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In homological algebra, a δ-functor between two abelian categories ''A'' and ''B'' is a collection of functors from ''A'' to ''B'' together with a collection of morphisms that satisfy properties generalising those of derived functors. A universal δ-functor is a δ-functor satisfying a specific universal property related to extending morphisms beyond "degree 0". These notions were introduced by Alexander Grothendieck in his "Tohoku paper" to provide an appropriate setting for derived functors.〔Grothendieck 1957〕 In particular, derived functors are universal δ-functors. The terms homological δ-functor and cohomological δ-functor are sometimes used to distinguish between the case where the morphisms "go down" (''homological'') and the case where they "go up" (''cohomological''). In particular, one of these modifiers should always be used, but is often dropped. ==Definition== Given two abelian categories ''A'' and ''B'' a covariant cohomological δ-functor between ''A'' and ''B'' is a family of covariant additive functors ''T''n : ''A'' → ''B'' indexed by the non-negative integers, and for each short exact sequence : a family of morphisms : indexed by the non-negative integers satisfying the following two properties: 1. For each short exact sequence as above, there is a long exact sequence : 2. For each morphism of short exact sequences : and for each non-negative ''n'', the induced square : is commutative (the δn on the top is that corresponding to the short exact sequence of ''M'' The second property expresses the ''functoriality'' of a δ-functor. The modifier "cohomological" indicates that the δn raise the index on the ''T''. A covariant homological δ-functor between ''A'' and ''B'' is similarly defined (and generally uses subscripts), but with δn a morphism ''T''n(''M'' 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「Delta-functor」の詳細全文を読む スポンサード リンク
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