Mathematics

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. ...

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. ...

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