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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 PROP (category theory) ： ウィキペディア英語版 PRO (category theory) In category theory, a PRO is a strict monoidal category whose objects are the natural numbers (including zero), and whose tensor product is given on objects by the addition on numbers. Some examples of PROs: * the discrete category $\backslash mathbb$ of natural numbers, * the category FinSet of natural numbers and functions between them, * the category Bij of natural numbers and bijections, * the category BijBraid of natural numbers, equipped with the braid group ''Bn ''as the automorphisms of each ''n ''(and no other morphisms). * the category Inj of natural numbers and injections, * the simplex category $\backslash Delta$ of natural numbers and monotonic functions. The name PRO is an abbreviation of "PROduct category". PROBs and PROPs are defined similarly with the additional requirement for the category to be braided, and to have a symmetry (that is, a permutation), respectively. All of the examples above are PROPs, except for the simplex category and BijBraid; the latter is a PROB but not a PROP, and the former is not even a PROB. == Algebras of a PRO == An algebra of a PRO $P$ in a monoidal category $C$ is a strict monoidal functor from $P$ to $C$. Every PRO $P$ and category $C$ give rise to a category $\backslash mathrm\_P^C$ of algebras whose objects are the algebras of $P$ in $C$ and whose morphisms are the natural transformations between them. For example: * an algebra of $\backslash mathbb$ is just an object of $C$, * an algebra of FinSet is a commutative monoid object of $C$, * an algebra of $\backslash Delta$ is a monoid object in $C$. More precisely, what we mean here by "the algebras of $\backslash Delta$ in $C$ are the monoid objects in $C$" for example is that the category of algebras of $P$ in $C$ is equivalent to the category of monoids in $C$. 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「PRO (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.