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. 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 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. Abelian categories are named after Niels Henrik Abel.
Definitions.
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.
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.
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 formula_7 is called semi-simple if there is a collection of objects formula_8 called simple objects (meaning the only sub-objects of any formula_9 are the zero object formula_10 and itself) such that an object formula_11 can be decomposed as a direct sum (denoting the coproduct of the abelian category)formula_12This technical condition is rather strong and excludes many natural examples of abelian categories found in nature. For example, most module categories over a ring formula_13 are not semi-simple; in fact, this is the case if and only if formula_13 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 formula_26 has the representationformula_27which only has one subrepresentation of dimension formula_28. In fact, this is true for any unipotent grouppg 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, formula_34 the algebra of upper-triangular formula_35 matrices over "k", and formula_36 the category of finite-dimensional formula_34-modules. Then each formula_36 is an abelian category and we have an inclusion functor formula_39 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".