Delta-functor explained

Delta-functor should not be confused with 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.^[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 of covariant additive functors Tⁿ : A → B indexed by the non-negative integers, and for each short exact sequence

0 → M ^{\prime →} M → M ^{\prime\prime} → 0 a family of morphisms\delta ^{n:T} ^{n(M} ^{\prime\prime} ) → T ^{n+1} (M ^{\prime)} indexed by the non-negative integers satisfying the following two properties:

Notes and References

1. [#Tohoku|Grothendieck 1957]