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. 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' ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Effaceable Functor
In mathematics, an effaceable functor is an additive functor ''F'' between abelian category, abelian categories ''C'' and ''D'' for which, for each object ''A'' in ''C'', there exists a monomorphism u: A \to M, for some ''M'', such that F(u) = 0. Similarly, a coeffaceable functor is one for which, for each ''A'', there is an epimorphism into ''A'' that is killed by ''F''. The notions were introduced in Grothendieck's Tohoku paper. A theorem of Grothendieck says that every effaceable Delta-functor, δ-functor (i.e., effaceable in each degree) is universal. References * External links Meaning of “efface” in “effaceable functor” and “injective effacement”
Functors {{categorytheory-stub ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]