Generalized Witt and Morita equivalences
Liang Kong, Yilong Wang, Hao Zheng
TL;DR
The paper addresses the coarse nature of Morita and Witt equivalences in the classification of (braided) fusion categories by introducing $S$-Witt and $S$-Morita equivalences, parameterized by submonoids $S$ of fusion categories. By quotienting the commutative monoid of fusion-category classes by these weakened relations, it constructs abelian groups $\mathds{W}_S$ and refines the Witt structure to expose internal Witt-class data; it also establishes exact sequences with the usual Witt group and analyzes saturation and group conditions. The framework yields a flexible toolkit for accessing internal structures within Witt classes and has potential applications in classifying braided/fusion categories and in the physics of 2+1D topological orders, including gapless edge phenomena. The authors also outline natural generalizations to fusion categories over a symmetric category $\mathcal{E}$ and to higher fusion categories, suggesting broad applicability to braided fusion higher categories and higher-dimensional topological phases.
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.