Equivalent definitions of fusion category arising from separability

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.

💡 Research Summary

The paper investigates the relationship between the simplicity of the unit object in a semisimple multiring category with left duals and the separability of tensor functors induced by non‑zero algebras. A multiring category is a locally finite k‑linear abelian monoidal category with a bilinear, biexact tensor product; if End(1)=k it is called a ring category. A fusion category is a multifusion category whose unit object is simple (i.e., End(1)=k and the category is indecomposable).

The authors begin by recalling the notion of a separable functor (introduced in