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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 Kan complex ： ウィキペディア英語版 Kan fibration In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category for simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan. ==Definition== For each ''n'' ≥ 0, recall that the standard $n$-simplex, $\backslash Delta^n$, is the representable simplicial set :$\backslash Delta^n(i)\; =\; \backslash mathrm\_\; \backslash rightarrow\; \backslash Delta^n$ corresponding to all the other faces of $\backslash Delta^n$.〔See Goerss and Jardine, page 7〕 Horns of the form $\backslash Lambda\_k^2$ sitting inside $\backslash Delta^2$ look like the black V at the top of the image to the right. If $X$ is a simplicial set, then maps : $s:\; \backslash Lambda\_k^n\; \backslash to\; X$ correspond to collections of $n+1$ $n$-simplices satisfying a compatibility condition. Explicitly, this condition can be written as follows. Write the $n$-simplices as a list $(s\_0,\backslash dots,s\_,s\_,\backslash dots,s\_)$ and require that : $d\_i\; s\_j\; =\; d\_\; s\_i\backslash ,$ for all with $i,j\; \backslash neq\; k$.〔See May, page 2〕 These conditions are satisfied for the $(n-1)$-simplices of $\backslash Lambda\_k^n$ sitting inside $\backslash Delta^n$. A map of simplicial sets $f:\; X\backslash rightarrow\; Y$ is a Kan fibration if, for any $n\backslash ge\; 1$ and $0\backslash le\; k\backslash le\; n$, and for any maps $s:\backslash Lambda^n\_k\backslash rightarrow\; X$ and $y:\backslash Delta^n\backslash rightarrow\; Y\backslash ,$ such that $f\; \backslash circ\; s=y\; \backslash circ\; i$, there exists a map $x:\backslash Delta^n\; \backslash rightarrow\; X$ such that $s=x\; \backslash circ\; i$ and $y=f\; \backslash circ\; x$. Stated this way, the definition is very similar to that of fibrations in topology (see also homotopy lifting property), whence the name "fibration". Using the correspondence between $n$-simplices of a simplicial set $X$ and morphisms $\backslash Delta^n\; \backslash to\; X$ (a consequence of the Yoneda lemma), this definition can be written in terms of simplices. The image of the map $fs:\; \backslash Lambda\_k^n\; \backslash to\; Y$ can be thought of as a horn as described above. Asking that $fs$ factors through $yi$ corresponds to requiring that there is an $n$-simplex in $Y$ whose faces make up the horn from $fs$ (together with one other face). Then the required map $x:\; \backslash Delta^n\backslash to\; X$ corresponds to a simplex in $X$ whose faces include the horn from $s$. The diagram to the right is an example in two dimensions. Since the black V in the lower diagram is filled in by the blue $2$-simplex, if the black V above maps down to it then the striped blue $2$-simplex has to exist, along with the dotted blue $1$-simplex, mapping down in the obvious way.〔May uses this simplicial definition; see page 25〕 A simplicial set ''X'' is called a Kan complex if the map from ''X'' to 1, the one-point simplicial set, is a Kan fibration. In the model category for simplicial sets, $1$ is the terminal object and so a Kan complex is exactly the same as a fibrant object. 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「Kan fibration」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.