Title: Descent and Galois theory for Hopf categories,
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.