nLab Frobenius pseudomonoid

Frobenius pseudomonoids


Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory

Category theory

Frobenius pseudomonoids


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.


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

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

  2. AA is Frobenius if it is equipped with a specified 2-adjunction AAA\dashv A, with counit p:AAIp:A\otimes A \to I, and an isomorphism p(μ1)p(1μ)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 KK is a compact closed bicategory, then the 2-adjunction AAA\dashv A can equivalently be expressed as an equivalence AA A\simeq A^\circ from AA to its specified dual object.

Relation to *\ast-autonomy


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


See Day-Street 03 and Street04.

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


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.


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.


  • 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 January 25, 2023 at 13:37:56. See the history of this page for a list of all contributions to it.