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 nice properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. Mathematics is the study of quantity, structure, space and change. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ... In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ... In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ...

Definitions

A category is abelian if

• it has a zero object,
• it has all pullbacks and pushouts, and
• all monomorphisms and epimorphisms are normal.

By a theorem of Peter Freyd, this definition is equivalent to the following "piecemeal" definition: In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ... In category theory, a branch of mathematics, the pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In the context of abstract algebra or universal algebra, an epimorphism is simply a homomorphism onto or surjective homomorphism. ... In category theory and its applications to mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of morphism. ...

• A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear.
• A preadditive category is additive if every finite set of objects has a biproduct. This means that we can form finite direct sums and direct products.
• An additive category is preabelian if every morphism has both a kernel and a cokernel.
• Finally, a preabelian category is abelian if every monomorphism and every epimorphism 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-sets is a consequence of the three axioms of the first definition. A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... In category theory and its applications to mathematics, an enriched category is a category whose hom-sets are replaced by objects from some other category, in a well-behaved manner. ... In mathematics, a monoidal category (or tensor category) is a category equipped with a binary tensor functor and a unit object . ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A1,...,An of C have a biproduct A1 ⊕ ··· ⊕ An in C. (Recall that a category C is preadditive if all its morphism sets are Abelian groups and morphism composition is... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In category theory and its applications to mathematics, a biproduct is a generalisation of the notion of direct sum that makes sense in any preadditive category. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... In mathematics, specifically in category theory, a pre-Abelian category is an additive category that has all kernels and cokernels. ... In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ... In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In the context of abstract algebra or universal algebra, an epimorphism is simply a homomorphism onto or surjective homomorphism. ... In category theory and its applications to mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of morphism. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... Axiom - Wikipedia /**/ @import /skins-1. ...

Examples

• As mentioned above, the category of all abelian groups is an abelian category. The category of all finitely generated abelian groups 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 abelian category is equivalent to a full subcategory of such a category of modules (Mitchell's embedding theorem).
• If K is a commutative noetherian ring, then the category of finitely generated modules over K is abelian. In this way, abelian categories show up in commutative algebra.
• As special cases of the two previous examples: the category of vector spaces over a fixed field k is abelian, as is the category of finite-dimensional vector spaces over k.
• If R is a ring, then the category of all finitely presented left (or right) modules over R is an abelian category. (The category of finitely generated modules over R is not always abelian.)
• If X is a topological 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 in algebraic topology and algebraic geometry.
• If C is a small category and A is an abelian category, then the category of all functors from C to A forms an abelian category (the morphisms of this category are the natural transformations between functors). If C is small and preadditive, then the category of all additive functors 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.

In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In abstract algebra, a module is a generalization of a vector space. ... A subcategory in Wikipedia is a category that depends on another category. ... In mathematics, Mitchells embedding theorem is an important result about abelian categories; it states that these categories, while rather abstractly defined, are all quite concrete categories of modules. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ... In mathematics, a module is a finitely-generated module if it has a finite generating set. ... In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, a module is a finitely-generated module if it has a finite generating set. ... In mathematics, a module is a finitely-generated module if it has a finite generating set. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In category theory, a functor is a special type of mapping between categories. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... A preadditive category is a category that is enriched over the monoidal category of abelian groups. ...

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 category theory, a zero morphism is a special kind of trivial morphism. ... 0 (zero) or nought is both a number and a numeral. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there...

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. In mathematics, particularly in algebra, the coimage of a homomorphism f: A → B is the quotient coim f = A/ker f of the domain and kernel. ... Given a category C and a morphism in C, the image of f is a monomorphism satisfying the following universal property: There exists a morphism such that f = hg. ...

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. In category theory, there is a general definition of subobject extending the idea of subset and subgroup. ... In category theory, there is a general definition of subobject extending the idea of subset and subgroup. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ...

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. In abstract algebra, a module is a generalization of a vector space. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics, a comodule is a concept dual to a module. ... In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ... In mathematics or logic, a finitary operation is one, like those of arithmetic, that take a number of input values to produce an output. ... The enriched limit of a substance is the maximum percentage of ions in an element. ...

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). Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ... In mathematics, especially homological algebra and other applications of Abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. ... In mathematics, especially homological algebra and other applications of Abelian category theory, the short five lemma is a special case of the five lemma. ... In mathematics, particularly homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the long exact sequences that are ubiquitous in homological algebra and its applications, for instance in algebraic topology. ... In mathematics, the nine lemma is a statement about commutative diagrams and exact sequences valid in any abelian category, as well as in the category of groups. ...

History

Abelian categories were introduced by Alexander Grothendieck in the middle of the 1950s 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 of category theory was developed as a language to study these similarities. Grothendieck managed to unify the two theories: they both arise as derived functors 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. Alexander Grothendieck (born March 28, 1928, Berlin) was one of the most important mathematicians active in the 20th century. ... 1950 was a common year starting on Sunday (link will take you to calendar). ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...

References

• P. Freyd. Abelian Categories, Harper and Row, New York, 1964. Available online.
• Barry Mitchell: Theory of Categories, New York, Academic Press, 1965.
• N. Popescu: Abelian categories with applications to rings and modules, Academic Press, London, 1973.