|
In category theory, a Kleisli category is a category naturally associated to any monad ''T''. It is equivalent to the category of free ''T''-algebras. The Kleisli category is one of two extremal solutions to the question ''Does every monad arise from an adjunction?'' The other extremal solution is the Eilenberg–Moore category. Kleisli categories are named for the mathematician Heinrich Kleisli. ==Formal definition== Let〈''T'', η, μ〉be a monad over a category ''C''. The Kleisli category of ''C'' is the category ''C''''T'' whose objects and morphisms are given by : That is, every morphism ''f: X → T Y'' in ''C'' (with codomain ''TY'') can also be regarded as a morphism in ''C''''T'' (but with codomain ''Y''). Composition of morphisms in ''C''''T'' is given by : where ''f: X → T Y'' and ''g: Y → T Z''. The identity morphism is given by the monad unit η: :. An alternative way of writing this, which clarifies the category in which each object lives, is used by Mac Lane.〔Mac Lane(1998) p.147〕 We use very slightly different notation for this presentation. Given the same monad and category as above, we associate with each object in a new object , and for each morphism in a morphism . Together, these objects and morphisms form our category , where we define : Then the identity morphism in is : 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「Kleisli category」の詳細全文を読む スポンサード リンク
|