# Delta-functor

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 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 covariant additive functors

*T*

^{n}:

*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:

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

2. For each morphism of short exact sequences

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

is commutative (the δ^{n} on the top is that corresponding to the short exact sequence of *M*'s whereas the one on the bottom corresponds to the short exact sequence of *N*'s).

The second property expresses the *functoriality* of a δ-functor. The modifier "cohomological" indicates that the δ^{n} 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**and

*A*and*B***contravariant homological δ-functor between**can also be defined by "reversing the arrows" accordingly.

*A*and*B*### 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*_{n} : S^{n} → T^{n} of natural transformations such that for every short exact sequence

the following diagram commutes:

### Universal δ-functor

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 *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

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

## See also

## References

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- Section XX.7 of Template:Lang Algebra

- Section 2.1 of Template:Weibel IHA