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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 Monoidal monad ： ウィキペディア英語版 Monoidal monad In category theory, a monoidal monad $(T,\backslash eta,\backslash mu,m)$ is a monad $(T,\backslash eta,\backslash mu)$ on a monoidal category $(C,\backslash otimes,I)$ such that the functor :$T:(C,\backslash otimes,I)\backslash to(C,\backslash otimes,I)$ is a lax monoidal functor and the natural transformations $\backslash eta,\backslash mu$ are monoidal natural transformations. In other words, $T$ is equipped with coherence maps :$m\_:TA\backslash otimes\; TB\backslash to\; T(A\backslash otimes\; B)$ and :$m:I\backslash to\; TI$ satisfying certain properties, and its structure maps :$\backslash eta:\; id\; \backslash Rightarrow\; T$ and :$\backslash mu:T^2\backslash Rightarrow\; T$ must be monoidal with respect to $(C,\backslash otimes,I)$. By monoidality of $\backslash eta$, the morphisms $m$ and $\backslash eta\_I$ are necessarily equal. This is equivalent to saying that a monoidal monad is a monad in the 2-category MonCat of monoidal categories, monoidal functors, and monoidal natural transformations. ==Hopf monads and bimonads== Ieke Moerdijk introduced the notion of a Hopf monad, which is an opmonoidal monad, that is, a monad with coherence morphisms $m^:T(A\backslash otimes\; B)\; \backslash to\; TA\backslash otimes\; TB$ and $m^0:TI\backslash to\; I$ and opmonoidal natural transformations as multiplication and left/right units. An easy example for the category $\backslash operatorname$ of vector spaces is the monad $-\; \backslash otimes\; A$, where $A$ is a bialgebra.〔 The multiplication in $A$ then defines the multiplication of the monad, while the comultiplication gives rise to the opmonoidal structure. The algebras of this monad are just right $A$-modules. In works of Bruguières and Virelizier, this concept has been renamed bimonad, by analogy to "bialgebra". They reserve the term "Hopf monad" for bimonads with an antipode, in analogy to "Hopf algebras". 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「Monoidal 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.