Descent and forms of tensor categories

TL;DR

This paper develops a comprehensive theory of descent and forms of tensor categories over arbitrary fields, providing classification schemes, explicit examples, and insights into categorification of weak fusion rings.

Contribution

It introduces a general framework for classifying forms of tensor categories using algebraic and homotopical methods, and explores categorifiability of weak fusion rings.

Findings

Classification scheme for forms of tensor categories

Explicit examples of form classifications

Criteria for categorifiability of weak fusion rings

Abstract

We develop a theory of descent and forms of tensor categories over arbitrary fields. We describe the general scheme of classification of such forms using algebraic and homotopical language, and give examples of explicit classification of forms. We also discuss the problem of categorification of weak fusion rings, and for the simplest families of such rings, determine which ones are categorifiable.