nLab Frobenius pseudomonoid

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.

Remark

This characterisation of $\ast$-autonomous algebras in terms of pseudo-Frobenius algebras can further be refined to characterise autonomous categories. Namely, an autonomous category is a representable pseudo-Frobenius algebra in Prof whose pseudomonoid and pseudocomonoid structure not only are adjoint, but which also satsify “locality” coherence equations Bartlett-Vicary 10.

References

• Aaron Lauda, Frobenius algebras and ambidextrous adjunctions, 2006; TAC

• Aaron Lauda, Frobenius algebras and planar open string topological field theories, 2005; arxiv

• Brian Day and Ross Street, Quantum categories, star autonomy, and quantum groupoids, 2003, arxiv

• Bruce Bartlett and Jamie Vicary, Compact categories as dagger-Frobenius pseudoalgebras, 2010, QPL2010

• Ross Street, Frobenius monads and pseudomonoids, 2004, DOI, pdf

• Jeff Egger, The Frobenius relations meet linear distributivity, 2010 TAC