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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 Eckmann–Hilton duality ： ウィキペディア英語版 Eckmann–Hilton duality In the mathematical disciplines of algebraic topology and homotopy theory, Eckmann–Hilton duality in its most basic form, consists of taking a given diagram for a particular concept and reversing the direction of all arrows, much as in category theory with the idea of the opposite category. It is named after Beno Eckmann and Peter Hilton. For example, the fact that the dual notion of a limit is a colimit allows us to change the Eilenberg–Steenrod axioms for homology to give axioms for cohomology. Another example is given by currying, which tells us that for any object $X$, a map $X\; \backslash times\; I\; \backslash to\; Y$ is the same as a map $X\; \backslash to\; Y^I$, where $Y^I$ is the exponential object, given by all maps from $I$ to $Y$. In the case of topological spaces, if we take $I$ to be the unit interval, this leads to a duality between $X\; \backslash times\; I$ and $Y^I$ which then gives a duality between the reduced suspension $\backslash Sigma\; X$ which is a quotient of $X\; \backslash times\; I$ and the loop space $\backslash Omega\; Y$ which is a subspace of $Y\; ^\; I$. This then leads to the adjoint relation $\backslash langle\; \backslash Sigma\; X,\; Y\; \backslash rangle\; =\; \backslash langle\; X,\; \backslash Omega\; Y\; \backslash rangle$ which allows the study of spectra which give rise to cohomology theories. We can also directly relate fibrations and cofibrations: a fibration $p\; \backslash colon\; E\; \backslash to\; B$ is defined by having the homotopy lifting property, represented by the following diagram and a cofibration $i\; \backslash colon\; A\; \backslash to\; X$ is defined by having the dual homotopy extension property, represented by dualising the previous diagram: The above considerations also apply when looking at the sequences associated to a fibration or a cofibration, as given a fibration $F\; \backslash to\; E\; \backslash to\; B$ we get the sequence : $\backslash cdots\; \backslash to\; \backslash Omega^2\; B\; \backslash to\; \backslash Omega\; F\; \backslash to\; \backslash Omega\; E\; \backslash to\; \backslash Omega\; B\; \backslash to\; F\; \backslash to\; E\; \backslash to\; B\; \backslash ,$ and given a cofibration $A\; \backslash to\; X\; \backslash to\; X/A$ we get the sequence : $A\; \backslash to\; X\; \backslash to\; X/A\; \backslash to\; \backslash Sigma\; A\; \backslash to\; \backslash Sigma\; X\; \backslash to\; \backslash Sigma\; \backslash left\; (X/A\; \backslash right\; )\; \backslash to\; \backslash Sigma^2\; A\; \backslash to\; \backslash cdots.\; \backslash ,$ This also allows us to relate homotopy and cohomology: we know that homotopy groups are homotopy classes of maps from the ''n''-sphere to our space, written $\backslash pi\_n(X,p)\; \backslash cong\; \backslash langle\; S^n,X\; \backslash rangle$, and we know that the sphere has a single nonzero (reduced) cohomology group. On the other hand, cohomology groups are homotopy classes of maps to spaces with a single nonzero homotopy group. This is given by the Eilenberg–MacLane spaces $K(G,n)$ and the relation $H^n(X;G)\; \backslash cong\; \backslash langle\; X,K(G,n)\; \backslash rangle$. == References == * 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「Eckmann–Hilton duality」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.