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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 extension ： ウィキペディア英語版 Kan extension Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan, who constructed certain (Kan) extensions using limits in 1960. An early use of (what is now known as) a Kan extension from 1956 was in homological algebra to compute derived functors. In ''Categories for the Working Mathematician'' Saunders Mac Lane titled a section "All Concepts Are Kan Extensions", and went on to write that :The notion of Kan extensions subsumes all the other fundamental concepts of category theory. Kan extensions generalize the notion of extending a function defined on a subset to a function defined on the whole set. The definition, not surprisingly, is at a high level of abstraction. When specialised to posets, it becomes a relatively familiar type of question on 'constrained optimization'. ==Definition== A Kan extension proceeds from the data of three categories :$\backslash mathbf,\; \backslash mathbf,\; \backslash mathbf$ and two functors :$X\; \backslash colon\; \backslash mathbf\; \backslash to\; \backslash mathbf,\; F\; \backslash colon\; \backslash mathbf\; \backslash to\; \backslash mathbf$, and comes in two varieties: the "left" Kan extension and the "right" Kan extension of $X$ along $F$. It amounts to finding the dashed arrow and the 2-cell $\backslash eta$ in the following diagram: : Right Kan Extension Formally, the ''right Kan extension of $X$ along $F$'' consists of a functor $R\; \backslash colon\; \backslash mathbf\; \backslash to\; \backslash mathbf$ and a natural transformation $\backslash eta\; \backslash colon\; RF\; \backslash to\; X$ which is couniversal with respect to the specification, in the sense that for any functor $M\; \backslash colon\; \backslash mathbf\; \backslash to\; \backslash mathbf$ and natural transformation $\backslash mu\; \backslash colon\; MF\; \backslash to\; X$, a unique natural transformation $\backslash delta\; \backslash colon\; M\; \backslash to\; R$ is defined and fits into a commutative diagram : (where $\backslash delta\_F$ is the natural transformation with $\backslash delta\_F(a)\; =\; \backslash delta(Fa)\; \backslash colon\; MF(a)\; \backslash to\; RF(a)$ for any object $a$ of $\backslash mathbf$). The functor ''R'' is often written $\backslash operatorname\_FX$. As with the other universal constructs in category theory, the "left" version of the Kan extension is dual to the "right" one and is obtained by replacing all categories by their opposites. The effect of this on the description above is merely to reverse the direction of the natural transformations (recall that a natural transformation $T$ between the functors $F,G\; \backslash colon\; \backslash mathbf\; \backslash to\; \backslash mathbf$ consists of the data of an arrow $T(a)\; \backslash colon\; F(a)\; \backslash to\; G(a)$ for every object $a$ of $\backslash mathbf$, satisfying a "naturality" property. When we pass to the opposite categories, the source and target of $T(a)$ are swapped, causing $T$ to act in the opposite direction). This gives rise to the alternate description: the ''left Kan extension of $X$ along $F$'' consists of a functor $L\; \backslash colon\; \backslash mathbf\; \backslash to\; \backslash mathbf$ and a natural transformation $\backslash epsilon\; \backslash colon\; X\; \backslash to\; L\; F$ which are universal with respect to this specification, in the sense that for any other functor $M\; \backslash colon\; \backslash mathbf\; \backslash to\; \backslash mathbf$ and natural transformation $\backslash alpha\; \backslash colon\; X\; \backslash to\; M\; F$, a unique natural transformation $\backslash sigma\; \backslash colon\; L\; \backslash to\; M$ exists and fits into a commutative diagram: : (where $\backslash sigma\_F$ is the natural transformation with $\backslash sigma\_F(a)\; =\; \backslash sigma(Fa)\; \backslash colon\; LF(a)\; \backslash to\; MF(a)$ for any object $a$ of $\backslash mathbf$). The functor ''L'' is often written $\backslash operatorname\_FX$. The use of the word "the" (as in "the left Kan extension") is justified by the fact that, as with all universal constructions, if the object defined exists, then it is unique up to unique isomorphism. In this case, that means that (for left Kan extensions) if $L,\; M$ are two left Kan extensions of $X$ along $F$, and $\backslash epsilon,\; \backslash alpha$ are the corresponding transformations, then there exists a unique ''isomorphism'' of functors $\backslash sigma\; \backslash colon\; L\; \backslash to\; M$ such that the second diagram above commutes. Likewise for right Kan extensions. 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 extension」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.