# nLab Frobenius pseudomonoid

Frobenius pseudomonoids

### Context

#### Monoidal categories

monoidal categories

category theory

# Frobenius pseudomonoids

## Idea

The concept of Frobenius pseudomonoid is the categorification of that of Frobenius algebra. It can be defined in any monoidal bicategory. Since Frobenius pseudomonoids are closely related to star-autonomous categories, they are sometimes called $\ast$-autonomous pseudomonoids.

## Definitions

Like a Frobenius algebra, a Frobenius pseudomonoid can be defined in many essentially equivalent ways. Let $(K,\otimes,I)$ be a monoidal bicategory, and let $(A,\mu,\eta)$ be a pseudomonoid in $K$.

1. $A$ is Frobenius if it is equipped with a morphism $\epsilon : A \to I$ such that the composite $A\otimes A \xrightarrow{\mu} A \xrightarrow{\epsilon} I$ is the counit of a specified 2-adjunction $A\dashv A$. (Lauda06), (Street04)

2. $A$ is Frobenius if it is equipped with a specified 2-adjunction $A\dashv A$, with counit $p:A\otimes A \to I$, and an isomorphism $p\circ (\mu\otimes 1) \cong p\circ (1\otimes \mu)$. (Day-Street 03)

There should also be a definition in terms of an interacting pseudomonoid and pseudocomonoid structure, but I have not been able to find this in the literature.

Note that if $K$ is a compact closed bicategory, then the 2-adjunction $A\dashv A$ can equivalently be expressed as an equivalence $A\simeq A^\circ$ from $A$ to its specified dual object.

## Relation to $\ast$-autonomy

###### Theorem

A star-autonomous category is equivalent to a Frobenius pseudomonoid in the monoidal bicategory Prof whose multiplication $A\otimes A \to A$, unit $I\to A$, and duality equivalence $A\simeq A^\circ$ are representable profunctors (i.e. functors).

###### Proof

See Day-Street 03 and Street04.

Note that the morphisms $\epsilon : A\to I$, $p:A\otimes A\to I$, and the induced comultiplication $A\to A\otimes A$ are not representable. A general Frobenius pseudomonoid in $Prof$, without any representability conditions, may be called a pro-$\ast$-autonomous category.

###### Remark

There is another relation between Frobenius algebras and $\ast$-autonomous categories: (Egger10) shows that cocomplete $\ast$-autonomous posets are equivalently Frobenius algebras in the $\ast$-autonomous category Sup.

## References

• Aaron Lauda, Frobenius algebras and ambidextrous adjunctions, 2006; TAC
• Aaron Lauda, Frobenius algebras and planar open string topological field theories, 2005; arxiv
• Jeff Egger, The Frobenius relations meet linear distributivity, 2010 TAC

Last revised on October 12, 2017 at 14:01:47. See the history of this page for a list of all contributions to it.