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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] |
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-set Hom(''A'',''B'') in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas: f\circ (g + h) = (f\circ g) + (f\circ h) and (f + g)\circ h = (f\circ h) + (g\circ h), where + is the group operation. Some authors have used the term ''additive category'' for preadditive categories, but here we follow the current trend of reserving this term for certain special preadditive categories (see below). Examples The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. Note that commutativity is crucial here; it ensures that the sum of two gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Category
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 abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very ''stable'' categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow such that for all objects and all morphisms , : f \circ g_1 = f \circ g_2 \implies g_1 = g_2. Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a category ''C'' is an epimorphism in the dual category ''C''op. Every section is a monomorphism, and every retraction is an epimorphism. Relation to invertibility Left-invertible morphisms are necessarily monic: if ''l'' is a left inverse for ''f'' (meani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
