{{Short description|Functor between abelian categories}} {{distinguish|Delta-function}} 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.<ref>Grothendieck 1957</ref> 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 is always implicit, although often left unstated.

==Definition== Given two abelian categories ''A'' and ''B'' a '''covariant cohomological δ-functor between ''A'' and ''B''''' is a family {''T''<sup>n</sup>} of covariant additive functors ''T''<sup>n</sup> : ''A'' → ''B'' indexed by the non-negative integers, and for each short exact sequence :<math>0\rightarrow M^\prime\rightarrow M\rightarrow M^{\prime\prime}\rightarrow0</math> a family of morphisms :<math>\delta^n:T^n(M^{\prime\prime})\rightarrow T^{n+1}(M^\prime)</math> indexed by the non-negative integers satisfying the following two properties: {{ordered list | 1 = For each short exact sequence as above, there is a long exact sequence

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| 2 = For each morphism of short exact sequences

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and for each non-negative ''n'', the induced square

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is commutative (the δ<sup>n</sup> on the top is that corresponding to the short exact sequence of ''M''<nowiki>'</nowiki>s whereas the one on the bottom corresponds to the short exact sequence of ''N''<nowiki>'</nowiki>s). }} The second property expresses the ''functoriality'' of a δ-functor. The modifier "cohomological" indicates that the δ<sup>n</sup> raise the index on the ''T''. A '''covariant homological δ-functor between ''A'' and ''B''''' is similarly defined (and generally uses subscripts), but with δ<sub>n</sub> a morphism ''T''<sub>n</sub>(''M'' <nowiki>''</nowiki>) → ''T''<sub>n-1</sub>(''M'''). The notions of '''contravariant cohomological δ-functor between ''A'' and ''B''''' and '''contravariant homological δ-functor between ''A'' and ''B''''' can also be defined by "reversing the arrows" accordingly.

===Morphisms of δ-functors=== A '''morphism of δ-functors''' is a family of natural transformations that, for each short exact sequence, commute with the morphisms δ. For example, in the case of two covariant cohomological δ-functors denoted ''S'' and ''T'', a morphism from ''S'' to ''T'' is a family ''F''<sub>n</sub> : S<sup>n</sup> → T<sup>n</sup> of natural transformations such that for every short exact sequence :<math>0\rightarrow M^\prime\rightarrow M\rightarrow M^{\prime\prime}\rightarrow0</math> the following diagram commutes:

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===Universal δ-functor=== A '''universal δ-functor''' is characterized by the (universal) property that giving a morphism ''F'' from it to any other δ-functor (between ''A'' and ''B'') is equivalent to giving just ''F''<sub>0</sub>. If ''S'' denotes a covariant cohomological δ-functor between ''A'' and ''B'', then ''S'' is universal if given any other (covariant cohomological) δ-functor ''T'' (between ''A'' and ''B''), and given any natural transformation :<math>F_0:S^0\rightarrow T^0</math> there is a unique sequence ''F''<sub>n</sub> indexed by the positive integers such that the family { ''F''<sub>n</sub> }<sub>n ≥ 0</sub> is a morphism of δ-functors.

==See also== *Effaceable functor

==Notes== <references/>

==References== * {{citation | last=Grothendieck | first=Alexander | author-link=Alexander Grothendieck | title=Sur quelques points d'algèbre homologique | journal=The Tohoku Mathematical Journal |series=Second Series | volume=9 | issue=2–3 | year=1957 | mr=0102537 |ref=Tohoku}}

* Section XX.7 of {{Lang Algebra|edition=3r}} * Section 2.1 of {{Weibel IHA}}

Category:Homological algebra