Generalized Witt and Morita equivalences

TL;DR

This paper introduces new, less refined equivalence relations among fusion categories and more refined relations among braided fusion categories, leading to new abelian groups that enhance the classification of these mathematical structures.

Contribution

It defines novel equivalence relations that refine existing classifications, resulting in new abelian groups for fusion and braided fusion categories, aiding their structural understanding.

Findings

Defined new equivalence relations among fusion categories

Constructed abelian groups from these equivalence classes

Provided tools for classification of (braided) fusion categories

Abstract

In this work, we introduce a family of new equivalence relations among fusion categories that are less refined than the usual Morita equivalence. We obtain abelian groups by quotienting these new equivalence relations from the commutative monoids of the equivalence classes of all fusion categories. Moreover, we upgrade them to equivalence relations among nondegenerate braided fusion categories that are more refined than the usual Witt equivalence. As a consequence, we obtain new abelian groups that are more refined than the usual Witt group. These new groups allow us to access the internal structures within Witt classes. We expect that they are useful in the classification program of (braided) fusion (higher) categories and in the study of gapless edges of 2+1D topological orders.