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{{Category theory}} |
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{{Short description|
In [[mathematics]], an '''abelian category''' is a [[Category (mathematics)|category]] in which [[morphism]]s and [[Object (category theory)|objects]] can be added and in which [[Kernel (category theory)|kernel]]s and [[cokernel]]s exist and have desirable properties.
The motivating prototypical example of an abelian category is the [[category of abelian groups]], {{math|1='''Ab'''}}. Abelian categories are very ''stable'' categories; for example they are [[regular category|regular]] and they satisfy the [[snake lemma]]. The [[Class (set theory)|class]] of abelian categories is closed under several categorical constructions, for example, the category of [[chain complex]]es of an abelian category, or the category of [[functor]]s 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 [[Saunders Mac Lane|Mac Lane]]<ref>{{Cite book |last=Mac Lane |first=Saunders |title=Categories for the Working Mathematician |date=2013-04-17|publisher=Springer Science+Business Media |isbn=978-1-4757-4721-8 |edition=second |series=[[Graduate Texts in Mathematics]] |volume=5 |pages=205}}</ref> says [[Alexander Grothendieck]]<ref>{{harvtxt|Grothendieck|1957}}</ref> defined the abelian category, but there is a reference<ref>{{Cite web |author=David Eisenbud and Jerzy Weyman |title=MEMORIAL TRIBUTE Remembering David Buchsbaum |url=https://www.ams.org/journals/notices/202201/rnoti-p76.pdf?adat=January%202022&trk=2410&cat=interest&galt=none&_zs=kq3BH1&_zl=7ckX6 |access-date=2023-12-22 |publisher=[[American Mathematical Society]]}}</ref> that says [[Samuel Eilenberg|Eilenberg]]'s disciple, [[David Buchsbaum|Buchsbaum]], proposed the concept in his PhD thesis,<ref>{{harvtxt|Buchsbaum|1955}}</ref> and Grothendieck popularized it under the name "abelian category".
==Definitions==
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This definition is equivalent<ref>Peter Freyd, [http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html Abelian Categories]</ref> to the following "piecemeal" definition:
* A category is ''[[Preadditive category|preadditive]]'' if it is [[enriched category|enriched]] over the [[monoidal category]] {{math|1='''Ab'''}} of [[abelian group]]s. This means that all [[hom-set]]s are abelian groups and the composition of morphisms is [[bilinear operator|bilinear]].
* A preadditive category is ''[[Additive category|additive]]'' if every [[finite set]] of objects has a [[biproduct]]. This means that we can form finite [[direct sum of modules|direct sum]]s and [[direct product]]s. In <ref>Handbook of categorical algebra, vol. 2, F. Borceux</ref> Def. 1.2.6, it is required that an additive category
* An additive category is ''[[preabelian category|preabelian]]'' if every morphism has both a [[kernel (category theory)|kernel]] and a [[cokernel]].
* Finally, a preabelian category is '''abelian''' if every [[monomorphism]] and every [[epimorphism]] is [[normal morphism|normal]]. This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism.
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* AB3) For every indexed family (''A''<sub>''i''</sub>) of objects of '''A''', the [[coproduct]] *''A''<sub>i</sub> exists in '''A''' (i.e. '''A''' is [[cocomplete]]).
* AB4) '''A''' satisfies AB3), and the coproduct of a family of monomorphisms is a monomorphism.
* [[AB5 category|AB5]]) '''A''' satisfies AB3), and [[filtered colimit]]s of [[exact sequence]]s are exact.
and their duals
* AB3*) For every indexed family (''A''<sub>''i''</sub>) of objects of '''A''', the [[Product (category theory)|product]] P''A''<sub>''i''</sub> exists in '''A''' (i.e. '''A''' is [[Complete category|complete]]).
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The abelian category is also a [[comodule]]; Hom(''G'',''A'') can be interpreted as an object of '''A'''.
If '''A''' is [[Complete category|complete]], then we can remove the requirement that ''G'' be finitely generated; most generally, we can form [[finitary]] [[enriched limit]]s in '''A'''.
Given an object <math>A</math> in an abelian category, '''flatness''' refers to the idea that <math>- \otimes A</math> is an [[exact functor]]. See [[flat module]] or, for more generality, [[flat morphism]].
==Related concepts==
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Some Abelian categories found in nature are semi-simple, such as
* Category of [[vector space]]s <math>\text{Vect}(k)</math> over a fixed field <math>k</math>.
* By [[Maschke's theorem]] the category of representations <math>\text{Rep}_k(G)</math> of a finite group <math>G</math> over a field <math>k</math> whose characteristic does not divide <math>|G|</math> is a semi-simple abelian category.
* The category of [[Coherent sheaf|coherent sheaves]] on a [[Noetherian scheme|Noetherian]] [[Scheme (mathematics)|scheme]] is semi-simple if and only if <math>X</math> is a finite disjoint union of irreducible points. This is equivalent to a finite coproduct of categories of vector spaces over different fields. Showing this is true in the forward direction is equivalent to showing all <math>\text{Ext}^1</math> groups vanish, meaning the [[cohomological dimension]] is 0. This only happens when the skyscraper sheaves <math>k_x</math> at a point <math>x \in X</math> have [[Zariski tangent space]] equal to zero, which is isomorphic to <math>\text{Ext}^1(k_x,k_x)</math> using [[local algebra]] for such a scheme.<ref>{{Cite web|title=algebraic geometry - Tangent space in a point and First Ext group|url=https://math.stackexchange.com/questions/75673/tangent-space-in-a-point-and-first-ext-group|access-date=2020-08-23|website=Mathematics Stack Exchange}}</ref>
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*{{Citation | last1=Buchsbaum | first1=David A. | author-link=David Buchsbaum| title=Exact categories and duality | jstor=1993003 | mr=0074407 | year=1955 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=80 | issue=1 | pages=1–34 | doi=10.1090/S0002-9947-1955-0074407-6| doi-access=free }}
* {{Citation | last1=Freyd | first1=Peter | author1-link=Peter Freyd | title=Abelian Categories | url=http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html | publisher=Harper and Row | location=New York | year=1964}}
* {{Citation |
* {{Citation | last1=Mitchell | first1=Barry | title=Theory of Categories | publisher=[[Academic Press]] | location=Boston, MA | year=1965}}
* {{Citation | last1=Popescu | first1=Nicolae | author-link=Nicolae Popescu| title=Abelian categories with applications to rings and modules | publisher=[[Academic Press]] | location=Boston, MA | year=1973}}
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{{Category theory}}
{{Authority control}}
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