Editing Monad (category theory) (section)

{{Short description|Operation in algebra and mathematics}}
{{distinguish|Monad (homological algebra)}}
{{for|the uses of monads in computer software|monads in functional programming}}

In [[category theory]], a branch of [[mathematics]], a '''monad''' is a triple <math>(T, \eta, \mu)</math> consisting of a [[functor]] ''T'' from a category to itself and two [[natural transformation]]s <math>\eta, \mu</math> that satisfy the conditions like associativity. For example, if <math>F, G</math> are functors [[adjoint functor|adjoint]] to each other, then <math>T = G \circ F</math> together with <math>\eta, \mu</math> determined by the adjoint relation is a monad.

In concise terms, a monad is a [[Monoid (category theory)|monoid]] in the [[Category (mathematics)|category]] of [[endofunctor]]s of some fixed category (an endofunctor is a [[functor]] mapping a category to itself). According to [[John C. Baez|John Baez]], a monad can be considered at least in two ways: <ref name="Baez"> https://golem.ph.utexas.edu/category/2009/07/the_monads_hurt_my_head_but_no.html</ref>
# A monad as a generalized monoid; this is clear since a monad is a monoid in a certain category,
# A monad as a tool for studying [[Algebraic gadget|algebraic gadgets]]; for example, a [[group (mathematics)|group]] can be described by a certain monad.

Monads are used in the theory of pairs of [[adjoint functors]], and they generalize [[closure operator]]s on [[partially ordered set]]s to arbitrary categories. Monads are also useful in the [[Type theory|theory of datatypes]], the [[denotational semantics]] of [[Imperative programming language|imperative programming languages]], and in [[functional programming language]]s, allowing languages without mutable state to do things such as simulate [[for-loop]]s; see [[Monad (functional programming)]].

A monad is also called, especially in old literature, a '''triple''', '''triad''', '''standard construction''' and '''fundamental construction'''.<ref>{{citation|url=http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf | title=Toposes, Triples and Theories | year=1985 | first1=Michael | last1=Barr | first2=Charles | last2= Wells | publisher=Springer-Verlag | isbn=0-387-96115-1 | volume=278 | work= Grundlehren der mathematischen Wissenschaften |pages=82 and 120 |postscript=.}}</ref>