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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 functor ： ウィキペディア英語版 Monoidal functor In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two ''coherence maps''—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors * The coherence maps of lax monoidal functors satisfy no additional properties; they are not necessarily invertible. * The coherence maps of strong monoidal functors are invertible. * The coherence maps of strict monoidal functors are identity maps. Although we distinguish between these different definitions here, authors may call any one of these simply monoidal functors. == Definition == Let $(\backslash mathcal\; C,\backslash otimes,I\_)$ and $(\backslash mathcal\; D,\backslash bullet,I\_)$ be monoidal categories. A monoidal functor from $\backslash mathcal\; C$ to $\backslash mathcal\; D$ consists of a functor $F:\backslash mathcal\; C\backslash to\backslash mathcal\; D$ together with a natural transformation :$\backslash phi\_:FA\backslash bullet\; FB\backslash to\; F(A\backslash otimes\; B)$ between functors $\backslash mathcal\backslash times\backslash mathcal\backslash to\backslash mathcal$ and a morphism :$\backslash phi:I\_\backslash to\; FI\_$, called the coherence maps or structure morphisms, which are such that for every three objects $A$, $B$ and $C$ of $\backslash mathcal\; C$ the diagrams :, :    and    commute in the category $\backslash mathcal\; D$. Above, the various natural transformations denoted using $\backslash alpha,\; \backslash rho,\; \backslash lambda$ are parts of the monoidal structure on $\backslash mathcal\; C$ and $\backslash mathcal\; D$. 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 functor」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.