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In the mathematical field category theory, an allegory is a category that has some of the structure of the category of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in this sense the theory of allegories is a generalization of relation algebra to relations between different sorts. Allegories are also useful in defining and investigating certain constructions in category theory, such as exact completions. In this article we adopt the convention that morphisms compose from right to left, so ''RS'' means "first do ''S'', then do ''R''". ==Definition== An allegory is a category in which * every morphism ''R'':''X→Y'' is associated with an anti-involution, i.e. a morphism ''R''°:''Y→X''; and * every pair of morphisms ''R'',''S'':''X''→''Y'' with common domain/codomain is associated with an intersection, i.e. a morphism ''R''∩''S'':''X''→''Y'' all such that * intersections are idempotent (''R''∩''R''=''R''), commutative (''R''∩''S''=''S''∩''R''), and associative (''R''∩''S'')∩''T''=''R''∩(''S''∩''T''); * anti-involution distributes over composition ((''RS'')°=''S''°''R''°) and intersection ((''R''∩''S'')°=''S''°∩''R''°); * composition is semi-distributive over intersection (''R''(''S''∩''T'')⊆''RS''∩''RT'', (''R''∩''S'')''T''⊆''RT''∩''ST''); and * the modularity law is satisfied: (''RS''∩''T''⊆(''R''∩''TS''°)''S''). Here, we are abbreviating using the order defined by the intersection: "''R''⊆''S''" means "''R''=''R''∩''S''". A first example of an allegory is the category of sets and relations. The objects of this allegory are sets, and a morphism ''X→Y'' is a binary relation between ''X'' and ''Y''. Composition of morphisms is composition of relations; intersection of morphisms is intersection of relations. 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「Allegory (category theory)」の詳細全文を読む スポンサード リンク
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