nLab Frobenius monad

Idea

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

Properties

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

Proposition

If $M$ is a Frobenius monad on a category $C$, then the usual free-forgetful adjunction $F \dashv U \colon Alg_M \to C$ on its Eilenberg-Moore category is an ambidextrous adjunction whose associated Frobenius monad is isomorphic to $M$.

(cf. Street (2004), Section II)

Proof

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

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

Examples

Example

Example

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

(co)monad name	underlying endofunctor	(co)monad structure induced by
reader monad	$W \to (\text{-})$ on cartesian types	unique comonoid structure on $W$
coreader comonad	$W \times (\text{-})$ on cartesian types	unique comonoid structure on $W$
writer monad	$A \otimes (\text{-})$ on monoidal types	chosen monoid structure on $A$
cowriter comonad	$\array{A \to (\text{-}) \\ \\ A \otimes (\text{-})}$ on monoidal types	chosen monoid structure on $A$ chosen comonoid structure on $A$
Frobenius (co)writer	$\array{A \to (\text{-}) \\ A \otimes (\text{-})}$ on monoidal types	chosen Frobenius monoid structure

Example

(linear (co)reader monad)
For $W$ a finite set and $\mathbb{K}$ any ground field, the reader monad $\bigcirc_W \;\colon\; Vect_{\mathbb{K}} \to Vect_{\mathbb{K}}$ on $\mathbb{K}$-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 $\bigcirc_W \,\simeq\, \bigstar_W \,\colon\, Vect \to Vect$ is isomorphic to the Frobenius writer monad (of Exp. ) corresponding to the Frobenius algebra $\mathrm{Q}W \,\equiv\,\oplus_W \mathbb{K}$ (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.

Related concepts

• Wirthmüller context

• infinitesimal cohesion

References

The notion of Frobenius monads appears briefly in

• William Lawvere, pp. 151 of: Ordinal sums and equational doctrines, in Seminar on Triples and Categorical Homology Theory LNM 80 (1969) 141-155 [doi:10.1007/BFb0083085]

The relation of Frobenius monads to ambidextrous adjunctions:

in Cat

• Ross Street, Frobenius monads and pseudomonoids, J. Math. Phys. 45 3930 (2004) [doi:10.1063/1.1788852]

and in general 2-categories:

• Aaron Lauda, Frobenius algebras and ambidextrous adjunctions, Theory and Applications of Categories 16 4 (2006) 84-122 [arXiv:math/0502550, tac:16-04]

  (motivated by 2d TQFT)

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

• F. Castaño Iglesias, José Gómez Torrecillas, C. Nastasescu, Frobenius functors: applications, Comm. Alg. 27 10 (1998) 4879-4900 [doi:10.1080/00927879908826735]

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

• Martti Karvonen, Frobenius algebras in functor categories, Oxford (2014) [pdf, pdf]

and analogous discussion in dagger-categories:

• Chris Heunen, Martti Karvonen, Monads on dagger categories, Theory and Applications of Categories 31 35 (2016) 1016-1043 [arXiv:1602.04324, tac:31-35]