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Abelian category AI simulator
(@Abelian category_simulator)
Hub AI
Abelian category AI simulator
(@Abelian category_simulator)
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 prototypical example of an abelian category is the category of abelian groups, Ab.
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.
Mac Lane says Alexander Grothendieck defined abelian categories in 1957, but there is a reference that says Eilenberg's disciple, Buchsbaum, had proposed the concept in his 1955 PhD thesis, and Grothendieck popularized it under the name "abelian category".
A category is abelian if it is preadditive and
This definition is equivalent to the following "piecemeal" definition:
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, forming a very special case of regular categories.
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 prototypical example of an abelian category is the category of abelian groups, Ab.
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.
Mac Lane says Alexander Grothendieck defined abelian categories in 1957, but there is a reference that says Eilenberg's disciple, Buchsbaum, had proposed the concept in his 1955 PhD thesis, and Grothendieck popularized it under the name "abelian category".
A category is abelian if it is preadditive and
This definition is equivalent to the following "piecemeal" definition:
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, forming a very special case of regular categories.