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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 Simplicial presheaf ： ウィキペディア英語版 Simplicial presheaf In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s.〔http://ncatlab.org/nlab/files/ToenStacksNAC.pdf〕 Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site. Example: Let us consider, say, the étale site of a scheme ''S''. Each ''U'' in the site represents the presheaf $\backslash operatorname(-,\; U)$. Thus, a simplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf). Example: Let ''G'' be a presheaf of groupoids. Then taking nerves section-wise, one obtains a simplicial presheaf $BG$. For example, one might set $B\backslash operatorname\; =\; \backslash varinjlim\; B\backslash operatorname$. These types of examples appear in K-theory. If $f:\; X\; \backslash to\; Y$ is a local weak equivalence of simplicial presheaves, then the induced map $\backslash mathbb\; f:\; \backslash mathbb\; X\; \backslash to\; \backslash mathbb\; Y$ is also a local weak equivalence. == Homotopy sheaves of a simplicial presheaf == Let ''F'' be a simplicial presheaf on a site. The homotopy sheaves $\backslash pi\_$ * F of ''F'' is defined as follows. For any $f:X\; \backslash to\; Y$ in the site and a 0-simplex ''s'' in ''F''(''X''), set $(\backslash pi\_0^\backslash text\; F)(X)\; =\; \backslash pi\_0\; (F(X))$ and $(\backslash pi\_i^\backslash text\; (F,\; s))(f)\; =\; \backslash pi\_i\; (F(Y),\; f^$ *(s)). We then set $\backslash pi\_i\; F$ to be the sheaf associated with the pre-sheaf $\backslash pi\_i^\backslash text\; F$. 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「Simplicial presheaf」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.