Monad (category Theory) - Formal Definition

Formal Definition

If is a category, a monad on consists of a functor together with two natural transformations: (where denotes the identity functor on ) and (where is the functor from to ). These are required to fulfill the following conditions (sometimes called coherence conditions):

• (as natural transformations );
• (as natural transformations ; here denotes the identity transformation from to ).

We can rewrite these conditions using following commutative diagrams:

See the article on natural transformations for the explanation of the notations and, or see below the commutative diagrams not using these notions:

The first axiom is akin to the associativity in monoids, the second axiom to the existence of an identity element. Indeed, a monad on can alternatively be defined as a monoid in the category whose objects are the endofunctors of and whose morphisms are the natural transformations between them, with the monoidal structure induced by the composition of endofunctors.