Monad

A monad is an endofunctor $T$ with unit and multiplication satisfying associativity and unit laws.

An adjunction $F⊣G$ with unit $\eta$ and counit $\epsilon$ induces the monadwith unit $\eta$.

An algebra for a monad $(T,\eta ,\mu )$ is an object $A$ with a morphism $a:TA\to A$ satisfying $a{\eta }_A=1_A$ and $aT(a)=a{\mu }_A$.

The Eilenberg-Moore category ${\mathcal{C}}^T$ consists of algebras for a monad $T$ and their structure-preserving morphisms.

For $F⊣G:\mathcal{D}\to \mathcal{C}$ with induced monad $T=GF$, the Eilenberg-Moore comparison functor is

When each comparison functor has a left adjoint, iterating the Eilenberg-Moore comparison functor produces the monadic tower. Its monadic length is the least number of comparison steps needed to reach an equivalence; an equivalence has length zero and a non-equivalence that is already monadic has length one.

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 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.