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In [[homological algebra]], a '''δ-functor''' between two [[abelian categories]] ''A'' and ''B'' is a collection of [[functor]]s from ''A'' to ''B'' together with a collection of [[morphism]]s that satisfy properties generalising those of [[derived functor]]s. 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 famous "Tohoku" paper to provide an appropriate setting for derived functors.<ref>[[#Tohoku|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 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 {''T''n} of [[functor|covariant]] [[additive functor]]s ''T''n : ''A'' → ''B'' [[Indexed family|indexed]] by the [[non-negative integer]]s, 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:

1. For each short exact sequence as above, there is a [[long exact sequence]]

:[[Image:DeltaFunctorLongExactSequence.png|480px]]

2. For each morphism of short exact sequences

:[[Image:Morphism of short exact sequences.png|400px]]

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

:[[Image:DeltaFunctorFunctoriality.png|250px]]

is commutative (the δn 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 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'' <nowiki>''</nowiki>) → ''T''n-1(''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 transformation]]s 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''n : Sn → Tn 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:

:[[Image:MorphismOfDeltaFunctors.png|720px]]

===Universal δ-functor===
A '''universal δ-functor''' is characterized by the ([[Universal property|universal]]) property that giving a morphism from it to any other δ-functor (between ''A'' and ''B'') is equivalent to giving just ''F''0. For example, 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''n indexed by the positive integers such that the family { ''F''n }n ≥ 0 is a morphism of δ-functors.

==Notes==
<references/>

==See also==
*[[Effaceable functor]]

==References==
* <cite id=Tohoku>{{Citation
| last=Grothendieck
| first=Alexander
| author-link=Alexander Grothendieck
| title=Sur quelques points d'algèbre homologique
| journal=The Tôhoku Mathematical Journal. Second series
| volume=9
| issue=2–3
| year=1957
| mr=0102537
}}</cite>

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

* Section 2.1 of {{Weibel IHA}}

[[Category:Homological algebra]]

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