Monadic adjunction

An adjunction is monadic when its comparison functor from the right-hand category to the Eilenberg-Moore category of the induced monad is an equivalence.

The induced monad determines this comparison; Beck's monadicity theorem gives a criterion for it to be an equivalence of categories.

A useful form of the crude monadicity theorem says that a right adjoint is monadic if it reflects isomorphisms and its source has and the functor preserves coequalizers of reflexive pairs.

An adjunction is comonadic when the comparison from its left-hand category to coalgebras for the induced comonad is an equivalence. The dual Beck theorem tests this using reflected isomorphisms and suitable equalizers.

The relevant criterion is the Beck comonadicity theorem, dual to Beck's monadicity theorem.

The Beck comonadicity theorem, dual to the Beck monadicity theorem, says that a left adjoint is comonadic when it reflects isomorphisms and preserves the required equalizers of pairs whose images admit split equalizers. A stronger convenient hypothesis is that it preserves all equalizers.

The precise monadicity theorem says that a right adjoint is monadic exactly when it reflects isomorphisms and creates coequalizers of the pairs whose images have split coequalizers.