nLab Frobenius monad



Category theory

2-Category theory

Higher algebra



A Frobenius monad [Lawvere 1969] is a monad which as a monoid in endofunctors is a Frobenius monoid.


The monad induced from an ambidextrous adjunction is Frobenius. The converse is also true:


If MM is a Frobenius monad on a category CC, then the usual free-forgetful adjunction FU:Alg MCF \dashv U \colon Alg_M \to C on its Eilenberg-Moore category is an ambidextrous adjunction whose associated Frobenius monad is isomorphic to MM.

(cf. Street (2004), Section II)

In general (see this Prop. at adjoint monad), if a monad MM admits a right adjoint KK, then KK carries a comonad structure mated to the monad structure of MM, and there is an adjoint equivalence Alg MCoAlg KAlg_M \simeq CoAlg_K between their Eilenberg-Moore categories (considered as categories over CC, via the usual forgetful functors).

Now if MM is Frobenius, then MMM \dashv M and the comonad structure mated to the monad is indeed the given comonad structure of MM (a proof is given at Frobenius algebra (see there). Hence the left adjoint CAlg MC \to Alg_M to the forgetful functor may be identified with the right adjoint CCoAlg MC \to CoAlg_M of the forgetful functor, each being unique (up to isomorphism) lifts of M:CCM \colon C \to C through the forgetful functors.




(Frobenius writer monad)
For AA a Frobenius algebra (over some ground field 𝕂\mathbb{K}) the writer monad/cowriter comonad A():A \otimes (-) \,\colon\, is canonically a Frobenius monad, with (co)product and (co)unit induced from the corresponding operations on AA.

(co)monad nameunderlying endofunctor(co)monad structure induced by
reader monadW(-)W \to (\text{-}) on cartesian typesunique comonoid structure on WW
coreader comonadW×(-)W \times (\text{-}) on cartesian typesunique comonoid structure on WW
writer monadA(-)A \otimes (\text{-}) on monoidal typeschosen monoid structure on AA
cowriter comonadA(-) A(-)\array{A \to (\text{-}) \\ \\ A \otimes (\text{-})} on monoidal typeschosen monoid structure on AA

chosen comonoid structure on AA
Frobenius (co)writerA(-) A(-)\array{A \to (\text{-}) \\ A \otimes (\text{-})} on monoidal typeschosen Frobenius monoid structure


(linear (co)reader monad)
For WW a finite set and 𝕂\mathbb{K} any ground field, the reader monad W:Vect 𝕂Vect 𝕂\bigcirc_W \;\colon\; Vect_{\mathbb{K}} \to Vect_{\mathbb{K}} on 𝕂\mathbb{K}-VectorSpaces is Frobenius and isomorphic (see there for details) with the corresponding comonad being the linear cowriter comonad, this reflecting that finite products of vector spaces are biproducts, namely direct sums.

Finally, this Frobenius monad W W:VectVect\bigcirc_W \,\simeq\, \bigstar_W \,\colon\, Vect \to Vect is isomorphic to the Frobenius writer monad (of Exp. ) corresponding to the Frobenius algebra QW W𝕂\mathrm{Q}W \,\equiv\,\oplus_W \mathbb{K} (see there).

As such, it has been proposed [Coecke & Pavlović 2008] to reflect aspects of quantum measurement in the context of quantum information via dagger-compact categories and is used as such in the zxCalculus (where the Frobenius property is embodied by “spider diagrams”). Various authors discuss the Frobenius monads in this context, see the references there.


The notion of Frobenius monads appears briefly in

The relation of Frobenius monads to ambidextrous adjunctions:

in Cat

and in general 2-categories:

Discussion of ambidextrous adjunctions between abelian categories under the name Frobenius morphisms:

On Frobenius (co)writer monads induced form Frobenius monoids with special attiotion to their strength (following Wolff 1973):

and analogous discussion in dagger-categories:

Last revised on August 14, 2023 at 17:53:24. See the history of this page for a list of all contributions to it.