Nakayama functors for coalgebras and their applications to Frobenius tensor categories

TL;DR

This paper introduces Nakayama functors for coalgebras and Frobenius tensor categories, providing a categorical Radford $S^4$-formula and generalizing key results in tensor category theory.

Contribution

It defines Nakayama functors for coalgebras and Frobenius tensor categories using (co)ends, extending finite case concepts to a broader categorical setting.

Findings

Established the categorical Radford $S^4$-formula for Frobenius tensor categories

Generalized results from finite tensor categories to Frobenius tensor categories

Connected Nakayama functors with known properties of co-Frobenius Hopf algebras

Abstract

We introduce Nakayama functors for coalgebras and investigate their basic properties. These functors are expressed by certain (co)ends as in the finite case discussed by Fuchs, Schaumann, and Schweigert. This observation allows us to define Nakayama functors for Frobenius tensor categories in an intrinsic way. As applications, we establish the categorical Radford $S^4$-formula for Frobenius tensor categories and obtain some related results. These are generalizations of works of Etingof, Nikshych, and Ostrik on finite tensor categories and some known facts on co-Frobenius Hopf algebras.