Monad (category theory)

In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is an endofunctor (a functor mapping a category to itself), together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories. In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is an endofunctor (a functor mapping a category to itself), together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories. A monad is a certain type of endofunctor. For example, if F {displaystyle F} and G {displaystyle G} are a pair of adjoint functors, with F {displaystyle F} left adjoint to G {displaystyle G} , then the composition G ∘ F {displaystyle Gcirc F} is a monad. If F {displaystyle F} and G {displaystyle G} are inverse functors, the corresponding monad is the identity functor. In general, adjunctions are not equivalences—they relate categories of different natures. The monad theory matters as part of the effort to capture what it is that adjunctions 'preserve'. The other half of the theory, of what can be learned likewise from consideration of F ∘ G {displaystyle Fcirc G} , is discussed under the dual theory of comonads. Throughout this article C {displaystyle C} denotes a category. A monad on C {displaystyle C} consists of an endofunctor T : C → C {displaystyle Tcolon C o C} together with two natural transformations: η : 1 C → T {displaystyle eta colon 1_{C} o T} (where 1 C {displaystyle 1_{C}} denotes the identity functor on C {displaystyle C} ) and μ : T 2 → T {displaystyle mu colon T^{2} o T} (where T 2 {displaystyle T^{2}} is the functor T ∘ T {displaystyle Tcirc T} from C {displaystyle C} to C {displaystyle C} ). These are required to fulfill the following conditions (sometimes called coherence conditions): We can rewrite these conditions using the following commutative diagrams: See the article on natural transformations for the explanation of the notations T μ {displaystyle Tmu } and μ T {displaystyle mu T} , or see below the commutative diagrams not using these notions: The first axiom is akin to the associativity in monoids if we think of μ {displaystyle mu } as the monoid's binary operation, and the second axiom is akin to the existence of an identity element (which we think of as given by η {displaystyle eta } ). Indeed, a monad on C {displaystyle C} can alternatively be defined as a monoid in the category E n d C {displaystyle mathbf {End} _{C}} whose objects are the endofunctors of C {displaystyle C} and whose morphisms are the natural transformations between them, with the monoidal structure induced by the composition of endofunctors. The power set monad is a monad on the category S e t {displaystyle mathbf {Set} } : For a set A {displaystyle A} let T ( A ) {displaystyle T(A)} be the power set of A {displaystyle A} and for a function f : A → B {displaystyle fcolon A o B} let T ( f ) {displaystyle T(f)} be the function between the power sets induced by taking direct images under f {displaystyle f} . For every set A {displaystyle A} , we have a map η A : A → T ( A ) {displaystyle eta _{A}colon A o T(A)} , which assigns to every a ∈ A {displaystyle ain A} the singleton { a } {displaystyle {a}} . The function takes a set of sets to its union. These data describe a monad. The axioms of a monad are formally similar to the monoid axioms. In fact, monads are special cases of monoids, namely they are precisely the monoids among endofunctors End ⁡ ( C ) {displaystyle operatorname {End} (C)} , which is equipped with the multiplication given by composition of endofunctors.