(abelian category)
An abelian category is a category \(\mca\) such that
for all objects \(A, B\), \(\Hom(A, B)\) is an abelian group;
composition distributes: \(f \circ (g+h) = (f\circ g) + (f \circ h)\) and \((f+g)\circ h = (f \circ h) + (g\circ h)\);
biproducts \(A \oplus B\) exist, hence so do all finite sums;
for every \(f: A \to B\), \(\ker f\) and \(\coker f\) exist;
every monomorphism is the kernel of its cokernel, and every epimorphism is the cokernel of its kernel;
every morphism \(f: A \to B\) factors as \(f = g \circ h\) with \(h\) an epimorphism and \(g\) a monomorphism, so that \(f\) surjects onto its image, which injects into the target.
Axioms (i)–(iii) say that \(\mca\) is additive. Axiom (v) is the first isomorphism theorem in categorical form: with (iv), it makes the canonical map \(\coim f\to\im f\) an isomorphism for every \(f\), so a morphism that is both a monomorphism and an epimorphism is an isomorphism. The additive category of topological abelian groups has kernels and cokernels but is not abelian: the identity map from \(\RR\) with the discrete topology to \(\RR\) with the Euclidean topology is a monomorphism and an epimorphism but not an isomorphism.
Examples include \(\Ab\), \(\mods{A}\) for a ring \(A\), and \(\mods{\OO_X}\) on a ringed space. Sheaves of abelian groups on a space form an abelian category, whose cokernel is the sheafification of the presheaf cokernel (D-A7LCT). The functor \(\Gamma(X,\wait)\) preserves kernels but not cokernels: on \(X=\CC\sm\ts{0}\), \(\exp\colon\OO_X\to\OO_X^\times\) is an epimorphism of sheaves, and \(z\in\OO_X^\times(X)\) is not the image of any section of \(\OO_X\) over \(X\).
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D-DEFADDFNAdditive functorsusesD-DEFCPLXComplexes, morphisms of complexes, and chain homotopyusesD-DEFEXACTExactness, and left, right, and exact functorsusesD-DEFINJOBInjective objects, injective resolutions, and enough injectivesusesD-DEFPROJOProjective objects and projective modulesuses