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

TL;DR

This paper establishes a connection between group quotients of dg-categories and stable $ abla$-categories, introducing skew group dg-algebras and their Morita equivalences, extending to ring spectra and $ abla$-categorical quotients.

Contribution

It introduces a framework relating group actions on dg-algebras and dg-categories to skew group dg-algebras and $ abla$-categorical quotients, including new constructions for ring spectra.

Findings

Skew group dg-algebra is Morita equivalent to the dg-categorical homotopy group quotient.

Extension of skew group algebra concepts to ring spectra.

Description of orbit dg-categories and their relation to group quotients.

Abstract

We relate 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 $\infty$-categorical group quotients.