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.
These notions were introduced by Alexander Grothendieck in his "Tohoku paper" to provide an appropriate setting for derived functors.
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 a family of morphisms indexed by the non-negative integers satisfying the following two properties: and for each non-negative n, the induced square 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 Tn(M '') → Tn-1(M').
For example, in the case of two covariant cohomological δ-functors denoted S and T, a morphism from S to T is a family Fn : Sn → Tn of natural transformations such that for every short exact sequence the following diagram commutes: A universal δ-functor is characterized by the (universal) property that giving a morphism from it to any other δ-functor (between A and B) is equivalent to giving just F0.