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Skeleton (category theory)
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Skeleton (category theory) : ウィキペディア英語版
Skeleton (category theory)

In mathematics, a skeleton of a category is a subcategory which, roughly speaking, does not contain any extraneous isomorphisms. In a certain sense, the skeleton of a category is the "smallest" equivalent category which captures all "categorical properties". In fact, two categories are equivalent if and only if they have isomorphic skeletons. A category is called skeletal if isomorphic objects are necessarily identical.
== Definition ==

A skeleton of a category ''C'' is a full, isomorphism-dense subcategory ''D'' in which no two distinct objects are isomorphic. In detail, a skeleton of ''C'' is a category ''D'' such that:
*Every object of ''D'' is an object of ''C''.
*(Fullness) For every pair of objects ''d''1 and ''d''2 of ''D'', the morphisms in ''D'' are precisely the morphisms in ''C'', i.e.
:hom_D(d_1, d_2) = hom_C(d_1, d_2)
*For every object ''d'' of ''D'', the ''D''-identity on ''d'' is the ''C''-identity on ''d''.
*The composition law in ''D'' is the restriction of the composition law in ''C'' to the morphisms in ''D''.
*(Isomorphism-dense) Every ''C''-object is isomorphic to some ''D''-object.
*No two distinct ''D''-objects are isomorphic.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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