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.^[1] 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 {Tⁿ} of covariant additive functors Tⁿ : A → B indexed by the non-negative integers, and for each short exact sequence

${\displaystyle 0\rightarrow M^{\prime }\rightarrow M\rightarrow M^{\prime \prime }\rightarrow 0}$

a family of morphisms

${\displaystyle \delta ^{n}:T^{n}(M^{\prime \prime })\rightarrow T^{n+1}(M^{\prime })}$

indexed by the non-negative integers satisfying the following two properties:

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 δ_n a morphism T_n(M '') → 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.

See also

• Effaceable functor

Notes

References

• Grothendieck, Alexander (1957), "Sur quelques points d'algèbre homologique", The Tohoku Mathematical Journal, Second Series, 9 (2–3), MR 0102537

• Section XX.7 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001
• Section 2.1 of Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.