Equivalent definitions of fusion category arising from separability
Zhenbang Zuo
TL;DR
This work establishes a unified framework linking fusion category structure to separability properties of tensor functors by nonzero algebras in a semisimple multiring category with left duals. The central result shows that the unit's simplicity is equivalent to the separability (and its variants) of $- ens A$ for all nonzero algebras $A$, yielding multiple equivalent characterizations of fusion categories and enabling diverse applications. By analyzing algebras in component subcategories and proving a separable Frobenius pair between inclusion and projection, the authors derive a comprehensive network of equivalences among unit-simplicity, separability, faithfulness, Maschke-type conditions, and related morphism properties. The paper then demonstrates concrete applications to weak Hopf algebras, transfer of simplicity via monoidal functors, semisimple indecomposable module categories, and Grothendieck rings, and provides explicit examples to illustrate the theory in classical settings such as Vec and Rep$(H)$. Overall, the results offer a robust criterion set for recognizing fusion categories through tensorial separability, with broad implications for representation theory and categorical algebra.
Abstract
For a semisimple multiring category with left duals, we prove that the unit object is simple if and only if the tensor functors by any non-zero algebra are separable (resp. faithful, resp. Maschke, resp. dual Maschke, resp. conservative). This induces a list of equivalent definitions of fusion category. As an application, we describe the connectness of a class of weak Hopf algebras by the separability of tensor functors. We also consider applications to transfer of simplicity between the unit objects, semisimple indecomposable module category and Grothendieck ring.