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category theory

## Applications

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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

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 $A$.

reader monad$W \to (\text{-})$ on cartesian typesunique comonoid structure on $W$
coreader comonad$W \times (\text{-})$ on cartesian typesunique comonoid structure on $W$
writer monad$A \otimes (\text{-})$ on monoidal typeschosen monoid structure on $A$
cowriter comonad$\array{A \to (\text{-}) \\ \\ A \otimes (\text{-})}$ on monoidal typeschosen monoid structure on $A$

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

###### Example

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.

The notion of Frobenius monads appears briefly in