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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 strong monad ： ウィキペディア英語版 strong monad In category theory, a strong monad over a monoidal category (''C'', ⊗, I) is a monad (''T'', η, μ) together with a natural transformation ''t''''A,B'' : ''A'' ⊗ ''TB'' → ''T''(''A'' ⊗ ''B''), called (''tensorial'') ''strength'', such that the diagrams :, , :, and : commute for every object ''A'', ''B'' and ''C'' (see Definition 3.2 in ). If the monoidal category (''C'', ⊗, I) is closed then a strong monad is the same thing as a ''C''-enriched monad. == Commutative strong monads == For every strong monad ''T'' on a symmetric monoidal category, a ''costrength'' natural transformation can be defined by :$t\text{'}\_=T(\backslash gamma\_)\backslash circ\; t\_\backslash circ\backslash gamma\_\; :\; TA\backslash otimes\; B\backslash to\; T(A\backslash otimes\; B)$. A strong monad ''T'' is said to be commutative when the diagram : commutes for all objects $A$ and $B$. One interesting fact about commutative strong monads is that they are "the same as" symmetric monoidal monads. More explicitly, * a commutative strong monad $(T,\backslash eta,\backslash mu,t)$ defines a symmetric monoidal monad $(T,\backslash eta,\backslash mu,m)$ by :$m\_=\backslash mu\_\backslash circ\; Tt\text{'}\_\backslash circ\; t\_:TA\backslash otimes\; TB\backslash to\; T(A\backslash otimes\; B)$ * and conversely a symmetric monoidal monad $(T,\backslash eta,\backslash mu,m)$ defines a commutative strong monad $(T,\backslash eta,\backslash mu,t)$ by :$t\_=m\_\backslash circ(\backslash eta\_A\backslash otimes\; 1\_):A\backslash otimes\; TB\backslash to\; T(A\backslash otimes\; B)$ and the conversion between one and the other presentation is bijective. 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「strong monad」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.