Mathematics

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 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: Objects of $\mathsf{coSpan}(\mathcal{C})$ are the objects of $\mathcal{C}$. ...

Definition Definition 1. Let $\kappa$ be a regular cardinal (for example $\kappa = \omega$ or $\kappa = \aleph_1$). A 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. Let $\mathcal{C}$ 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 ...

The goal of this note is to study the dualizable stable categories — the compactly assembled categories. The reasons to care about them are several: In 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 $X$. Given 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 ...

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. Let \(\mathcal{C} \in \mathsf{CAlg}(\mathsf{Pr}^L)\) 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 ...

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. Informally, algebraic pattern generalizes the active and inert morphisms in operads and chooses certain objects to control the Segal condition. Definition 1. An algebraic pattern is a category $\mathcal{O}$ equipped with: A 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 ...

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 $\mathrm{K}$. The 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. The 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. ...

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 $0\mathsf{Pr} \coloneqq \mathsf{An}$. Since 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. ...

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. Quasicoherent Sheaves of Higher Categories

Conventions. Category 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 “space” $X$ 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 $\Gamma(X; \Lambda)$. ...

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’s Spectral Algebraic Geometry ([lurie-sag] ) and Mathew’s work on Galois groups in stable homotopy theory ([mathew-galois] ). Conventions Throughout we work in the derived setting and drop the $\mathrm{R}$-prefix on all functors. Concretely: All 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. ...