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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 Reflective subcategory ： ウィキペディア英語版 Reflective subcategory In mathematics, a full subcategory ''A'' of a category ''B'' is said to be reflective in ''B'' when the inclusion functor from ''A'' to ''B'' has a left adjoint. This adjoint is sometimes called a ''reflector''. Dually, ''A'' is said to be coreflective in ''B'' when the inclusion functor has a right adjoint. Informally, a reflector acts as a kind of completion operation. It adds in any "missing" pieces of the structure in such a way that reflecting it again has no further effect. ==Definition== A full subcategory A of a category B is said to be reflective in B if for each B-object ''B'' there exists an A-object $A\_B$ and a B-morphism $r\_B\; \backslash colon\; B\; \backslash to\; A\_B$ such that for each B-morphism $f\backslash colon\; B\backslash to\; A$ to an A-object $A$ there exists a unique A-morphism $\backslash overline\; f\; \backslash colon\; A\_B\; \backslash to\; A$ with $\backslash overline\; f\backslash circ\; r\_B=f$. :File:Refl1.png The pair $(A\_B,r\_B)$ is called the A-reflection of ''B''. The morphism $r\_B$ is called A-reflection arrow. (Although often, for the sake of brevity, we speak about $A\_B$ only as about the A-reflection of ''B''). This is equivalent to saying that the embedding functor $E\backslash colon\; \backslash mathbf\; \backslash hookrightarrow\; \backslash mathbf$ is adjoint. The coadjoint functor $R\; \backslash colon\; \backslash mathbf\; B\; \backslash to\; \backslash mathbf\; A$ is called the reflector. The map $r\_B$ is the unit of this adjunction. The reflector assigns to $B$ the A-object $A\_B$ and $Rf$ for a B-morphism $f$ is determined by the commuting diagram :File:Reflsq1.png If all A-reflection arrows are (extremal) epimorphisms, then the subcategory A is said to be (extremal) epireflective. Similarly, it is bireflective if all reflection arrows are bimorphisms. All these notions are special case of the common generalization — $E$-reflective subcategory, where $E$ is a class of morphisms. The $E$-reflective hull of a class A of objects is defined as the smallest $E$-reflective subcategory containing A. Thus we can speak about reflective hull, epireflective hull, extremal epireflective hull, etc. An anti-reflective subcategory is a full subcategory A such that the only objects of B that have an A-reflection arrow are those that are already in A. Dual notions to the above mentioned notions are coreflection, coreflection arrow, (mono)coreflective subcategory, coreflective hull, anti-coreflective subcategory. 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「Reflective subcategory」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.