Higher Algebra

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 ...

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 ...

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. ...

Unified exposition of the stable Dold–Kan correspondence and its application to descent theory in stable categories.