A generalization of Gabriel's Galois covering functors II: 2-categorical Cohen-Montgomery duality

TL;DR

This paper extends Cohen-Montgomery duality to a 2-categorical setting, establishing equivalences between categories with group actions and graded categories, thus broadening the theoretical framework of Galois coverings.

Contribution

It introduces 2-categorical structures on categories with group actions and graded categories, proving they are 2-equivalent via extended orbit and smash product constructions.

Findings

2-categorical orbit and smash product constructions are 2-equivalences.

Extension of Cohen-Montgomery duality to 2-categories.

Provides a new framework for Galois coverings in higher category theory.

Abstract

Given a group $G$, we define suitable 2-categorical structures on the class of all small categories with $G$-actions and on the class of all small $G$-graded categories, and prove that 2-categorical extensions of the orbit category construction and of the smash product construction turn out to be 2-equivalences (2-quasi-inverses to each other), which extends the Cohen-Montgomery duality.