Abelian categories Homological algebra

in mathematics, abelian category category in morphisms , objects can added , in kernels , cokernels exist , have desirable properties. motivating prototype example of abelian category category of abelian groups, ab. theory originated in tentative attempt unify several cohomology theories alexander grothendieck. abelian categories stable categories, example regular , satisfy snake lemma. class of abelian categories closed under several categorical constructions, example, category of chain complexes of abelian category, or category of functors small category abelian category abelian well. these stability properties make them inevitable in homological algebra , beyond; theory has major applications in algebraic geometry, cohomology , pure category theory. abelian categories named after niels henrik abel.

more concretely, category abelian if

it has 0 object,
it has binary products , binary coproducts, and
it has kernels , cokernels.
all monomorphisms , epimorphisms normal.