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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 Diagonal functor ： ウィキペディア英語版 Diagonal functor In category theory, for any object $a$ in any category $\backslash mathcal$ where the product $a\backslash times\; a$ exists, there exists the diagonal morphism :$\backslash delta\_a\; :\; a\; \backslash rightarrow\; a\; \backslash times\; a$ satisfying :$\backslash pi\_k\; \backslash circ\; \backslash delta\_a\; =\; id\_a$ for $k\; \backslash in\; \backslash$, where $\backslash pi\_k$ is the canonical projection morphism to the $k$-th component. The existence of this morphism is a consequence of the universal property which characterizes the product (up to isomorphism). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly for arbitrary products. The image of a diagonal morphism in the category of sets, as a subset of the Cartesian product, is a relation on the domain, namely equality. For concrete categories, the diagonal morphism can be simply described by its action on elements $x$ of the object $a$. Namely, $\backslash delta\_a(x)\; =\; \backslash langle\; x,x\; \backslash rangle$, the ordered pair formed from $x$. The reason for the name is that the image of such a diagonal morphism is diagonal (whenever it makes sense), for example the image of the diagonal morphism $\backslash mathbb\; \backslash rightarrow\; \backslash mathbb^2$ on the real line is given by the line which is a graph of the equation $y=x$. The diagonal morphism into the infinite product $X^\backslash infty$ may provide an injection into the space of sequences valued in $X$; each element maps to the constant sequence at that element. However, most notions of sequence spaces have convergence restrictions which the image of the diagonal map will fail to satisfy. In particular, the category of small categories has products, and so one finds the diagonal functor $\backslash mathcal\; \backslash rightarrow\; \backslash mathcal\; \backslash times\; \backslash mathcal$ given by $\backslash Delta(a)\; =\; \backslash langle\; a,a\; \backslash rangle$, which maps objects as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects ''within'' the category $\backslash mathcal$: a product $a\; \backslash times\; b$ is a universal arrow from $\backslash Delta$ to $\backslash langle\; a,b\; \backslash rangle$. The arrow comprises the projection maps. More generally, in any functor category $\backslash mathcal^\backslash mathcal$ (here $\backslash mathcal$ should be thought of as a small index category), for each object $a$ in $\backslash mathcal$, there is a constant functor with fixed object $a$: $\backslash Delta(a)\; \backslash in\; \backslash mathcal^\backslash mathcal$. The diagonal functor $\backslash Delta\; :\; \backslash mathcal\; \backslash rightarrow\; \backslash mathcal^\backslash mathcal$ assigns to each object of $\backslash mathcal$ the functor $\backslash Delta(a)$, and to each morphism $f:\; a\; \backslash rightarrow\; b$ in $\backslash mathcal$ the obvious natural transformation $\backslash eta$ in $\backslash mathcal^\backslash mathcal$ (given by $\backslash eta\_j\; =\; f$). In the case that $\backslash mathcal$ is a discrete category with two objects, the diagonal functor $\backslash mathcal\; \backslash rightarrow\; \backslash mathcal\; \backslash times\; \backslash mathcal$ is recovered. Diagonal functors provide a way to define limits and colimits of functors. The limit of any functor $\backslash mathcal\; :\; \backslash mathcal\; \backslash rightarrow\; \backslash mathcal$ is a universal arrow from $\backslash Delta$ to $\backslash mathcal$ and a colimit is a universal arrow $F\; \backslash rightarrow\; \backslash Delta$. If every functor from $\backslash mathcal$ to $\backslash mathcal$ has a limit (which will be the case if $\backslash mathcal$ is complete), then the operation of taking limits is itself a functor from $\backslash mathcal^\backslash mathcal$ to $\backslash mathcal$. The limit functor is the right-adjoint of the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor. For example, the diagonal functor $\backslash mathcal\; \backslash rightarrow\; \backslash mathcal\; \backslash times\; \backslash mathcal$ described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor. 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「Diagonal functor」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.