The Hopf category of Frobenius algebras

TL;DR

This paper introduces a Hopf category structure on Frobenius algebras using universal measuring and comeasuring coalgebras, extending classical isomorphism results and exploring implications for Topological Quantum Field Theories.

Contribution

It establishes a Hopf category framework for Frobenius algebras via universal measuring coalgebras, generalizing classical isomorphism results and exploring duality and compatibility.

Findings

Frobenius algebras form a Hopf category under universal measuring coalgebras.

Frobenius algebras form a Hopf opcategory under universal comeasuring algebras.

The theory applies to finite characteristic cases with explicit examples.

Abstract

We show that the universal measuring coalgebras between Frobenius algebras turn the category of Frobenius algebras into a Hopf category (in the sense of Batista-Caenepeel-Vercruysse), and the universal comeasuring algebras between Frobenius algebras turn the category Frobenius algebras into a Hopf opcategory. We also discuss duality and compatibility results between these structures. Our theory vastly generalizes the well-known fact that any homomorphism beween Frobenius algebras is an isomorphism, but also allows to go beyond classical (iso)morphisms between Frobenius algebras, especially in finite characteristic, as we show by some explicit examples. The paper is concluded with some open questions and considerations about Topological Quantum Field Theories.