Monad (category Theory) - Algebras For A Monad

Algebras For A Monad

Suppose that is a given monad on a category .

A -algebra is an object of together with an arrow of called the structure map of the algebra such that the diagrams

commute.

A morphism of -algebras is an arrow of such that the diagram

commutes.

The category of -algebras and their morphisms is called the Eilenberg-Moore category or category of (Eilenberg-Moore) algebras of the monad . The forgetful functor → has a left adjoint → taking to the free algebra .

Given the monad, there exists another "canonical" category called the Kleisli category of the monad . This category is equivalent to the category of free algebras for the monad, i. e. the full subcategory of whose objects are of the form, for an object of .