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$\infty$-categorical group quotients via skew group algebras

Merlin Christ

Abstract

We describe group quotients of dg-categories and linear stable $\infty$-categories. Given a group acting on a dg-algebra, we prove that the skew group dg-algebra is Morita equivalent to the dg-categorical homotopy group quotient. We also treat the cases of group actions on dg-categories, with corresponding skew group dg-categories, and of orbit dg-categories. Finally, we describe a version of the skew group algebra in the setting of ring spectra and relate it with the $\infty$-categorical group quotient.

$\infty$-categorical group quotients via skew group algebras

Abstract

We describe group quotients of dg-categories and linear stable -categories. Given a group acting on a dg-algebra, we prove that the skew group dg-algebra is Morita equivalent to the dg-categorical homotopy group quotient. We also treat the cases of group actions on dg-categories, with corresponding skew group dg-categories, and of orbit dg-categories. Finally, we describe a version of the skew group algebra in the setting of ring spectra and relate it with the -categorical group quotient.
Paper Structure (12 sections, 21 theorems, 40 equations)

This paper contains 12 sections, 21 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Infinity-categorical group actions
  3. The categorical group algebra
  4. Group actions and group quotients
  5. Skew group dg-categories
  6. Skew group dg-categories as $\infty$-categorical group quotients
  7. Skew group dg-categories as homotopy colimits of cofibrant diagrams
  8. Orbit categories
  9. General orbit $\infty$-categories
  10. Orbit dg-categories
  11. Example: Periodic derived categories
  12. Spectral skew group algebras

Key Result

Proposition 1.1

Fix a base field $k$. Let $A$ be a dg-category with a strict action by a group $G$. We consider the $G$-action as a functor where $BG$ is the classifying space of $G$. The homotopy colimitWith respect to the Morita model structure on the category $\operatorname{dgCat}_k$ of dg-categories. of $\rho$ is Morita equivalent to the skew group dg-category, denoted $A\ast G$. If $A$ has a single object,

Theorems & Definitions (59)

  • Proposition 1.1: \ref{['prop:skewgroupcatiscolim']} and \ref{['rem:htpycolim']}
  • Theorem 1.2
  • Proposition 1.3: $\!\!$Dem11
  • Corollary 1.4
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Example 2.5
  • Definition 2.6
  • ...and 49 more