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In category theory, a regular category is a category with finite limits and coequalizers of a pair of morphisms called kernel pairs, satisfying certain ''exactness'' conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of ''images'', without requiring additivity. At the same time, regular categories provide a foundation for the study of a fragment of first-order logic, known as regular logic. == Definition == A category ''C'' is called regular if it satisfies the following three properties:〔Pedicchio & Tholen (2004) p.177〕 * ''C'' is finitely complete. * If ''f:X→Y'' is a morphism in ''C'', and : is a pullback, then the coequalizer of ''p0,p1'' exists. The pair (''p0,p1'') is called the kernel pair of ''f''. Being a pullback, the kernel pair is unique up to a unique isomorphism. * If ''f:X→Y'' is a morphism in ''C'', and : is a pullback, and if ''f'' is a regular epimorphism, then ''g'' is a regular epimorphism as well. A regular epimorphism is an epimorphism which appears as a coequalizer of some pair of morphisms. 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「Regular category」の詳細全文を読む スポンサード リンク
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