Monad (category Theory) - Monads and Adjunctions

Monads and Adjunctions

An adjunction between two categories and (where is left adjoint to and and are respectively the unit and the counit) always defines a monad .

Conversely, it is interesting to consider the adjunctions which define a given monad this way. Let be the category whose objects are the adjunctions such that and whose arrows are the morphisms of adjunctions which are the identity on . Then this category has

• an initial object, where is the Kleisli category,
• a terminal object, where is the Eilenberg-Moore category.

An adjunction between two categories and is a monadic adjunction when the category is equivalent to the Eilenberg-Moore category for the monad . By extension, a functor is said to be monadic if it has a left adjoint forming a monadic adjunction. Beck's monadicity theorem gives a characterization of monadic functors.