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. 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 ${\displaystyle F}$ and ${\displaystyle G}$ are a pair of adjoint functors, with ${\displaystyle F}$ left adjoint to ${\displaystyle G}$, then the composition ${\displaystyle G\circ F}$ is a monad. If ${\displaystyle F}$ and ${\displaystyle G}$ 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 ${\displaystyle F\circ G}$, is discussed under the dual theory of comonads.

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