nLab
Frobenius monad

Contents

Context

Category theory

2-Category theory

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

Proof

In general, 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 (considered as categories over CC, via the usual forgetful functors). 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 here). 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: 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.