- Abelian category
In

mathematics , an**abelian category**is a category in whichmorphism s and objects can be added and in which kernels andcokernel s exist and have desirable properties. The motivating prototype example of an abelian category is thecategory of abelian groups ,**Ab**. The theory originated in a tentative attempt to unify several cohomology theories byAlexander Grothendieck . Abelian categories are very "stable" categories, for example they are regular and they satisfy theSnake lemma . The class of Abelian categories is closed under several categorical constructions, for example, the category ofchain complex es of an Abelian category, or the category offunctor s from asmall category to an Abelian category are Abelian as well. These stability properties make them inevitable inhomological algebra and beyond; the theory has major applications inalgebraic geometry ,cohomology and purecategory theory .**Definitions**A category is

**abelian**if

*it has azero object ,

*it has all "pullbacks" and "pushouts", and

*allmonomorphism s andepimorphism s are normal.By a theorem of

Peter Freyd , this definition is equivalent to the following "piecemeal" definition:

* A category is "preadditive" if it is enriched over themonoidal category **Ab**ofabelian group s. This means that allhom-set s are abelian groups and the composition of morphisms is bilinear.

* A preadditive category is "additive" if everyfinite set of objects has abiproduct . This means that we can form finitedirect sum s anddirect product s.

* An additive category is "preabelian" if every morphism has both a kernel and acokernel .

* Finally, a preabelian category is**abelian**if everymonomorphism and everyepimorphism 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-set s is a "consequence" of the threeaxiom s of the first definition. This highlights the foundational relevance of the category ofAbelian group s in the theory and its canonical nature.The concept of

exact sequence arises naturally in this setting, and it turns out thatexact functor s, 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.**Examples*** As mentioned above, the category of all abelian groups is an abelian category. The category of all

finitely generated abelian group s is also an abelian category, as is the category of all finite abelian groups.

* If "R" is a ring, then the category of all left (or right) modules over "R" is an abelian category. In fact, it can be shown that any small abelian category is equivalent to afull subcategory of such a category of modules ("Mitchell's embedding theorem ").

* If "R" is a left-noetherian ring , then the category of finitely generated left modules over "R" is abelian. In particular, the category of finitely generated modules over a noetheriancommutative ring is abelian; in this way, abelian categories show up incommutative algebra .

* As special cases of the two previous examples: the category ofvector space s over a fixed field "k" is abelian, as is the category of finite-dimensional vector spaces over "k".

* If "X" is atopological space , then the category of all (real or complex)vector bundles on "X" is not usually an abelian category, as there can be monomorphisms that are not kernels.

* If "X" is atopological space , then the category of all sheaves of abelian groups on "X" is an abelian category. More generally, the category of sheaves of abelian groups on a Grothendieck site is an abelian category. In this way, abelian categories show up inalgebraic topology andalgebraic geometry .

* If**C**is a small category and**A**is an abelian category, then the category of allfunctor s from**C**to**A**forms an abelian category (the morphisms of this category are thenatural transformation s between functors). If**C**is small and preadditive, then the category of alladditive functor s from**C**to**A**also forms an abelian category. The latter is a generalization of the "R"-module example, since a ring can be understood as a preadditive category with a single object.**Grothendieck's Axioms**In his Tôhoku article, Grothendieck listed four additional axioms (and their duals) that an abelian category

**A**might satisfy. These axioms are still in common use to this day. They are the following:

* AB3) For every set {A_{i}} of objects of**A**, thecoproduct ∐A_{i}exists in**A**(i.e.**A**iscocomplete ).

* AB4)**A**satisfies AB3), and the coproduct of a family of monomorphisms is a monomorphism.

* AB5)**A**satisfies AB3), andfiltered colimit s ofexact sequence s are exact.and their duals

* AB3*) For every set {A_{i}} of objects of**A**, the product ΠA_{i}exists in**A**(i.e.**A**is complete).

* AB4*)**A**satisfies AB3*), and the product of a family of epimorphisms is an epimorphism.

* AB5*)**A**satisfies AB3*), andfiltered limit s of exact sequences are exact.Axioms AB1) and AB2) were also given. They are what make an additive category abelian. Specifically:

* AB1) Every morphism has a kernel and a cokernel.

* AB2) For every morphism "f", the canonical morphism from coim "f" to im "f" is an isomorphism.Grothendieck also gave axioms AB6) and AB6*).

**Elementary properties**Given any pair "A", "B" of objects in an abelian category, there is a special

zero morphism from "A" to "B".This can be defined as the zero element of thehom-set Hom("A","B"), since this is an abelian group.Alternatively, it can be defined as the unique composition "A" → 0 → "B", where 0 is thezero object of the abelian category.In an abelian category, every morphism "f" can be written as the composition of an epimorphism followed by a monomorphism.This epimorphism is called the "

coimage " of "f", while the monomorphism is called the "image" of "f".Subobject s andquotient object s arewell-behaved in abelian categories.For example, theposet of subobjects of any given object "A" is abounded lattice .Every abelian category

**A**is a module over the monoidal category of finitely generated abelian groups; that is, we can form atensor product of a finitely generated abelian group "G" and any object "A" of**A**.The abelian category is also acomodule ; Hom("G","A") can be interpreted as an object of**A**.If**A**is complete, then we can remove the requirement that "G" be finitely generated; most generally, we can formfinitary enriched limits in**A**.**Related concepts**Abelian categories are the most general setting for

homological algebra .All of the constructions used in that field are relevant, such as exact sequences, and especiallyshort exact sequence s, andderived functor s.Important theorems that apply in all abelian categories include thefive lemma (and theshort five lemma as a special case), as well as thesnake lemma (and thenine lemma as a special case).**History**Abelian categories were introduced by Alexander Grothendieck in his famous Tôhoku paper in the middle of the

1950 s in order to unify various cohomology theories. At the time, there was a cohomology theory for sheaves, and a cohomology theory for groups. The two were defined completely differently, but they had formally almost identical properties. In fact, much ofcategory theory was developed as a language to study these similarities. Grothendieck managed to unify the two theories: they both arise asderived functor s on abelian categories; on the one hand the abelian category of sheaves of abelian groups on a topological space, on the other hand the abelian category of "G"-modules for a given group "G".**References*** P. Freyd. "Abelian Categories," Harper and Row, New York, 1964. [

*http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html Available online.*]

* Alexandre Grothendieck, "Sur quelques points d'algèbre homologique", Tôhoku Mathematics Journal, 1957

* Barry Mitchell: "Theory of Categories", New York, Academic Press, 1965.

* N. Popescu: "Abelian categories with applications to rings and modules", Academic Press, London, 1973.

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