# nLab Frobenius monad

Contents

category theory

## Applications

#### 2-Category theory

2-category theory

## Structures on 2-categories

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

A Frobenius monad is a monad which as a monoid in endofunctors is a Frobenius monoid.

## Examples

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 $F \dashv U: Alg_M \to C$ is an ambidextrous adjunction whose associated Frobenius monad is isomorphic to $M$.

###### Proof

In general, 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 $Alg_M \simeq CoAlg_K$ (considered as categories over $C$, via the usual forgetful functors). 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 here). 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 $M: C \to C$ through the forgetful functors.

## References

• Ross Street, Frobenius monads and pseudomonoids, J. Math. Phys. 45.(2004) (web)

Last revised on March 6, 2017 at 09:28:07. See the history of this page for a list of all contributions to it.