Coring categories and Villamayor-Zelinsky sequence for symmetric finite tensor categories

TL;DR

This paper interprets a specific cohomology group in a Villamayor-Zelinsky sequence for symmetric finite tensor categories, introducing coring categories and establishing their relation to Azumaya quasi coring categories, generalizing classical theorems.

Contribution

It introduces coring categories and links the middle cohomology group to Azumaya quasi coring categories, extending classical results to a categorical setting.

Findings

The middle cohomology group is isomorphic to the group of Azumaya quasi coring categories.

Constructs colimits of relative groups of Azumaya quasi coring categories over symmetric tensor categories.

Proves the isomorphism between the colimit group and the full group of Azumaya quasi coring categories over vec.

Abstract

In the preceeding paper we constructed an infinite exact sequence a la Villamayor-Zelinsky for a symmetric finite tensor category. It consists of cohomology groups evaluated at three types of coefficients which repeat periodically. In the present paper we interpret the middle cohomology group in the second level of the sequence. We introduce the notion of coring categories and we obtain that the mentioned middle cohomology group is isomorphic to the group of Azumaya quasi coring categories. This result is a categorical generalization of the classical Crossed Product Theorem, which relates the relative Brauer group and the second Galois cohomology group with respect to a Galois field extension. We construct the colimit over symmetric finite tensor categories of the relative groups of Azumaya quasi coring categories and the full group of Azumaya quasi coring categories over $vec$. We prove that the latter two groups are isomorphic.