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 the abelian category, but there is a reference that says Eilenberg's disciple, Buchsbaum, proposed the concept in his PhD thesis, and Grothendieck popularized it under the name "abelian category".

Definitions

A category is abelian if it is preadditive and This definition is equivalent to the following "piecemeal" definition: Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear. 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.

Examples

Grothendieck's axioms

In his Tōhoku 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: and their duals Axioms AB1) and AB2) were also given. They are what make an additive category abelian. Specifically: 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 the hom-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 the zero 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. Subobjects and quotient objects are well-behaved in abelian categories. For example, the poset of subobjects of any given object A is a bounded lattice. Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product of a finitely generated abelian group G and any object A of A. The abelian category is also a comodule; 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 form finitary enriched limits in A. Given an object A in an abelian category, flatness refers to the idea that - \otimes A is an exact functor. See flat module or, for more generality, flat morphism.

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 especially short exact sequences, and derived functors. Important theorems that apply in all abelian categories include the five lemma (and the short five lemma as a special case), as well as the snake lemma (and the nine lemma as a special case).

Semi-simple Abelian categories

An abelian category \mathbf{A} is called semi-simple if there is a collection of objects called simple objects (meaning the only sub-objects of any X_i are the zero object 0 and itself) such that an object can be decomposed as a direct sum (denoting the coproduct of the abelian category)""This technical condition is rather strong and excludes many natural examples of abelian categories found in nature. For example, most module categories over a ring R are not semi-simple; in fact, this is the case if and only if R is a semisimple ring.

Examples

Some Abelian categories found in nature are semi-simple, such as

Non-examples

There do exist some natural counter-examples of abelian categories which are not semi-simple, such as certain categories of representations. For example, the category of representations of the Lie group has the representation which only has one subrepresentation of dimension 1. In fact, this is true for any unipotent group pg 112.

Subcategories of abelian categories

There are numerous types of (full, additive) subcategories of abelian categories that occur in nature, as well as some conflicting terminology. Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor. Here is an explicit example of a full, additive subcategory of an abelian category that is itself abelian but the inclusion functor is not exact. Let k be a field, T_n the algebra of upper-triangular n\times n matrices over k, and \mathbf A_n the category of finite-dimensional T_n-modules. Then each \mathbf A_n is an abelian category and we have an inclusion functor identifying the simple projective, simple injective and indecomposable projective-injective modules. The essential image of I is a full, additive subcategory, but I is not exact.

History

Abelian categories were introduced by (under the name of "exact category") and 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 differently, but they had similar properties. In fact, much of category theory was developed as a language to study these similarities. Grothendieck unified the two theories: they both arise as derived functors on abelian categories; the abelian category of sheaves of abelian groups on a topological space, and the abelian category of G-modules for a given group G.