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Concept
Abelian category
Summary
In mathematics, an abelian category is a in which morphisms and can be added and in which s and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the , 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 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 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 .
A category is abelian if it is and
it has a zero object,
it has all binary biproducts,
it has all and cokernels, and
all monomorphisms and epimorphisms are normal.
This definition is equivalent to the following "piecemeal" definition:
A category is if it is over the Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear.
A preadditive category is 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 if every morphism has both a 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.
Official source
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