# 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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative attempt to unify several cohomology theories by Alexander Grothendieck. 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 are named after Niels Henrik Abel.

Read more about Abelian Category:  Definitions, Examples, Grothendieck's Axioms, Elementary Properties, Related Concepts, History

### Other articles related to "abelian category, abelian, category":

Abelian Category - History
... Abelian categories were introduced by Buchsbaum (1955) (under the name of "exact category") and Grothendieck (1957) in order to unify various cohomology theories ... In fact, much of category theory was developed as a language to study these similarities ... two theories they both arise as derived functors on abelian categories the abelian category of sheaves of abelian groups on a topological space, and the abelian category of G-modules for a given ...
Mapping Cone (homological Algebra) - Properties
... Suppose now that we are working over an abelian category, so that the cohomology of a complex is defined ... triangle where the maps are the projections onto the direct summands (see Homotopy category of chain complexes) ... This fact recalls the usual alternative characterization of isomorphisms in an abelian category as those maps whose kernel and cokernel both vanish ...
Category Of Abelian Groups
... In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms ... This is the prototype of an abelian category ... Note that Ab is a full subcategory of Grp, the category of all groups ...
Exact Functor - Examples
... The most important examples of left exact functors are the Hom functors if A is an abelian category and A is an object of A, then FA(X) = HomA(A,X) defines a ... This yields a contravariant exact functor from the category of k-vector spaces to itself ... topological space, we can consider the abelian category of all sheaves of abelian groups on X ...
Grothendieck Group - Examples
... In the abelian category of finite dimensional vector spaces over a field k, two vector spaces are isomorphic if and only if they have the same dimension ... usual construction is as follows For a (not necessarily commutative) ring R, one defines the category to be the category of all finitely generated projective modules ... A ringed-space-version of the latter example works as follows Choose to be the category of all locally free sheaves over X ...

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