Frobenius Objects

Frobenius Object

A Frobenius object in a monoidal category $(\mathsf{C},\otimes ,1)$ is an object $A$ equipped with four morphisms
$\eta :\text{one}\to A,\mu :A\otimes A\to A,\delta :A\to A\otimes A,\epsilon :A\to \text{one},$
such that $(A,\mu ,\eta )$ is a monoid object, $(A,\delta ,\epsilon )$ is a comonoid object, and the Frobenius relation holds:

e.g. A Frobenius object in $({\mathsf{Vect}}_{\mathbb{K}},\otimes ,\mathbb{K})$ is precisely a Frobenius algebra.

Frobenius Homomorphism

A Frobenius homomorphism between two Frobenius objects $(A,{\mu }_A,{\eta }_A,{\delta }_A,{\epsilon }_A)$ and $(B,{\mu }_B,{\eta }_B,{\delta }_B,{\epsilon }_B)$ is a morphism $f:A\to B$ that is both a monoid homomorphism and a comonoid homomorphism.
Thus the Frobenius objects together with Frobenius homomorphisms form a category, denoted by $\mathsf{Frob}(\mathsf{C})$.

The Universal Frobenius Structure