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Stable $\infty$-categories for representation theorists

Gustavo Jasso

Abstract

This survey is intended as an invitation to the theory of stable $\infty$-categories, addressed primarily to mathematicians working in the representation theory of algebras and related subjects.

Stable $\infty$-categories for representation theorists

Abstract

This survey is intended as an invitation to the theory of stable -categories, addressed primarily to mathematicians working in the representation theory of algebras and related subjects.
Paper Structure (13 sections, 24 theorems, 93 equations)

This paper contains 13 sections, 24 theorems, 93 equations.

Table of Contents

  1. Primer on $\infty$-category theory
  2. From sets to $\infty$-groupoids
  3. Basic $\infty$-category theory
  4. Lurie's stable $\infty$-categories
  5. Definition and first properties
  6. Idempotent-complete stable $\infty$-categories
  7. Stable $\infty$-categories form Frobenius exact categories
  8. Compactly-generated and presentable stable $\infty$-categories
  9. Smooth and proper stable $\infty$-categories
  10. $t$-structures on stable $\infty$-categories and realisation functors
  11. $t$-injective objects and cosilting objects
  12. The universal property of the bounded derived $\infty$-category
  13. Recollements of stable $\infty$-categories

Key Result

theorem 1.2.2

Let $f\colon X\to Y$ be a functor between $\infty$-groupoids. Then, $f$ is an equivalence if and only if the induced map is bijective and, for each point $x\in X$ and each integer $n\geq1$, the induced map is an isomorphism of groups.

Theorems & Definitions (96)

  • remark 1.2.1
  • theorem 1.2.2: Cis19
  • remark 1.2.4
  • remark 1.2.5
  • definition 1.2.6
  • remark 1.2.7
  • definition 1.2.8: Lur09
  • remark 1.2.9
  • example 1.2.10
  • definition 1.2.11: Lur09
  • ...and 86 more