On normal tensor functors and coset decompositions for fusion categories

TL;DR

This paper develops a framework for understanding double cosets and normal tensor functors in fusion categories, providing new criteria and descriptions for their structure and images, with applications to equivariantizations.

Contribution

It introduces the concept of double cosets relative to fusion subcategories and characterizes normal tensor functors through equivalence relations on simple objects.

Findings

Equivalent classes of the relation are cosets.

Normal tensor functors have images characterized by colinearity.

Criteria for normality in composed tensor functors.

Abstract

We introduce the notion of double cosets relative to two fusion subcategories of a fusion category. Given a tensor functor $F : \C \to \D$ between fusion categories, we introduce an equivalence relation $\approx^F$ on the set $\Lambda_\C$ of isomorphism classes of simple objects of $\C$, and when $F$ is dominant, an equivalence relation $\approx_F$ on $\Lambda_\D$. We show that the equivalent classes of $\approx^F$ are cosets. We also give a description of the image of $F$ when it is a normal tensor functor, and we show that $F$ is normal if and only if the images of $\approx^F$ equivalent elements of $\Lambda_\C$ are colinear. We study the situation where the composition of two tensor functors $F=F'F"$ is normal, and we give a criterion of normality for $F"$, with an application to equivariantizations. Lastly, we introduce the radical of a fusion subcategory and compare it to its commutator in the case of a normal subcategory. We also give a description for the image of a normal tensor functor between any two fusion categories.