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.
Read more about this topic: Monad (category Theory)
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