homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

, a δ-functor between two

abelian categories In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abel ...

''A'' and ''B'' is a collection of

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

s from ''A'' to ''B'' together with a collection of

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

s that satisfy properties generalising those of

derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...

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 " Tohoku paper" to provide an appropriate setting for derived functors. Grothendieck 1957 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 functor In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom- ...

s ''T''ⁿ : ''A'' → ''B'' indexed by the

non-negative integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...

s, and for each

short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...

0\rightarrow M^\prime\rightarrow M\rightarrow M^\rightarrow0

a family of morphisms :

\delta^n:T^n(M^)\rightarrow T^(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.

Morphisms of δ-functors

A morphism of δ-functors is a family of

natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...

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 :

0\rightarrow M^\prime\rightarrow M\rightarrow M^\rightarrow0

the following diagram commutes: :

Universal δ-functor

A universal δ-functor is characterized by the (

universal Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a t ...

) property that giving a morphism from it to any other δ-functor (between ''A'' and ''B'') is equivalent to giving just ''F''₀. 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 :

F_0:S^0\rightarrow T^0

there is a unique sequence ''F''_n indexed by the positive integers such that the family _{n ≥ 0} is a morphism of δ-functors.

Notes

References

* * Section XX.7 of * Section 2.1 of {{Weibel IHA Homological algebra

Definition

Morphisms of δ-functors

Universal δ-functor

See also

Notes

References