Higher Category Theory

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 ...

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 ...

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. ...

This page aims to explain how type theory can be understood within the framework of synthetic category theory. The 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. We focus on the following two questions: Question 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. ...