Algebraic K-theory

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 $\mathrm{K}(\mathcal{C})$.

Connective K-theory

Since $\mathsf{coSpan}(\mathcal{C})$ 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.

Intuition

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 $[\varnothing] = 0$.

To 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 &\coloneqq (X \to X \leftarrow \varnothing) \qquad \text{(a morphism } X \to \varnothing\text{),} \ b_X &\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 $p_X$).

More generally:

• For 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.

This 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.

For this second construction we need $\mathcal{C}$ to be pointed with finite colimits. Before giving the simplicial object itself, here is the motivating picture.

We 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:

1. a basepoint $0$;

2. for each $X \in \mathcal{C}$ a loop in $W$, representing $[X] \in \pi_1(W) \simeq \mathrm{k}_0(\mathcal{C})$;

3. for 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 $Y/X \coloneqq \mathrm{cofib}(f)$;

4. higher coherence data: for a sequence $X \to Y \to Z$ we want $[Z] = [X] + [Y/X] + [Z/Y]$. Two natural routes derive this:

   • via $X \to Z \to Z/X$ 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

   

5. 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.

We next show that $\mathcal{C}$ and its pointed completion have the same algebraic K-theory.

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.

There 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

is 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.

In 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.

The 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:

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.

$\square$

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.

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

It is natural to ask whether K-theory detects essential surjectivity: given $D \in \mathcal{D}$, is $D$ in $\mathcal{C} \subseteq \mathcal{D}$?

Clearly 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’s insight is that simply connected finitely dominated anima are automatically finite.

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.

Verdier sequences

The stable case

Recall 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.

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.

Returning 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 $\mathsf{Cat}$.

Concretely, 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: $K \simeq \mathsf{Ind}(\mathcal{D}/\mathcal{C})$.

To 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 $p\colon \mathcal{D} \to K$.

This 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 $p$.

Under the further assumptions that

• $F$ 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’s mapping-space formula; in the stable case both assumptions are automatic. The description is compatible with idempotent completion, since $\mathsf{Ind}(\mathcal{C}^{\mathrm{idem}}) \simeq \mathsf{Ind}(\mathcal{C})$.

What 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.

Proof.

See [sheaves-on-manifolds, Prop. 3.2.3] .

We 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.

$\square$

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 $\mathrm{k}_0(\mathsf{Ind}(\mathcal{C})^{\aleph_1} / \mathcal{C}) = 0$.

The 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.

For 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:

The key tool for lifting connective results to the non-connective world is:

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 $\mathsf{Pr}_{\mathrm{ca}}^L$.

The 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.

Crucially, 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

Consider 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 $\hat{y}$’s right adjoint is $k$, and $k$’s right adjoint is $y$), limits can be computed in $\mathsf{Cat}$; the limit identifies $y$’s limit as the right adjoint of the right adjoint of $\phi$, establishing strongness.

Fully faithfulness and retract-closedness pass through filtered colimits by below.

(2) Assume $\mathcal{C} \to \mathcal{D} \to \mathcal{E}$ induces a $\mathrm{k}$-long-exact sequence. Consider the $3 \times 3$ diagram

All 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.

(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

Both 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 .

(4) For a filtered diagram $\mathcal{C}_{\bullet}$ with colimit $\mathcal{C}$, consider

The 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.

$\square$

Proof.

Full faithfulness: mapping spaces in a filtered colimit of categories are filtered colimits of the levelwise mapping spaces.

Retract-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 $d = \alpha_{\infty}(\lambda_i(c_i'))$.

$\square$

The main corollary lifts connective properties of $\mathrm{k}$ to non-connective $\mathrm{K}$.

Proof.

(1) Definition plus cofinality.

(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).

(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 $\mathrm{K}(\mathcal{C}) \simeq 0$.

(4) By Proposition 23 (2).

(5) Combine Proposition 23 (3) with the connective equivalences Proposition 6 and Proposition 9 .

$\square$

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.