Frobenius Algebra - Category-theoretical Definition

Category-theoretical Definition

In category theory, the notion of Frobenius object is an abstract definition of a Frobenius algebra in a category. A Frobenius object in a monoidal category consists of an object A of C together with four morphisms

such that

• is a monoid object in C,
• is a comonoid object in C,

• the diagrams

and

commute (for simplicity the diagrams are given here in the case where the monoidal category C is strict).

More compactly, a Frobenius algebra in C is a so-called Frobenius monoidal functor A:1 → C, where 1 is the category consisting of one object and one arrow.