# Descent and Galois theory for Hopf categories

## TL;DR

This paper develops descent and Galois theory for linear categories and Hopf categories, establishing conditions under which categories of descent data are equivalent to representation categories, and introducing Hopf-Galois category extensions.

## Contribution

It extends descent and Galois theory to linear and Hopf categories, including the development of Hopf-Galois category extensions and dual actions.

## Key findings

- Category of descent data is isomorphic to representation category under flatness.
- Hopf-Galois category extensions generalize classical Galois extensions.
- Strongly graded linear categories correspond to Hopf-Galois extensions over groupoid algebras.

## Abstract

Descent theory for linear categories is developed. Given a linear category as an extension of a diagonal category, we introduce descent data, and the category of descent data is isomorphic to the category of representations of the diagonal category, if some flatness assumptions are satisfied. Then Hopf-Galois descent theory for linear Hopf categories, the Hopf algebra version of a linear category, is developed. This leads to the notion of Hopf-Galois category extension. We have a dual theory, where actions by dual linear Hopf categories on linear categories are considered. Hopf-Galois category extensions over groupoid algebras correspond to strongly graded linear categories.