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 are named after Niels Henrik Abel. Definitions A category is abelian if it is preadditive and it has a zero object, it has all binary biproducts, it has all kernels and cokernels, and all monomorphisms and epimorphisms are normal. This definition is equivalent to the following "piecemeal" definition: A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear. A preadditive category is additive if every finite set of objects has a biproduct. This means that we can form finite direct sums and direct products. In Def. 1.2.6, it is required that an additive category have a zero object (empty biproduct). An additive category is preabelian if every morphism has both a kernel and a cokernel. Finally, a preabelian category is abelian if every monomorphism and every epimorphism is normal. This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism. Note that the enriched structure on hom-sets is a consequence of the first three axioms of the first definition. This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature. The concept of exact sequence arises naturally in this setting, and it turns out that exact functors, i.e. the functors preserving exact sequences in various senses, are the relevant functors between abelian categories. This exactness concept has been axiomatized in the theory of exact categories,