operatorname{Ext}^m_R(C^{\\otimes m}, R) = 0$ for $m \u003e n$.\nConcluding descendability. The factorisation $\\rho^{(m+1)}\\colon K^{\\otimes(m+1)} \\xrightarrow{\\mathrm{id} \\otimes \\rho^{(m)}} K \\xrightarrow{\\rho} R$ produces a cofibre sequence \\[ K^{\\otimes m} \\otimes_R S \\to \\mathrm{cofib}(\\rho^{(m+1)}) \\to \\mathrm{cofib}(\\rho^{(m)}). \\] Inductively, $\\mathrm{cofib}(\\rho^{(m)}) \\in \\langle S \\rangle$ for every $m$, since $K^{\\otimes m} \\otimes_R S \\in \\langle S \\rangle$ and $\\langle S \\rangle$ is closed under cofibres. When $\\rho^{(m)}$ is null-homotopic, \\[ \\mathrm{cofib}(\\rho^{(m)}) \\simeq R \\oplus K^{\\otimes m}[1], \\] exhibiting $R$ as a retract of an object in $\\langle S \\rangle$. Thus $R \\in \\langle S \\rangle$, i.e. $S$ is descendable.\n$\\square$ Proof of the main theorem Proof of Theorem on quasi-coherent modules. Set $S \\coloneqq \\prod_i R_{f_i}$ for a Zariski cover $(f_i)_{i \\in I}$ — i.e. elements generating the unit ideal of $R$. (We may reduce to finite $I$, since the unit ideal is generated by finitely many of the $f_i$.) The ring map $R \\to S$ is faithfully flat: each $R_{f_i}$ is flat over $R$, and $\\Spec(S) = \\coprod_i \\Spec(R_{f_i}) \\to \\Spec(R)$ is surjective by the unit-ideal hypothesis. The cardinality bound of Theorem 12\nis automatic.\nTheorem 12 gives an equivalence \\[ \\mathrm{Mod}_R \\xrightarrow{\\;\\sim\\;} \\lim_{\\mathbf{\\Delta}} \\mathrm{Mod}_{S^{\\otimes_R(\\bullet+1)}}. \\] Tracing $M$ through this equivalence: the unit of the adjunction sends $M$ to the cosimplicial $S^{\\otimes_R(\\bullet+1)}$-module $[n] \\mapsto M \\otimes_R S^{\\otimes_R(n+1)}$, and the equivalence reads \\[ M \\xrightarrow{\\;\\sim\\;} \\lim_{[n] \\in \\mathbf{\\Delta}} \\bigl(M \\otimes_R S^{\\otimes_R(n+1)}\\bigr). \\] This is exactly the Zariski sheaf condition for $\\mathcal{F}_M$ along the cover $(\\Spec R_{f_i})_i$: at level $n$, $S^{\\otimes_R(n+1)} = \\prod_{i_0, \\dots, i_n} R_{f_{i_0} \\cdots f_{i_n}}$, so the displayed limit is the equaliser (in the appropriate cosimplicial sense) corresponding to $\\mathcal{F}_M$ on the cover. Hence $\\mathcal{F}_M$ is a Zariski sheaf, and $L\\mathcal{F}_M \\simeq \\mathcal{F}_M$.\n$\\square$ Naïve Čech as a special case Theorem 1 immediately clarifies the relationship between derived and classical Čech cohomology.\nDerived Čech is sheaf cohomology, always. Let $\\mathcal{F}$ be a sheaf on $X$ and $\\mathfrak{U} = (U_i \\to X)_{i \\in I}$ any cover. The sheaf condition along $\\mathfrak{U}$ is the equivalence \\[ \\Gamma(X, \\mathcal{F}) \\xrightarrow{\\;\\sim\\;} \\lim_{[n] \\in \\mathbf{\\Delta}} \\Gamma(U_n, \\mathcal{F}). \\] No truncation, no spectral sequence, no acyclicity hypothesis. The right-hand side is the derived Čech complex.\nNaïve Čech recovers derived Čech when intersections are acyclic. The classical naïve Čech complex is the cosimplicial discrete abelian group $[n] \\mapsto H^0(U_n, \\mathcal{F})$ — the degree-zero truncation of $\\Gamma(U_n, \\mathcal{F})$. By Corollary 2 , this truncation loses nothing precisely when each $U_n$ is affine (so $\\Gamma(U_n, \\widetilde{M})$ is concentrated in degree zero). By the affine intersection property of separated schemes (diagonal is a closed immersion), this is automatic for affine covers of separated schemes:\nCorollary 13 (Naïve Čech for separated schemes). Let $X$ be a separated scheme, $\\mathfrak{U}$ an affine open cover, and $\\mathcal{F}$ a quasi-coherent sheaf on $X$. Then naïve and derived Čech cohomology along $\\mathfrak{U}$ coincide, and both equal sheaf cohomology: \\[ \\check{H}^i(\\mathfrak{U}, \\mathcal{F}) \\xrightarrow{\\;\\sim\\;} H^i(X, \\mathcal{F}) \\qquad \\text{for all } i \\ge 0. \\] Proof. Since $X$ is separated, the diagonal $X \\to X \\times X$ is a closed immersion, so every finite intersection of affine opens is affine; in particular each $U_n$ is affine. Corollary 2 then makes $\\Gamma(U_n, \\mathcal{F})$ concentrated in degree zero, where it agrees with $H^0(U_n, \\mathcal{F})$. Hence the cosimplicial object $[n] \\mapsto \\Gamma(U_n, \\mathcal{F})$ is discrete, and its limit is the Moore complex of the cosimplicial abelian group $[n] \\mapsto H^0(U_n, \\mathcal{F})$ — i.e. the naïve Čech complex. $\\square$ Cartan–Leray, reinterpreted. The classical Cartan–Leray spectral sequence \\[ E_2^{p,q} = \\check{H}^p(\\mathfrak{U}, \\mathcal{H}^q(\\mathcal{F})) \\Longrightarrow H^{p+q}(X, \\mathcal{F}) \\] is precisely the spectral sequence associated to the Bousfield–Kan filtration on $\\lim_{\\mathbf{\\Delta}} \\Gamma(U_{\\bullet}, \\mathcal{F})$. The Cartan–Leray hypothesis — vanishing of higher $H^q$ on the cover — collapses the spectral sequence to its $E_2^{p,0}$ row, recovering naïve Čech.\nConnection with the locally ringed space picture The category $\\mathrm{Mod}_X$ defined as a limit \\[ \\mathrm{Mod}_X = \\lim_{(R,\\, x\\colon \\Spec R \\to X)} \\mathrm{Mod}_R \\] agrees, when $X$ is a scheme, with the classical category of quasi-coherent $\\mathcal{O}_X$-modules on the locally ringed space $(|X|, \\mathcal{O}_X)$. Under this identification, the derived global sections $\\Gamma(X, \\mathcal{F})$ defined here match the classical $\\mathrm{R}\\Gamma$ of an $\\mathcal{O}_X$-module computed via injective resolutions. All the classical formalism (injective resolutions, derived $\\Hom$, hypercohomology, …) gives an equivalent answer; the derived viewpoint adopted here is simply a way to skip the resolution machinery and work directly with the universal property.\nReferences J. Lurie. Spectral Algebraic Geometry. Book draft. PDF. A. Mathew. The Galois group of a stable homotopy theory. Adv. Math. 291 (2016), 403–541. arXiv:1404.2156. J. Lurie. Higher Topos Theory. Ann. Math. Stud. 170, Princeton Univ. Press, 2009. J. Lurie. Higher Algebra. PDF. ","permalink":"https://ou-liu-red-sugar.github.io/notes/notes/sheaf-cohomology-as-sheafification/","summary":"\u003cp\u003e\u003cem\u003eThis note is not part of the original lecture course; it grew out of\ndiscussions about understanding sheaf cohomology from a derived / animated\nperspective. The treatment follows Lurie\u0026rsquo;s\u003c/em\u003e Spectral Algebraic Geometry\n\u003cem\u003e(\u003ca class=\"citation\" href=\"#ref-lurie-sag\"\u003e[lurie-sag]\u003c/a\u003e\n) and Mathew\u0026rsquo;s work on Galois groups in stable\nhomotopy theory (\u003ca class=\"citation\" href=\"#ref-mathew-galois\"\u003e[mathew-galois]\u003c/a\u003e\n).\u003c/em\u003e\u003c/p\u003e\n\u003ch2 id=\"conventions\"\u003eConventions\u003c/h2\u003e\n\u003cp\u003eThroughout we work in the derived setting and \u003cstrong\u003edrop the $\\mathrm{R}$-prefix\u003c/strong\u003e\non all functors. Concretely:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eAll limits, colimits, and tensor products are derived. The symbol $\\otimes$\ndenotes derived tensor product; the classical tensor product is recovered\nas $\\pi_0(- \\otimes -)$.\u003c/li\u003e\n\u003cli\u003e$\\Gamma(X, \\mathcal{F})$ denotes derived global sections; the classical\n$\\Gamma$ is $H^0(X, \\mathcal{F}) \\coloneqq \\pi_0\\,\\Gamma(X, \\mathcal{F})$.\u003c/li\u003e\n\u003cli\u003e$\\Hom_R(M, N)$ denotes derived $\\Hom$, so\n$\\operatorname{Ext}^i_R(M, N) = \\pi_{-i}\\Hom_R(M, N)$.\u003c/li\u003e\n\u003cli\u003e$\\mathrm{Mod}_R$ denotes the $\\infty$-category of (left) module spectra\nover a connective $\\mathbb{E}_\\infty$-ring $R$ (equivalently,\n$\\mathsf{D}(R)$ when $R$ is discrete).\u003c/li\u003e\n\u003cli\u003ePresheaves take values in a stable presentable category $\\mathcal{D}$ —\ntypically $\\mathsf{D}(\\mathbb{Z})$, $\\mathsf{Sp}$, or $\\mathrm{Mod}_R$ for\na base ring.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThe classical $1$-categorical theory is recovered by passing to $\\pi_0$ at\nthe end.\u003c/p\u003e","title":"Sheaf Cohomology as Sheafification"},{"content":"This page aims to explain how type theory can be understood within the framework of synthetic category theory.\nThe content of this page is derived from my questions to Tashi during the second exercise class and Tashi’s responses. I would like to express my gratitude to Tashi here.\nWe focus on the following two questions:\nQuestion I. How should we understand the notion of isofibration (hereafter referred to as a fibration) in synthetic category theory? Question II. Do we still have a weak factorization system in this context? Next, we will answer these questions through the lens of type theory and the categorical perspective of synthetic category theory.\n\\tableofcontents\nType Theory (More precisely, dependent type theory.)\nI have not formally studied type theory, so the definitions here may differ significantly from those of a type theory expert. Please feel free to correct me if there are any issues.\nContext First, let us discuss what a context is.\nDefinition A context $\\Gamma$ is a finite sequence of typed variables \\[ x_1 : X_1,\\; x_2 : X_2(x_1),\\; \\dots,\\; x_n : X_n(x_1,\\dots,x_{n-1}), \\] satisfying the following condition: for any $1 \\le k \\le n$, we can form the judgment \\[ x_1 : X_1,\\dots,x_{k-1} : X_{k-1}(x_1,\\dots,x_{k-2}) \\;\\vdash\\; X_k(x_1,\\dots,x_{k-1})\\ \\mathrm{Type}. \\] The Empty Context A context of length $0$ is denoted by $[0]$ (sometimes also written as $()$) and is called the empty context.\nThis means that at this stage we have no prior assumptions or restrictions. Hence, for any type $X$ (in particular, a constant type not depending on undefined variables), we have the judgment \\[ [0] \\vdash X \\ \\mathrm{Type}. \\]This also implies that any sequence of length $1$, $(x_1 : X_1)$, forms a valid context if and only if $X_1$ is a type in the empty context.\nIntuitive Interpretation Intuitively, a context is a value-dependent iterative structure:\nFirst, we have a type $X_1$ in the empty context, which may be viewed as a base space. Given a variable $x_1 : X_1$, we can specify $X_2(x_1)$. Strictly speaking, $X_2(x_1)$ is not a single type but a family of types over $X_1$, depending on the value of $x_1$. This is why it is called a dependent type. Fixing $x_1 : X_1$ and then $x_2 : X_2(x_1)$, we may specify $X_3(x_1,x_2)$. This process continues inductively. If for a context $\\Gamma$ we can form the judgment \\[ \\Gamma \\vdash A \\ \\mathrm{Type}, \\] then $A$ is called a type (or dependent type) in the context $\\Gamma$.\nExample Consider the natural number type $\\mathbb{N}$ and the context \\[ \\Gamma = (x : \\mathbb{N}). \\]In this context, we would like to discuss the object “integers modulo $x$”, denoted $\\mathbb{Z}/x$. Clearly, its structure depends on the specific value of $x$. Therefore, $\\mathbb{Z}/x$ is a dependent type in the context $\\Gamma$, and we have the judgment \\[ x : \\mathbb{N} \\vdash \\mathbb{Z}/x \\ \\mathrm{Type}. \\]Geometrically, this corresponds to a fiber bundle over the discrete base space $\\mathbb{N}$:\nwhen $x = 2$, the fiber is $\\mathbb{Z}/2$; when $x = 0$, the fiber is $\\mathbb{Z}$ (if we define $\\mathbb{Z}/0 \\cong \\mathbb{Z}$). Context Extension Given a context $\\Gamma$ and a type $A$ in the context $\\Gamma$, we may form a new context $(\\Gamma, a : A)$.\nIf $\\Gamma$ has the form \\[ x_1 : X_1,\\dots,x_n : X_n, \\] then the extended context encodes the data \\[ x_1 : X_1,\\dots,x_n : X_n,\\ a : A(x_1,\\dots,x_n). \\]This is called the extension of $\\Gamma$ by $A$, and is also written as $\\Gamma.A$.\nGeometrically:\n$\\Gamma$ corresponds to the base space; $A$ corresponds to a fiber bundle over the base; $\\Gamma.A$ corresponds to the total space. Context Induction Contexts in dependent type theory are generated inductively by the following rules:\nBase: the empty context $[0]$ is valid. Induction: if $\\Gamma$ is a valid context and $\\Gamma \\vdash A \\ \\mathrm{Type}$, then the extension $\\Gamma.A$ is also a valid context. Contexts as $\\Sigma$-Types Once $\\Sigma$-types (dependent sums) are available, every context can be encoded as a single type:\nthe empty context corresponds to the unit type $\\mathbf{1}$; a context $(x : X)$ corresponds to $X$; a context $(x : X, y : Y(x))$ corresponds to $\\sum_{x:X} Y(x)$; in general, contexts correspond to iterated $\\Sigma$-types. Terms A term of type $A$ in a context $\\Gamma$ is expressed by the judgment \\[ \\Gamma \\vdash a : A. \\]From various perspectives:\nin type theory, $a$ is a term of $A$; in set theory, $a$ is an element of $A$; in homotopy theory or geometry: in the empty context, $a$ is a point of $A$; in a non-empty context, $a$ is a section of the bundle corresponding to $A$. Concretely, $a$ is a rule assigning to each choice of context variables a point in the corresponding fiber.\n(Synthetic) Category Theory We now consider a category $\\mathcal{C}$, often referred to as the syntactic category or the category of contexts.\nObjects: contexts $\\Gamma$. Morphisms: given contexts $\\Gamma$ and $\\Delta$, a morphism $\\Gamma \\to \\Delta$ is a tuple of terms representing substitution. Composition: substitution of substitutions. Display Maps and Isofibrations Given a context $\\Gamma$ and a type $A$ over it, the extension $\\Gamma.A$ comes with a canonical projection \\[ p : \\Gamma.A \\to \\Gamma. \\]In type theory, this is called a display map.\nIn synthetic category theory, such maps are called isofibrations.\nBase Change and Substitution Let $A \\twoheadrightarrow B$ be an isofibration corresponding to a dependent type $A(b)$ over $B$, \\[ A \\twoheadrightarrow B \\quad\\Longleftrightarrow\\quad \\sum_{b:B} A(b) \\to B. \\]Given any morphism $f : B' \\to B$, substitution in type theory corresponds to pullback in synthetic category theory. Explicitly, the following diagrams describe the same construction:\n\\(\\Leftrightarrow\\) Thus, pullback corresponds to substitution, and the object $A'$ is precisely the total space of the substituted dependent type.\nPath Objects For any object $A$, consider the diagonal morphism $\\Delta : A \\to A \\times A$. This admits a factorization\nHere:\n$p : \\operatorname{Iso}(A) \\twoheadrightarrow A \\times A$ is an isofibration; $r : A \\xrightarrow{\\sim} \\operatorname{Iso}(A)$ is a weak equivalence. The object $\\operatorname{Iso}(A)$ is called the path object of $A$.\nType-Theoretic Semantics In type-theoretic terms:\nThe isofibration $p$ corresponds to the identity type \\[ x : A,\\ y : A \\vdash (x \\simeq y)\\ \\mathrm{Type}. \\] The path object is the total space \\[ \\operatorname{Iso}(A) = \\sum_{x:A} \\sum_{y:A} (x \\simeq y). \\] The map $r$ corresponds to reflexivity, \\[ x \\mapsto (x,x,\\mathrm{refl}_x). \\] Mapping Path Spaces and Weak Factorization For any morphism $f : A \\to B$, we obtain a factorization\nThe intermediate object carries the type-theoretic meaning \\[ b : B \\vdash \\sum_{a:A} (f(a) \\simeq b)\\ \\mathrm{Type}, \\] which is precisely the homotopy fiber of $f$ over $b$.\nThis shows that synthetic category theory admits a weak factorization system, directly mirroring the situation in model categories.\n","permalink":"https://ou-liu-red-sugar.github.io/notes/notes/synthetic-category-theory-and-type-theory/","summary":"\u003cp\u003eThis page aims to explain how \u003cstrong\u003etype theory\u003c/strong\u003e can be understood within the framework of \u003cstrong\u003esynthetic category theory\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eThe content of this page is derived from my questions to Tashi during the second exercise class and Tashi’s responses. I would like to express my gratitude to Tashi here.\u003c/p\u003e\n\u003cp\u003eWe focus on the following two questions:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eQuestion I.\u003c/strong\u003e How should we understand the notion of \u003cstrong\u003eisofibration\u003c/strong\u003e (hereafter referred to as a \u003cem\u003efibration\u003c/em\u003e) in synthetic category theory?\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eQuestion II.\u003c/strong\u003e Do we still have a \u003cstrong\u003eweak factorization system\u003c/strong\u003e in this context?\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eNext, we will answer these questions through the lens of type theory and the categorical perspective of synthetic category theory.\u003c/p\u003e","title":"Synthetic category theory and type theory"},{"content":"Introduction Remark(Conventions). Throughout this talk, we adopt the following conventions:\nImplicit ∞-categories: For the sake of readability, we systematically omit the prefix “∞-” from our terminology. Thus, unless stated otherwise, the term category always refers to an ∞-category. 1-categories: To avoid confusion, we refer to ordinary categories (i.e., those enriched over sets) explicitly as 1-categories. Homological indexing: We use homological indexing for chain complexes. In particular, the differential on any chain complex, $\\partial_n: C_n \\to C_{n-1}$, lowers the degree by $1$. Simplex category: We use $\\Delta$ to denote the standard simplex category. Overview and Structure These notes accompany a talk given at the Goodwillie Calculus Seminar, held in the Winter Term 2025 at the University of Regensburg.\nOur primary objective is two-fold:\nTo provide a self-contained introduction to and proof of the stable Dold–Kan correspondence. To demonstrate its power by applying it to descent theory, specifically relating descendable algebras to nilpotent Adams towers. The notes are organized as follows:\nSection “Classical Dold–Kan”: We begin by warming up with the classical story. We review the correspondence between simplicial objects and chain complexes in additive 1-categories. Section “The stable setup”: We shift to the stable categorical setting. We explain why, in this context, the natural analogue of a chain complex is a $\\mathbb{Z}_{\\ge 0}$-filtration. Chapter “The proof”: This is the technical heart of the talk. We construct the “bridge categories” $\\mathcal{J}$ and $\\mathcal{J}_+$ to prove that simplicial objects are equivalent to filtrations. Appendix “Application to descent”: Finally, we reap the rewards of our hard work. We apply the stable Dold–Kan correspondence to the theory of descendable algebras. We will show how this correspondence translates the difficult problem of descent (convergence of the cobar construction) into a manageable problem of nilpotence (vanishing of the Adams tower), culminating in the descendable Barr–Beck theorem. Review of the Classical Dold–Kan Correspondence We begin with a motivating example from topology.\nConstruction 1. Let $X$ be a topological space. We can construct a simplicial set $\\operatorname{Sing}_{\\bullet}(X)$ as follows: for $[n] \\in \\Delta$, define $\\operatorname{Sing}_n(X) \\coloneqq \\operatorname{Hom}_{\\mathsf{Top}}(\\Delta^n_{\\operatorname{top}}, X)$, where $\\Delta^n_{\\operatorname{top}}$ denotes the topological $n$-simplex. This defines a functor \\[ \\operatorname{Sing} \\colon \\mathsf{Top} \\to \\mathsf{sSet}. \\] This functor induces an equivalence between classical homotopy theory and simplicial homotopy theory: \\[ \\mathsf{Top}[\\text{weak homotopy equivalence}^{-1}] \\simeq \\mathsf{An}. \\]However, for computational purposes, we often want to “linearize” this homotopy data. We can form the free abelian group generated by the $n$-simplices, denoted $\\mathbb{Z} \\operatorname{Sing}_n (X)$, obtaining a simplicial abelian group $\\mathbb{Z} \\operatorname{Sing}_{\\bullet} (X)$. While this object captures the homology of $X$, it contains redundant information: the degeneracy maps merely repeat lower-dimensional data, and the full collection of face maps is unwieldy.\nTo extract the homological data efficiently, we pass to the singular chain complex $\\operatorname{C}_*(X)$:\nFor each $n \\ge 0$, let $\\operatorname{C}_n(X) \\coloneqq \\mathbb{Z} \\operatorname{Sing}_n(X)$. The differential $\\partial_n \\colon \\operatorname{C}_n(X) \\to \\operatorname{C}_{n-1}(X)$ is defined as the alternating sum of the face maps: \\[ \\partial_n \\coloneqq \\sum_{i=0}^n (-1)^i d_i^n, \\] where $d_i^n$ is the $i$-th face map. This yields a functor \\[ \\operatorname{C}_* \\colon \\mathsf{sAb} \\to \\mathsf{Ch}_{\\geq 0}(\\mathsf{Ab}). \\] While this functor captures the correct homology, it is not an equivalence of categories because it retains the degenerate simplices in its object definition. However, if we quotient out the degenerate simplices, we obtain the normalized chain complex $\\operatorname{N}_*(X)$. The celebrated Dold–Kan correspondence asserts that this normalization functor is an equivalence of categories. Thus, in a very precise sense, singular homology theory is the linearization of the homotopy theory of topological spaces.\nMore formally, this correspondence establishes a fundamental relationship between connective chain complexes1 and simplicial objects in an idempotent-complete additive 1-category $\\mathcal{A}$.\nGiven a simplicial object $X_{\\bullet}$ in $\\mathcal{A}$, one can construct its unnormalized chain complex (or Moore complex), denoted $\\operatorname{C}_*(X)$, as follows:\nThe object in degree $n$ is simply $\\operatorname{C}_n(X) \\coloneqq X_n$. The differential $\\partial_n \\colon \\operatorname{C}_n(X) \\to \\operatorname{C}_{n-1}(X)$ is the alternating sum of face maps: \\[ \\partial_n \\coloneqq \\sum_{i=0}^n (-1)^i d_i^n. \\] This complex is “too large” as it contains redundant information from degenerate simplices. Let $\\operatorname{D}_n(X)$ be the subobject of $X_n$ generated by all degenerate $n$-simplices. A more efficient representation is given by the normalized chain complex, denoted $\\operatorname{N}_*(X)$, defined by $\\operatorname{N}_n(X) \\coloneqq \\bigcap_{i=1}^{n} \\ker(d_i^n)$. A fundamental result states that there is a canonical isomorphism $\\operatorname{N}_n(X) \\xrightarrow{\\sim} \\operatorname{C}_n(X)/\\operatorname{D}_n(X)$, and the inclusion $\\operatorname{N}_*(X) \\hookrightarrow \\operatorname{C}_*(X)$ is a quasi-isomorphism.\nConversely, we can construct a simplicial object from a connective chain complex $C_* \\in \\operatorname{Ch}(\\mathcal{A})_{\\geq 0}$. First, we form a semi-simplicial object $C_{\\bullet,\\operatorname{inj}}$: where the arrow corresponding to the $i$-th coface operator $\\delta_n^i \\colon [n-1] \\hookrightarrow [n]$ is $\\partial$ if $i=0$ (or specific indices depending on convention) and zero otherwise.\nWe then obtain the corresponding simplicial object via the left Kan extension along the inclusion $\\Delta_{\\operatorname{inj}} \\hookrightarrow \\Delta$: We refer to this functor $\\operatorname{DK}$ as the Dold–Kan construction. Intuitively, this process “freely adds” the necessary degenerate simplices to the chain complex.\nTheorem 2 (Classical Dold–Kan correspondence). Let $\\mathcal{A}$ be an additive 1-category. The functor \\[ \\operatorname{DK} \\colon \\operatorname{Ch}(\\mathcal{A})_{\\geq 0} \\to \\operatorname{Fun}(\\Delta^{\\operatorname{op}},\\mathcal{A}) \\] is fully faithful. Furthermore, if $\\mathcal{A}$ is idempotent-complete, then $\\operatorname{DK}$ and the normalization functor $\\operatorname{N}_*$ constitute an equivalence of categories.\nProof. We verify this by reducing it to the abelian case. Recall that for $\\mathcal{A} = \\mathsf{Ab}$, the classical Dold–Kan correspondence holds (c.f. [HA, Lemma 1.2.3.13] ).\nNow, let $\\mathcal{A}$ be a general additive 1-category. Consider the category of additive presheaves $\\mathcal{A}' = \\operatorname{Fun}(\\mathcal{A}^{\\operatorname{op}},\\mathsf{Ab})$. Via the additive Yoneda embedding $y \\colon \\mathcal{A} \\to \\mathcal{A}'$, we can embed $\\mathcal{A}$ into an abelian category. Since $y$ preserves finite limits and colimits, the following diagram commutes up to canonical isomorphism:\nNow consider the bottom map. By the exponential law (or currying), we have $\\mathsf{Ch}(\\mathcal{A}')_{\\geq 0} \\cong \\operatorname{Fun}(\\mathcal{A}^{\\operatorname{op}}, \\mathsf{Ch}(\\mathsf{Ab})_{\\geq 0})$. Thus, the bottom functor corresponds to post-composition with the classical equivalence $\\operatorname{DK}_{\\mathsf{Ab}}$. Since post-composition with an equivalence is an equivalence, the bottom map is an equivalence.\nSince $y$ is fully faithful, it follows that the top $\\operatorname{DK}$ functor is fully faithful. The essential surjectivity in the idempotent-complete case follows from the splitting of $\\operatorname{N}_n$ as a direct summand.\n$\\square$ What is the Stable Dold–Kan Correspondence? We now shift our focus to the setting of higher category theory. In modern homotopy theory, stable categories serve as the higher categorical analogue of abelian categories.\nConsider the homotopy category $\\operatorname{h}\\mathcal{C}$ of a stable category $\\mathcal{C}$. While $\\operatorname{h}\\mathcal{C}$ is typically not abelian, it carries a triangulated structure. This implies two key properties:\n$\\operatorname{h}\\mathcal{C}$ is an additive 1-category. It satisfies the following splitting property: $(*)$ If $f \\colon X \\to Y$ is a morphism in $\\operatorname{h}\\mathcal{C}$ which admits a left inverse, then there is an isomorphism $Y \\simeq X \\oplus X'$ such that $f$ is identified with the inclusion into the first factor. This property guarantees that the classical Dold–Kan correspondence applies perfectly to $\\operatorname{h}\\mathcal{C}$ (allowing us to identify chain complexes in $\\operatorname{h}\\mathcal{C}$ with simplicial objects).\nHowever, the stable correspondence is a much deeper statement at the full categorical level. The key insight is to relate simplicial objects to $\\mathbb{Z}_{\\geq 0}$-filtrations. A functor $F_{\\star} \\colon \\mathbb{Z}_{\\geq 0} \\to \\mathcal{C}$ represents a tower of objects: \\[ F_0 \\to F_1 \\to F_2 \\to \\cdots \\] Construction 3. Let $F_{\\star}$ be a filtration in stable category $\\mathcal{C}$, then for $s \\in \\Z$, one can define the $s$-th associative graded piece of $F_{\\star}$ to be the cofiber \\[ \\operatorname{gr}_{F}^{s} \\coloneqq \\operatorname{cofib}(F_{s} \\to F_{s+1}) = \\frac{F_{s+1}}{F_s}. \\] In [HA, Remark 1.2.2.3] , one can find that if $F_{\\star}$ is a filtration, then the graded objects form a kind of chain complex. Specifically, each fiber sequence \\[ \\operatorname{gr}^{s} \\to \\frac{F_{s+2}}{F_s} \\to \\operatorname{gr}^{s+1} \\] gives rise to a ‘differential’ $d \\colon \\operatorname{gr}^{s+1} \\to \\operatorname{gr}^s[1]$.\nThus, one obtain a chain complex \\[ \\cdots \\to \\operatorname{gr}^{2}[-2] \\to \\operatorname{gr}^1[-1] \\to \\operatorname{gr}^0 \\to \\operatorname{gr}^{-1}[1] \\to \\operatorname{gr}^{-2}[2] \\to \\cdots \\]Using the classical Dold–Kan correspondence on $\\operatorname{h}\\mathcal{C}$, this chain complex determines a simplicial object. The stable Dold–Kan correspondence asserts that this relationship lifts to an equivalence of $\\infty$-categories:\nTheorem 4 (Stable Dold–Kan correspondence). Let $\\mathcal{C}$ be a stable category. Then there exists an equivalence of categories: \\[ \\operatorname{Fun}(\\Delta^{\\operatorname{op}},\\mathcal{C}) \\simeq \\operatorname{Fun}(\\mathbb{Z}_{\\geq 0},\\mathcal{C}). \\] Acknowledgements I would like to express my sincere gratitude to the organizer, Prof. Marc Hoyois, for the opportunity to give this talk at the Goodwillie Calculus Seminar. I am also indebted to Yuchen Wu for providing the key insight regarding the model-independent proof of , which greatly clarified the combinatorial arguments. Finally, I thank Gemini for assisting with grammatical corrections and polishing the text.\nTechnical Lemmas In the proof of the stable Dold–Kan correspondence (Chapter “Stable-DK”), we relied on a crucial identification between left and right Kan extensions. The goal of this chapter is to provide a rigorous, model-independent proof of this fact.\nLemma 5 (Kan extension equivalence). Let $\\mathcal{C}$ be a stable category, let $n \\geq 0$, and let $F \\colon \\Delta_{+,\\leq n}^{\\operatorname{op}} \\to \\mathcal{C}$ be a functor. The following conditions are equivalent:\nThe functor $F$ is a left Kan extension of its restriction $F|_{\\Delta_{\\leq n}^{\\operatorname{op}}}$. The functor $F$ is a right Kan extension of its restriction $F|_{\\Delta_{+,\\leq n-1}^{\\operatorname{op}}}$. To prove this, we need to analyze the combinatorics of the simplex category. We adopt the strategy of Yuchen Wu, which avoids the use of topological barycentric subdivisions.\nCoinitial We first establish a general lemma regarding the contractibility of unions of posets.\nLemma 6 (Union of Contractible Posets). Let $V$ be a poset and let $\\mathcal{F} = \\{U_1, \\dots, U_m\\}$ be a non-empty finite collection of subposets of $V$. Suppose that:\nCovering: $V = U_1 \\cup \\cdots \\cup U_m$. Downward Closure Compatibility: For any $a, b \\in V$ with $a \\le b$, if $a \\in U_i$ and $b \\in U_j$, then either $a, b \\in U_i$ or $a, b \\in U_j$. (This holds in particular if each $U_i$ is a downward closed subposet). Intersection Contractibility: Any non-empty intersection of elements in $\\mathcal{F}$ is a weakly contractible subposet of $V$. Then $V$ itself is weakly contractible.\nProof. We proceed by induction on $m$. The case $m=1$ is trivial. Assume the statement holds for $m \\le k$. Let $W_k = \\bigcup_{i=1}^k U_i$. Consider the pushout $P$ in $\\mathsf{Cat}$ of the span:\nWe claim that $P$ is equivalent to $V$. By condition (2), the inclusion functors in the span are fully faithful. As shown in [haine2025fullyfaithfulfunctorspushouts] , fully faithful inclusions ensure that the categorical pushout behaves well:\nBy condition (1) and (2), the pushout $P$ has the same underlying set (anima) as the union $V = W_k \\cup U_{k+1}$. The mapping animae in $P$ agree with those in $V$: for $a, b \\in V$, if $a \\le b$, the mapping space is contractible; otherwise, it is empty. Thus, $P \\simeq V$.\nNow, by the inductive hypothesis, $W_k$ is weakly contractible. The intersection $W_k \\cap U_{k+1} = \\bigcup_{i=1}^k (U_i \\cap U_{k+1})$ satisfies the conditions of the lemma for the collection $\\{U_i \\cap U_{k+1}\\}$, so it is also weakly contractible. Since $U_{k+1}$ is contractible by (3), $V$ (being the homotopy pushout of contractible spaces) is weakly contractible.\n$\\square$ Now we apply this to the specific geometry of the simplex category.\nLemma 7. Let $\\Delta^{\\operatorname{inj}}$ be the subcategory of $\\Delta$ consisting of injective maps. The functor \\[ \\iota \\colon \\Delta^{\\operatorname{inj}}_{/[n]} \\to \\Delta_{\\leq n} \\] defined by sending $([m], [m] \\hookrightarrow [n])$ to $[m]$ is coinitial.\nProof. By Quillen\u0026rsquo;s Theorem A, it suffices to show that for any $[k] \\in \\Delta_{\\leq n}$, the slice category \\[ Q_{n,k} \\coloneqq \\Delta^{\\operatorname{inj}}_{/[n]} \\times_{\\Delta_{\\leq n}} (\\Delta_{\\leq n})_{/[k]} \\] is weakly contractible.\nExplicitly, objects in $Q_{n,k}$ are pairs $(\\beta, f)$, where $\\beta \\colon [m] \\hookrightarrow [n]$ is an injective map and $f \\colon [m] \\to [k]$ is a map in $\\Delta_{\\leq n}$. Since $\\beta$ is injective, it is uniquely determined by its image $I \\subseteq [n]$. Thus, we can identify objects of $Q_{n,k}$ with pairs $(I, f)$ where $I \\subseteq [n]$ is a subset and $f \\colon I \\to [k]$ is an order-preserving map (where we identify $I$ with $[|I|-1]$ via the unique order isomorphism).\nLet $O_{n,k}$ be the set of all maps $[n] \\to [k]$. We equip $O_{n,k}$ with the alphabetic order $\\le$ (a reverse lexicographical order): we say $g \u003c h$ if there exists some $i \\in [n]$ such that \\[ g(n)=h(n), \\dots, g(i+1)=h(i+1) \\quad \\text{and} \\quad g(i) \u003c h(i). \\] This defines a total order on $O_{n,k}$.\nFor each $\\phi \\in O_{n,k}$, let $U_{\\phi} \\subseteq Q_{n,k}$ be the subposet of elements $(I, f)$ such that the composite map $([n] \\twoheadrightarrow I \\xrightarrow{f} [k])$ is $\\le \\phi$ in the pointwise order. Each $U_{\\phi}$ has a terminal object (the pair corresponding to the largest subset $I$ compatible with the constraints) and is thus contractible.\nWe filter $Q_{n,k}$ by the subposets $V_{\\phi} \\coloneqq \\bigcup_{\\psi \\le \\phi} U_{\\psi}$. We prove that each $V_{\\phi}$ is contractible by induction on the alphabetic order.\nBase case: $V_{(0,\\dots,0)}$ is contractible. Inductive step: Let $\\phi$ be a map and $\\phi' = \\phi + 1$ be its successor in the alphabetic order. We have a pushout square: It suffices to show the intersection $V_{\\phi} \\cap U_{\\phi'}$ is contractible.\nAnalyzing the intersection:\nLet $i$ be the largest index such that $\\phi(i) \u003c k$ (the pivot for the successor). Then the successor $\\phi'$ is given by: \\[ \\phi'(t) = \\begin{cases} \\phi(t) \u0026 \\text{if } t \u003e i \\\\ \\phi(t) + 1 \u0026 \\text{if } t = i \\\\ 0 \u0026 \\text{if } t \u003c i. \\end{cases} \\] The intersection $V_{\\phi} \\cap U_{\\phi'}$ consists of pairs $(I, f)$ compatible with $\\phi'$ that are also pointwise $\\le \\psi$ for some $\\psi \\le \\phi$. As shown in Wu, this intersection decomposes nicely as a union of principal ideals: \\[ V_{\\phi} \\cap U_{\\phi'} = \\bigcup_{c \\in S(\\phi)} P_{\\le ([n] \\setminus \\{c\\})}, \\] where $S(\\phi) \\subseteq [n]$ is a specific set of indices determined by the descent of $\\phi'$. An index $c$ belongs to $S(\\phi)$ if and only if $c=i$, or $c \u003e i$ and $\\phi'(c-1) \u003c \\phi'(c)$.\nCrucially, since $k \\le n$, the set $S(\\phi)$ is not the entire set $[n]$. Thus, the total intersection of these principal ideals corresponds to the ideal generated by $[n] \\setminus S(\\phi)$, which is non-empty.\nTherefore, the collection of ideals $\\{ P_{\\le ([n] \\setminus \\{c\\})} \\}_{c \\in S(\\phi)}$ satisfies the conditions of Lemma “Union of Contractible Posets” (any intersection is a principal ideal, hence contractible). We conclude that $V_{\\phi} \\cap U_{\\phi'}$ is contractible, and by induction, $Q_{n,k}$ is contractible.\n$\\square$ The Cube Lemma in Stable Categories Definition 8 (Cubes in a stable category). Let $\\mathcal{C}$ be a stable category. A cube is a functor $D \\colon \\mathcal{P}(S) \\to \\mathcal{C}$ for some finite set $S$.\nWe say $D$ is cartesian (or a limit diagram) if the object $D(\\emptyset)$ is the limit of $D|_{\\mathcal{P}(S) \\setminus \\{\\emptyset\\}}$. We say $D$ is cocartesian (or a colimit diagram) if the object $D(S)$ is the colimit of $D|_{\\mathcal{P}(S) \\setminus \\{S\\}}$. Lemma 9 (The cube lemma). Let $\\mathcal{C}$ be a stable category and $D \\colon \\mathcal{P}(S) \\to \\mathcal{C}$ a cube. The following are equivalent:\n$D$ is cartesian. $D$ is cocartesian. Proof. This is a standard result in higher algebra. The core idea is to define the total fiber (denoted $\\operatorname{tfib}(D)$) and the total cofiber (denoted $\\operatorname{tcof}(D)$).\n$D$ is cartesian $\\iff \\operatorname{tfib}(D) \\simeq 0$. $D$ is cocartesian $\\iff \\operatorname{tcof}(D) \\simeq 0$. In a stable category, there is a natural equivalence $\\operatorname{tfib}(D) \\simeq \\operatorname{tcof}(D)[-|S|]$, where $|S|$ is the cardinality of $S$. Thus, the vanishing of one implies the vanishing of the other. $\\square$ Proof of Lemma “Kan extension equivalence” Proof. Let $F \\colon \\Delta_{+,\\leq n}^{\\operatorname{op}} \\to \\mathcal{C}$ be the functor. We interpret the two conditions:\n1. Analysis of the Left Kan Extension.\nCondition (1) states that $F$ is a left Kan extension at the object $[-1]$. By definition, this means \\[ F([-1]) \\simeq \\operatorname{colim}_{\\alpha: [k] \\to [-1]} F([k]), \\] where the colimit is over the slice category $\\Delta_{\\le n}^{\\operatorname{op}}$. By Lemma “Coinitiality of the Injective Slice”, the inclusion $\\Delta^{\\operatorname{inj}}_{/[n]} \\to \\Delta_{\\leq n}$ is coinitial. Note that $\\Delta^{\\operatorname{inj}}_{/[n]}$ is isomorphic to $\\mathcal{P}([n]) \\setminus \\{\\emptyset\\}$. Thus, condition (1) is equivalent to saying that the restriction of $F$ to the $(n+1)$-cube $\\mathcal{P}([n])$ is a cocartesian (where the colimit cone is the value at $\\emptyset \\subseteq [n]$, corresponding to $[-1]$).\n2. Analysis of the Right Kan Extension.\nCondition (2) states that $F$ is a right Kan extension at the object $[n]$. The relevant index category is the slice of $\\Delta_{+, \\le n-1}^{\\operatorname{op}}$ under $[n]$, which essentially corresponds to $\\Delta_{\\leq n}$ (mapping into $[n]$). Using Lemma “Coinitiality of the Injective Slice” again (in the dual or shifted context), we can identify this limit with a limit over the same combinatorial structure $\\Delta^{\\operatorname{inj}}_{/[n]} \\cong \\mathcal{P}([n])$. Thus, condition (2) is equivalent to saying that the restriction of $F$ to the cube is a limit diagram (where the limit cone is the value at $[n]$).\n3. Conclusion.\nWe have identified both conditions with properties of a single cube diagram formed by restricted values of $F$.\nLeft Kan Extension $\\iff$ The cube is cocartesian. Right Kan Extension $\\iff$ The cube is cartesian. By the Cube Lemma (stability), these two conditions are equivalent. $\\square$ The Proof of the Stable Dold–Kan Correspondence In this chapter, we provide a complete proof of the stable Dold–Kan correspondence.\nOur proof strategy relies on constructing a “bridge” between the two worlds. Specifically, we will:\nConstruct a chain of functors connecting the category of simplicial objects, $\\operatorname{Fun}(\\Delta^{\\operatorname{op}}, \\mathcal{C})$, with the category of filtered objects, $\\operatorname{Fun}(\\mathbb{Z}_{\\geq 0}, \\mathcal{C})$. Show that every functor in this chain is an equivalence of categories. Preliminaries: The Skeletal Filtration Before diving into the formal construction of the bridge categories, let us ground our intuition in the geometry of simplicial objects. This will explain why the proof takes the form it does.\nRecall that for any simplicial object $X_{\\bullet} \\in \\operatorname{Fun}(\\Delta^{\\operatorname{op}}, \\mathcal{C})$, we can define its $n$-skeleton $\\operatorname{sk}_n X$. Categorically, this is the Left Kan Extension of the restriction of $X$ to the truncated category $\\Delta_{\\le n}^{\\operatorname{op}}$ along the inclusion: \\[ \\operatorname{sk}_n X \\coloneqq \\operatorname{Lan}_{\\Delta_{\\le n}^{\\operatorname{op}} \\hookrightarrow \\Delta^{\\operatorname{op}}} (X|_{\\Delta_{\\le n}^{\\operatorname{op}}}). \\] The sequence of skeletons provides a natural filtration of $X$: \\[ \\operatorname{sk}_0 X \\to \\operatorname{sk}_1 X \\to \\operatorname{sk}_2 X \\to \\cdots \\to \\operatorname{colim}_n \\operatorname{sk}_n X \\simeq X. \\]The Geometric Intuition Why is this relevant to Dold–Kan? In the classical case (e.g., simplicial sets), $\\operatorname{sk}_n X$ is obtained from $\\operatorname{sk}_{n-1} X$ by attaching non-degenerate $n$-simplices via a pushout square: Step 1: Building the Bridge Categories To make the skeletal intuition precise, we introduce two index categories: $\\mathcal{J}$ and $\\mathcal{J}_+$.\nThe Category $\\mathcal{J}_+$ We define $\\mathcal{J}_+$ as the full subcategory of $\\mathbb{Z}_{\\geq 0} \\times \\Delta^{\\operatorname{op}}_+$ spanned by pairs $(n,[m])$ satisfying $m \\leq n$. Here, $\\Delta^{\\operatorname{op}}_+$ is the augmented simplex category (including $[-1]$).\nThe first coordinate $n \\in \\mathbb{Z}_{\\ge 0}$ represents the filtration stage (related to the $n$-skeleton). The second coordinate $[m] \\in \\Delta^{\\operatorname{op}}_+$ represents the simplicial degree. Intuitively, an object $(n,[m])$ corresponds to the term $(X_m)$ sitting inside the $n$-skeleton. The condition $m \\leq n$ reflects that the $n$-skeleton is determined by simplices of dimension up to $n$.\nThe picture to have in mind is a large commutative diagram: The Category $\\mathcal{J}$ and the Functor Chain In parallel, we define $\\mathcal{J}$ as the full subcategory of $\\mathcal{J}_+$ spanned by pairs $(n,[m])$ where $0 \\leq m \\leq n$ (excluding the bottom row $m=-1$). This encodes the skeleton data without the geometric realization.\nWe define $\\operatorname{Fun}^0(\\mathcal{J},\\mathcal{C})$ to be the full subcategory of $\\operatorname{Fun}(\\mathcal{J},\\mathcal{C})$ spanned by functors $F$ satisfying the following stability condition:\nFor every $s \\leq m \\leq n$, the image of the natural map $(m,[s]) \\to (n,[s])$ is an equivalence in $\\mathcal{C}$. Similarly, we define $\\operatorname{Fun}^0(\\mathcal{J}_+,\\mathcal{C})$ for functors $F_+ \\colon \\mathcal{J}_+ \\to \\mathcal{C}$ satisfying the same stability condition. Additionally, we require that $F_+$ is a Left Kan Extension of its restriction to $\\mathcal{J}$. This condition formally encodes that the bottom row objects $(n, [-1])$ are geometric realizations (colimits) of the columns above them.\nThis setup yields a diagram of categories: \\[ \\operatorname{Fun}(\\Delta^{\\operatorname{op}},\\mathcal{C}) \\xrightarrow{G} \\operatorname{Fun}^0(\\mathcal{J},\\mathcal{C}) \\xleftarrow{G'} \\operatorname{Fun}^0(\\mathcal{J}_+,\\mathcal{C}) \\xrightarrow{G''} \\operatorname{Fun}(\\mathbb{Z}_{\\geq 0},\\mathcal{C}). \\] Here:\n$G$ is induced by the projection $p \\colon \\mathcal{J} \\to \\Delta^{\\operatorname{op}}$. $G'$ is the restriction functor. $G''$ is the restriction to the bottom row $\\mathbb{Z}_{\\geq 0} \\hookrightarrow \\mathcal{J}_+$ via $n \\mapsto (n,[-1])$. Our goal is to show that $G$, $G'$, and $G''$ are equivalences.\nStep 2: Proving the Equivalences The Functor $G$ is an equivalence We first show that $G$ is an equivalence. The strategy is to express $G$ as a limit of equivalences $G_k$.\nDefine the “truncated” index categories:\n$\\mathcal{J}^{\\leq k}$: the full subcategory of $\\mathcal{J}$ spanned by pairs $(n,[m])$ where $m \\leq n \\leq k$. $\\mathcal{J}^{k}$: the full subcategory of $\\mathcal{J}$ spanned by pairs $(n,[m])$ where $m \\leq n = k$. Note that the projection $p$ restricts to an equivalence $\\mathcal{J}^k \\simeq \\Delta_{\\leq k}^{\\operatorname{op}}$.\nWe aim to show an equivalence: \\[ \\operatorname{Fun}^0(\\mathcal{J}^{\\leq k},\\mathcal{C}) \\xrightarrow{\\sim} \\operatorname{Fun}(\\mathcal{J}^k,\\mathcal{C}). \\] Consider the right Kan extension along the fully faithful inclusion $\\iota \\colon \\mathcal{J}^{k} \\hookrightarrow \\mathcal{J}^{\\leq k}$: This induces a fully faithful functor $\\iota_* \\colon \\operatorname{Fun}(\\mathcal{J}^{k},\\mathcal{C}) \\to \\operatorname{Fun}(\\mathcal{J}^{\\leq k},\\mathcal{C})$. It remains to identify the essential image of $\\iota_*$ with $\\operatorname{Fun}^0(\\mathcal{J}^{\\leq k},\\mathcal{C})$.\nFor any $s \\leq m \\leq n \\leq k$, observe that the under-categories satisfy $\\mathcal{J}^k_{(m,[s])/} \\simeq \\mathcal{J}^k_{(n,[s])/}$. Consequently, the limit diagrams defining the right Kan extension at $(m,[s])$ and $(n,[s])$ are isomorphic. Since $\\mathcal{C}$ is stable (and thus admits finite limits), the pointwise formula for the Right Kan Extension implies that the map \\[ \\operatorname{Ran}_{\\iota}H ((m,[s])) \\to \\operatorname{Ran}_{\\iota}H ((n,[s])) \\] is an equivalence. Thus, the image of $\\iota_*$ lies in $\\operatorname{Fun}^0(\\mathcal{J}^{\\leq k},\\mathcal{C})$. Conversely, for any $F \\in \\operatorname{Fun}^0(\\mathcal{J}^{\\leq k},\\mathcal{C})$, one can check that $F \\simeq \\operatorname{Ran}_{\\iota}(F|_{\\mathcal{J}^k})$.\nThus, we obtain a sequence of equivalences: \\[ G_k \\colon \\operatorname{Fun}(\\Delta_{\\leq k}^{\\operatorname{op}},\\mathcal{C}) \\xrightarrow{\\sim} \\operatorname{Fun}(\\mathcal{J}^{k},\\mathcal{C}) \\xrightarrow{\\sim} \\operatorname{Fun}^0(\\mathcal{J}^{\\leq k},\\mathcal{C}). \\] Taking the limit as $k \\to \\infty$, we have $G \\simeq \\lim_{k} G_k$. Since a limit of equivalences is an equivalence, $G$ is an equivalence.\nThe Functor $G'$ is an equivalence The inclusion $\\mathcal{J} \\hookrightarrow \\mathcal{J}_+$ is fully faithful. For any $(n, [-1]) \\in \\mathcal{J}_+$, the slice category $\\mathcal{J}_{/(n,[-1])}$ is finite. Since $\\mathcal{C}$ admits finite colimits, the Left Kan Extension exists. By definition, $\\operatorname{Fun}^0(\\mathcal{J}_+,\\mathcal{C})$ consists precisely of those functors which are Left Kan extensions of their restriction to $\\mathcal{J}$. Since fully faithful embeddings induce fully faithful restriction functors onto the subcategory of Kan extensions, $G'$ is an equivalence.\nThe Functor $G''$ is an equivalence Finally, we show that $G''$ is an equivalence. This is the subtlest part, which relies on our Technical Lemma.\nLet $\\mathcal{J}_+^{\\leq k}$ be the full subcategory of $\\mathcal{J}_+$ spanned by $(n,[m])$ where either $m \\leq n \\leq k$ or $m = -1$. Let $\\mathcal{D}(k) \\coloneqq \\operatorname{Fun}^0(\\mathcal{J}_+^{\\leq k}, \\mathcal{C})$ be the category of functors satisfying the standard stability condition and the Left Kan Extension condition at $(n,[-1])$ for $n \\le k$.\nWe have a limit decomposition: \\[ \\operatorname{Fun}^0(\\mathcal{J}_+,\\mathcal{C}) \\simeq \\lim \\left( \\cdots \\to \\mathcal{D}(k) \\to \\mathcal{D}(k-1) \\to \\cdots \\to \\mathcal{D}(-1) \\right), \\] where $\\mathcal{D}(-1) \\simeq \\operatorname{Fun}(\\mathbb{Z}_{\\geq 0}, \\mathcal{C})$. It suffices to show that the restriction map $\\mathcal{D}(k) \\to \\mathcal{D}(k-1)$ is an equivalence for all $k \\geq 0$.\nWe decompose this restriction into two steps: \\[ \\mathcal{D}(k) \\xrightarrow{\\theta} \\mathcal{D}'(k) \\xrightarrow{\\theta'} \\mathcal{D}(k-1). \\] Here, $\\mathcal{D}'(k)$ is defined on the domain $\\mathcal{J}_{0}^{\\leq k} \\coloneqq \\mathcal{J}_+^{\\leq k} \\setminus \\{(k,[k])\\}$.\nThe map $\\theta'$: A functor in $\\mathcal{D}'(k)$ is determined by its restriction to $\\mathcal{J}_+^{\\leq k-1}$ plus the Left Kan extension condition at $(k,[-1])$. Since the Kan extension is unique, $\\theta'$ is an equivalence. The map $\\theta$: This restricts a functor from $\\mathcal{J}_+^{\\leq k}$ to $\\mathcal{J}_0^{\\leq k}$. To show this is an equivalence, we need to show that the value at the missing point $(k,[k])$ is uniquely determined. By definition of $\\mathcal{D}(k)$, any $F \\in \\mathcal{D}(k)$ is a Left Kan Extension of $F|_{\\mathcal{J}^{\\leq k}}$. Observe that $\\mathcal{J}^k \\simeq \\Delta_{\\leq k}^{\\operatorname{op}}$ is coinitial in the diagram computing this Kan extension at $(k,[-1])$.\nCrucially, we invoke our Technical Lemma (“Kan extension equivalence”): Since $\\mathcal{C}$ is stable, being a Left Kan Extension from $\\Delta_{\\leq k}^{\\operatorname{op}}$ is equivalent to being a Right Kan Extension from $\\Delta_{+,\\leq k-1}^{\\operatorname{op}} \\simeq \\mathcal{J}_+^{k-1}$.\nNote that $\\mathcal{J}_+^{k-1} \\subseteq \\mathcal{J}_0^{\\leq k}$. This means the value at $(k,[k])$ is determined by a Right Kan extension from data we already possess in $\\mathcal{D}'(k)$. Thus, $\\theta$ is an equivalence.\nSince both steps are equivalences, $G''$ is an equivalence.\nAlternative Perspective on $G$ There is a more high-level way to see that $G$ is an equivalence using the language of localizations.\nWe can regard $\\operatorname{Fun}^0(\\mathcal{J},\\mathcal{C})$ as the functor category $\\operatorname{Fun}(\\mathcal{J}[W^{-1}],\\mathcal{C})$, where $W$ is the set of morphisms $\\{(m,[s]) \\to (n,[s]) \\mid s\\leq m \\leq n\\}$ that we require to be inverted.\nConsider the forgetful functor $p' \\colon \\mathcal{J} \\to \\mathbb{Z}_{\\geq 0}$ given by $(m,[n]) \\mapsto m$. Observe that $p'$ is a cocartesian fibration, and $W$ is precisely the collection of $p'$-cocartesian morphisms. By the fundamental theorem of $\\infty$-categorical colimits (or the description of localizations of cocartesian fibrations), we have an equivalence: \\[ \\mathcal{J}[W^{-1}] \\simeq \\operatorname{colim}_{n \\in \\mathbb{Z}_{\\ge 0}} (\\mathcal{J}_n), \\] where $\\mathcal{J}_n$ is the fiber over $n$. Since each fiber $\\mathcal{J}_n \\simeq \\Delta_{\\leq n}^{\\operatorname{op}}$, the colimit is precisely $\\Delta^{\\operatorname{op}}$. Thus, $\\operatorname{Fun}^0(\\mathcal{J},\\mathcal{C}) \\simeq \\operatorname{Fun}(\\Delta^{\\operatorname{op}}, \\mathcal{C})$.\nApplication: Descent and Comonadicity In the final part of this talk, we apply the stable Dold–Kan correspondence to prove a fundamental result in stable homotopy theory: the descendable Barr–Beck theorem. This result provides a powerful criterion for recovering a category $\\mathcal{C}$ from the category of modules over a “nice” algebra $A$.\nUnless otherwise specified, let $\\mathcal{C}$ be a stable symmetric monoidal category where the tensor product preserves colimits.\nThe Descent Problem Let $A \\in \\mathsf{CAlg}(\\mathcal{C})$ be a commutative algebra. We have a standard adjunction: The central question of descent theory is: When is the comparison functor from $\\mathcal{C}$ to the category of coalgebras over the associated comonad an equivalence? In geometric terms, this asks if the map $A \\to A \\otimes A$ satisfies effective descent.\nAccording to the Lurie–Barr–Beck theorem, this equivalence holds if and only if two conditions are met:\nConservativity: The functor $-\\otimes A$ is conservative (i.e., it reflects equivalences). Convergence: For any cosimplicial object split by $-\\otimes A$, the totalization converges to the original object. Descendable Algebras We focus on a broad class of algebras where “convergence” is guaranteed by “nilpotence”.\nDefinition 10 (Descendable). For every object $A \\in \\mathcal{C}$, we denote by $\\langle A \\rangle \\subseteq \\mathcal{C}$ (or $\\operatorname{Thick}^{\\otimes}(A)$) the smallest full subcategory containing $A$ which is stable under:\nFinite limits and colimits (cofiber sequences), Retracts, Tensor products with arbitrary objects of $\\mathcal{C}$. We say that a commutative algebra $A$ is descendable if $\\mathbb{1}_{\\mathcal{C}}$ is in $\\langle A \\rangle$. Key Tool: Stable Dold–Kan The convergence condition involves the cobar construction $\\operatorname{CB}^{\\bullet}(A)$: Checking whether $\\operatorname{Tot}(\\operatorname{CB}^{\\bullet}(A)) \\simeq \\mathbb{1}$ is typically difficult.\nHowever, the stable Dold–Kan correspondence allows us to translate this cosimplicial problem into a much simpler filtration problem.\nLet $A \\in \\mathsf{Alg}(\\mathcal{C})$ be an associative algebra of $\\mathcal{C}$. Construction 11 (Adams Tower). Let $M \\in \\mathcal{C}$ be an object. We can form a tower in $\\mathcal{C}$ \\[ \\cdots \\to T_2(A,M) \\to T_1(A,M) \\to T_0(A,M) \\simeq M \\] as follows:\n$T_1(A,M)$ is the fiber of the morphism $M \\to M \\otimes A$ induced by $\\mathbb{1}_{\\mathcal{C}} \\to A$, so that $T_1(A,M)$ admits a natural morphism to $M$. More generally, $T_i(A,M) \\coloneqq T_1(A,T_{i - 1}(A,M))$, which admits a natural morphism to $T_{i-1}(A,M)$. Inductively, this defines the functors $T_i$ and the desired tower. We will call this the $A$-Adams tower of $M$. Observe that the $A$-Adams tower of $M$ is simply the tensor product of $M$ with the $A$-Adams tower of $\\mathbb{1}_{\\mathcal{C}}$. Remark(Alternate description). We can write the construction of the Adams tower in another way. Let $I = \\operatorname{fib}(\\mathbb{1}_{\\mathcal{C}} \\to A)$, so that $I$ is a nonunital associative algebra in $\\mathcal{C}$ equipped with a morphism $I \\to \\mathbb{1}_{\\mathcal{C}}$. In fact, we can get a tower \\[ \\cdots \\to I^{\\otimes n} \\to I^{\\otimes (n-1)} \\to \\cdots \\to I^{\\otimes 2} \\to I \\to \\mathbb{1}_{\\mathcal{C}}, \\] and this is precisely the $A$-Adams tower $\\{T_i(A,\\mathbb{1}_{\\mathcal{C}})\\}_{i \\geq 0}$. The $A$-Adams tower for $M$ is obtained by tensoring this with $M$.\nTheorem 12 (DK translation: cobar ↔ Adams). Let $I = \\operatorname{fib}(\\mathbb{1} \\to A)$ be the fiber of the unit map. The cobar construction corresponds to the Adams tower via the stable Dold–Kan correspondence. Specifically, we have a natural equivalence: \\[ \\operatorname{Tot}_n(\\operatorname{CB}^{\\bullet}(A)) \\simeq \\operatorname{cofib}\\left(I^{\\otimes(n+1)} \\to \\mathbb{1}\\right), \\] where $\\operatorname{Tot}_n(\\operatorname{CB}^{\\bullet}(A)) \\coloneqq \\operatorname{Tot}(\\operatorname{CB}^{\\bullet}(A)\\mid_{\\Delta_{\\leq n}})$.\nThis translation implies a crucial fact: The totalization converges ($\\operatorname{Tot} \\simeq \\mathbb{1}$) if and only if the Adams tower vanishes (i.e., is contractible).\nFrom Descendability to Nilpotence We now establish the link between our algebraic definition (descendability) and the geometric convergence (Adams tower).\nTheorem 13 (Nilpotence theorem). A commutative algebra $A$ is descendable if and only if the Adams tower is nilpotent. That is, the tower $\\{I^{\\otimes s}\\}_{s \\ge 0}$ is pro-zero: there exists an integer $N$ such that the transition map $I^{\\otimes (s+N)} \\to I^{\\otimes s}$ is null-homotopic for any $s$. Sketch. Let\u0026rsquo;s check the two implications separately.\n($\\Rightarrow$) Descendability implies Nilpotence:\nLet $\\mathcal{C}_{\\text{nil}}$ be the class of objects $M$ for which the $A$-based Adams tower vanishes (i.e., acts like zero).\nFirst, observe that $A \\otimes I \\simeq 0$. As we discussed, the unit map $A \\to A \\otimes A$ is a split monomorphism (via the multiplication map), so its fiber $A \\otimes I$ is contractible. Consequently, $A \\in \\mathcal{C}_{\\text{nil}}$. One can verify that $\\mathcal{C}_{\\text{nil}}$ forms a thick tensor-ideal. Since $A$ is descendable, we know that the unit lies in the thick ideal generated by $A$, i.e., $\\mathbb{1} \\in \\langle A \\rangle \\subseteq \\mathcal{C}_{\\text{nil}}$. Therefore, the Adams tower for $\\mathbb{1}$ itself must be pro-zero. ($\\Leftarrow$) Nilpotence implies Descendability:\nSuppose the tower is nilpotent. This means there exists some large $N$ such that the map $I^{\\otimes N} \\to \\mathbb{1}$ is null-homotopic. Let\u0026rsquo;s look at the cofiber sequence associated with this map: \\[ I^{\\otimes N} \\xrightarrow{0} \\mathbb{1} \\longrightarrow \\operatorname{cofib}(I^{\\otimes N} \\to \\mathbb{1}). \\] Since the first map is null, the sequence splits (a standard property in triangulated categories), giving us an equivalence: \\[ \\operatorname{cofib}(I^{\\otimes N} \\to \\mathbb{1}) \\simeq \\mathbb{1} \\oplus \\left(I^{\\otimes N}[1]\\right). \\] In particular, $\\mathbb{1}$ is a retract of $\\operatorname{cofib}(I^{\\otimes N} \\to \\mathbb{1})$.\nBy Theorem “DK translation: cobar ↔ Adams”, this cofiber is precisely the partial totalization $\\operatorname{Tot}_{N-1}(\\operatorname{CB}^{\\bullet}(A))$. This object is built from finite limits of $A, A^{\\otimes 2}, \\dots, A^{\\otimes N}$, all of which live in $\\langle A \\rangle$.\nSince $\\langle A \\rangle$ is closed under retracts, we conclude that $\\mathbb{1} \\in \\langle A \\rangle$. Thus, $A$ is descendable.\n$\\square$ Geometric interpretation: Nilpotent thickening This result offers a profound geometric intuition for descent in stable homotopy theory. Classically, for a map to satisfy effective descent (like a faithfully flat map in algebraic geometry), we usually require the cobar construction to be acyclic. However, in the stable setting, descendability is a relaxation of this condition.\nIt asserts that the error term (the ideal $I$) is not necessarily zero, but it is nilpotent (the tower $\\{I^{\\otimes s}\\}$ is pro-zero). Geometrically, this means the map $\\operatorname{Spec}(A) \\to \\operatorname{Spec}(\\mathbb{1})$ behaves like a nilpotent thickening. In classical algebraic geometry, a scheme and its reduction share the same underlying topological space; nilpotent elements only add “infinitesimal” structure without changing the topology. Similarly, a stable category $\\mathcal{C}$ is essentially unchanged if we thicken the unit by a nilpotent ideal. The stable Dold–Kan correspondence is the essential dictionary that allows us to see this “nilpotence” hidden inside the simplicial structure of descent.\nThe Main Result: Descendable Barr–Beck Finally, combining these insights, we prove the main theorem. Theorem 14 (Descendable Barr–Beck theorem). Let $A \\in \\mathsf{CAlg}(\\mathcal{C})$ be a descendable commutative algebra. Then the adjunction $-\\otimes A : \\mathcal{C} \\rightleftarrows \\mathsf{Mod}_A(\\mathcal{C})$ exhibits $\\mathcal{C}$ as comonadic over $\\mathsf{Mod}_{A}(\\mathcal{C})$. In particular, for any $M \\in \\mathcal{C}$, we have a canonical equivalence: \\[ M \\xrightarrow{\\sim} \\operatorname{Tot}\\left( M \\otimes \\operatorname{CB}^{\\bullet}(A) \\right). \\] Proof. We verify the conditions of Lurie–Barr–Beck Theorem ([HA, Theorem 4.7.3.5] ):\nConservativity: Suppose $M \\otimes A \\simeq 0$. Let $\\mathcal{Z} = \\{X \\in \\mathcal{C} \\mid M \\otimes X \\simeq 0\\}$. One can verify that $\\mathcal{Z}$ is stable under finite limits/colimits, retracts, and tensor products. Since $A \\in \\mathcal{Z}$ (by assumption) and $A$ is descendable, we have $\\mathbb{1} \\in \\langle A \\rangle \\subseteq \\mathcal{Z}$. Thus $M \\simeq M \\otimes \\mathbb{1} \\simeq 0$.\nConvergence: We need to show that for any $M$, the natural map $M \\to \\operatorname{Tot}(M \\otimes \\operatorname{CB}^{\\bullet}(A))$ is an equivalence. By the Dold–Kan translation, the fiber of this map is the limit of the Adams tower: $\\lim_s (M \\otimes I^{\\otimes s})$. Since $A$ is descendable, the nilpotence theorem ensures that the tower $\\{I^{\\otimes s}\\}$ is pro-zero. Thus, the inverse limit of the tower is zero. Consequently, the map to the totalization is an equivalence.\n$\\square$ References Jacob Lurie. Higher Algebra. (2017). Link. Jacob Lurie. Higher Topos Theory (AM-170). (2009). Link. Jacob Lurie. Kerodon. (2018). Link. Peter J. Haine; Maxime Ramzi; Jan Steinebrunner. Fully faithful functors and pushouts of ∞-categories. (2025). Link. Claudius Heyer; Lucas Mann. 6-Functor Formalisms and Smooth Representations. (2024). Link. Akhil Mathew; Niko Naumann; Justin Noel. Nilpotence and descent in equivariant stable homotopy theory. (2017). Link. Akhil Mathew. The Galois group of a stable homotopy theory. (2016). Link. A chain complex $C_*$ in an additive category $\\mathcal{A}$ is called connective if it is concentrated in non-negative degrees, i.e., $C_n = 0$ for all $n \u003c 0$.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\n","permalink":"https://ou-liu-red-sugar.github.io/notes/notes/stable-doldkan-and-descent/","summary":"Unified exposition of the stable Dold–Kan correspondence and its application to descent theory in stable categories.","title":"Stable Dold–Kan and Descent"},{"content":"Ou Liu (刘欧) I am a master\u0026rsquo;s student in Mathematics at the University of Regensburg, with interests in motivic homotopy theory, derived algebraic geometry, and higher category theory. Before Regensburg, I studied at Shanxi University, where my undergraduate thesis applied the six-functor formalism to smooth representation theory.\nPersonal Name. Ou Liu (刘欧) Date of birth. 2 February 2004 (as commonly observed; the national ID card records 12 January 2004) Age. 22 Nationality. Chinese Languages. Chinese (native), English (working), German (elementary, improving) Education 2025 \u0026ndash; Present · M.Sc. in Mathematics\nUniversity of Regensburg, Germany.\nResearch focus on motivic homotopy theory, derived algebraic geometry and higher category theory.\n2021 \u0026ndash; 2025 · B.Sc. in Mathematics and Applied Mathematics\nShanxi University, China.\nThesis: Applications of the six-functor formalism to smooth representation theory.\nEarlier schooling 2018 \u0026ndash; 2021 · High school at The Second High School Attached to Hunan Normal University (湖南师范大学第二附属中学), Changsha.\n2015 \u0026ndash; 2018 · Junior high at Changsha Beiya Middle School (长沙市北雅中学), Changsha.\n2009 \u0026ndash; 2015 · Primary school at Shenzhen Shiyan Public School (深圳市石岩公学), Shenzhen.\nPublications Lili Liu, Xi Wang, Ou Liu, et al.\nValuation and Comparison of the Actual and Optimal Control Strategy in an Emerging Infectious Disease.\nInfectious Disease Modelling 9 (2024).\n[Journal] See the Papers page for a running list.\nTalks and Seminars Upcoming Low weights — Motivic Cohomology of Schemes (Oberseminar, Regensburg), 19 May 2026. Six functors and Gestalten — Gestalten (Oberseminar, Regensburg), 22 June 2026. Zariski descent for modules — Higher Zariski Geometry (Oberseminar, Regensburg), 30 June 2026. See the Notes page for accompanying write-ups.\nPast WiSe 2025 / 26\nFunctors from Spaces to Spectra — Goodwillie Calculus Seminar (Regensburg), 11 December 2025. The Stable Dold–Kan Correspondence — Goodwillie Calculus Seminar (Regensburg), 27 November 2025. Galois Covers — Galois Theory in Algebra and Topology Seminar (Regensburg), 18 November 2025. Writing elsewhere Banana Space (香蕉空间). I contribute to the Chinese-language math wiki Banana Space, primarily on higher algebra, derived algebraic geometry and K-theory. Some of these entries are being translated and reorganised into the Notes section of this site. Contact Email. Ou.Liu@stud.uni-regensburg.de ORCID. 0009-0006-5593-4339 Affiliation. Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany. ","permalink":"https://ou-liu-red-sugar.github.io/about/","summary":"\u003ch1 id=\"ou-liu-刘欧\"\u003eOu Liu (刘欧)\u003c/h1\u003e\n\u003cp\u003eI am a master\u0026rsquo;s student in Mathematics at the \u003cem\u003eUniversity of Regensburg\u003c/em\u003e,\nwith interests in \u003cem\u003emotivic homotopy theory\u003c/em\u003e, \u003cem\u003ederived algebraic geometry\u003c/em\u003e, and\n\u003cem\u003ehigher category theory\u003c/em\u003e. Before Regensburg, I studied at \u003cem\u003eShanxi University\u003c/em\u003e,\nwhere my undergraduate thesis applied the six-functor formalism to smooth\nrepresentation theory.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"personal\"\u003ePersonal\u003c/h2\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eName.\u003c/strong\u003e Ou Liu (刘欧)\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDate of birth.\u003c/strong\u003e 2 February 2004 \u003cem\u003e(as commonly observed; the national ID card records 12 January 2004)\u003c/em\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAge.\u003c/strong\u003e 22\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eNationality.\u003c/strong\u003e Chinese\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLanguages.\u003c/strong\u003e Chinese (native), English (working), German (elementary, improving)\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"education\"\u003eEducation\u003c/h2\u003e\n\u003cp\u003e\u003cstrong\u003e2025 \u0026ndash; Present\u003c/strong\u003e · \u003cstrong\u003eM.Sc. in Mathematics\u003c/strong\u003e\u003cbr\u003e\n\u003cem\u003eUniversity of Regensburg\u003c/em\u003e, Germany.\u003cbr\u003e\nResearch focus on motivic homotopy theory, derived algebraic geometry and\nhigher category theory.\u003c/p\u003e","title":"About me"},{"content":"","permalink":"https://ou-liu-red-sugar.github.io/wiki/graph/","summary":"","title":"Graph view"}]