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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Monad (category theory) ： ウィキペディア英語版 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. == Introduction == If $F$ and $G$ are a pair of adjoint functors, with $F$ left adjoint to $G$, then the composition $G\; \backslash circ\; F$ is a monad. Therefore, a monad is an endofunctor. If $F$ and $G$ 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 $F\; \backslash circ\; G$, is discussed under the dual theory of comonads. The monad axioms can be seen at work in a simple example: let $G$ be the forgetful functor from the category Grp of groups to the category Set of sets. Then let $F$ be the free group functor. This means that the monad :$T\; =\; G\; \backslash circ\; F$ takes a set $X$ and returns the underlying set of the free group $\backslash mathrm(X)$. In this situation, we are given two natural morphisms: :$X\; \backslash rightarrow\; T(X)$ by including any set $X$ into the set $\backslash mathrm(X)$ in the natural way, as strings of length 1. Further, :$T(T(X))\; \backslash rightarrow\; T(X)$ can be made out of a natural concatenation or 'flattening' of 'strings of strings'. This amounts to two natural transformations :$I\; \backslash rightarrow\; T$ and :$T\; \backslash circ\; T\; \backslash rightarrow\; T$ They will satisfy some axioms about identity and associativity that result from the adjunction properties. Those axioms are formally similar to the monoid axioms. They are taken as the definition of a general monad (not assumed ''a priori'' to be connected to an adjunction) on a category. If we specialize to categories arising from partially ordered sets $(P,\; \backslash le)$ (with a single morphism from $x$ to $y$ iff $x\; \backslash le\; y$), then the formalism becomes much simpler: adjoint pairs are Galois connections and monads are closure operators. Every monad arises from some adjunction, in fact typically from many adjunctions. Two constructions introduced below, the Kleisli category and the category of Eilenberg-Moore algebras, are extremal solutions of the problem of constructing an adjunction that gives rise to a given monad. The example about free groups given above can be generalized to any type of algebra in the sense of a variety of algebras in universal algebra. Thus, every such type of algebra gives rise to a monad on the category of sets. Importantly, the algebra type can be recovered from the monad (as the category of Eilenberg-Moore algebras), so monads can also be seen as generalizing universal algebras. Even more generally, any adjunction is said to be monadic (or tripleable) if it shares this property of being (equivalent to) the Eilenberg-Moore category of its associated monad. Consequently, Beck's monadicity theorem, which gives a criterion for monadicity, can be used to show that an arbitrary adjunction can be treated as a category of algebras in this way. The notion of monad was invented by Roger Godement in 1958 under the name "standard construction." In the 1960s and 1970s, many people used the name "triple." The now standard term "monad" is due to Saunders Mac Lane. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Monad (category theory)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.