Abelian category

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, an abelian category is a category in which morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory...

s and objects can be added and in which kernel

Kernel (category theory)

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...

s and cokernel

Cokernel

In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....

s exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups

Category of abelian groups

In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....

, Ab. The theory originated in a tentative attempt to unify several cohomology theories by Alexander Grothendieck

Alexander Grothendieck

Alexander Grothendieck is considered one of the greatest mathematicians of the 20th century.He is most famous for his revolutionary advances in algebraic geometry, but he has also made major contributions to algebraic topology, number theory, category theory, Galois theory, descent theory,...

. Abelian categories are very stable categories, for example they are regular

Regular category

In category theory, a regular category is a category with finite limits and coequalizers of kernel pairs, satisfying certain exactness conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of images, without requiring additivity...

 and they satisfy the snake lemma

Snake 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

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, an abelian category is a category in which morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory...

s and objects can be added and in which kernel

Kernel (category theory)

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...

s and cokernel

Cokernel

In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....

s exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups

Category of abelian groups

In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....

, Ab. The theory originated in a tentative attempt to unify several cohomology theories by Alexander Grothendieck

Alexander Grothendieck

Alexander Grothendieck is considered one of the greatest mathematicians of the 20th century.He is most famous for his revolutionary advances in algebraic geometry, but he has also made major contributions to algebraic topology, number theory, category theory, Galois theory, descent theory,...

. Abelian categories are very stable categories, for example they are regular

Regular category

In category theory, a regular category is a category with finite limits and coequalizers of kernel pairs, satisfying certain exactness conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of images, without requiring additivity...

 and they satisfy the snake lemma

Snake 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...

. The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complex

Chain complex

In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...

es of an Abelian category, or the category of functor

Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms in the category of small categories....

s from a small category to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra

Homological algebra

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...

 and beyond; the theory has major applications in algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...

, cohomology

Cohomology

In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...

 and pure category theory

Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....

.

Definitions

A category is abelian if

• it has a zero object,
• it has all pullbacks
  Pullback (category theory)
  In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain; it is the limit of the cospan...
  
  and pushouts
  Pushout (category theory)
  In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain: it is the colimit of the span .The pushout is the categorical dual of the...
  
  , and
• all monomorphism
  Monomorphism
  In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation ....
  
  s and epimorphism
  Epimorphism
  In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
  
  s are normal
  Normal morphism
  In category theory and its applications to mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of morphism.A normal category is a category in which morphisms are normal, whenever reasonable.-Definition:...
  
  .

By a theorem of Peter Freyd, this definition is equivalent to the following "piecemeal" definition:

• A category is preadditive
  Preadditive category
  In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
  
  if it is enriched
  Enriched category
  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.-Definition:...
  
   over the monoidal category
  Monoidal category
  In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative , and an object I which is both a left and right identity for ⊗,...
  
   Ab of abelian group
  Abelian group
  An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
  
  s. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear
  Bilinear operator
  In mathematics, a bilinear operator is a function combining elements of two vector space to yield an element of third vector space that is linear in each of its arguments. Matrix multiplication is an example.-Definition:...
  
  .
• A preadditive category is additive
  Additive category
  In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A₁,...,A_n of C have a biproduct A₁ ⊕ ⋯ ⊕ A_n in C.In mathematics, specifically in category theory, an additive category is...
  
  if every finite set
  Finite set
  In mathematics, finite set is a set that has a finite number of elements. For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set. A set that is not finite is called infinite...
  
   of objects has a biproduct
  Biproduct
  In category theory and its applications to mathematics, a biproduct of a finite collection of objects in a preadditive category is both a product and a coproduct. In fact, the notions of product and coproduct conincide for finite collections of objects in a preadditive category...
  
  . This means that we can form finite direct sums and direct product
  Direct product
  In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
  
  s.
• An additive category is preabelian if every morphism has both a kernel
  Kernel (category theory)
  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...
  
   and a cokernel
  Cokernel
  In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
  
  .
• Finally, a preabelian category is abelian if every monomorphism
  Monomorphism
  In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation ....
  
   and every epimorphism
  Epimorphism
  In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
  
   is normal
  Normal morphism
  In category theory and its applications to mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of morphism.A normal category is a category in which morphisms are normal, whenever reasonable.-Definition:...
  
  . 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 axiom

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision...

s of the first definition. This highlights the foundational relevance of the category of Abelian group

Abelian group

An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s in the theory and its canonical nature.

The concept of exact sequence

Exact sequence

In mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory, an exact sequence is a sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the...

 arises naturally in this setting, and it turns out that exact functor

Exact functor

In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...

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

Exact category

In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition...

, forming a very special case of regular categories

Regular category

In category theory, a regular category is a category with finite limits and coequalizers of kernel pairs, satisfying certain exactness conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of images, without requiring additivity...

.

Examples

• As mentioned above, the category of all abelian groups is an abelian category. The category of all finitely generated abelian group
  Finitely generated abelian group
  In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x₁,...,x_s in G such that every x in G can be written in the formwith integers n₁,...,n_s...
  
  s is also an abelian category, as is the category of all finite abelian groups.
• If R is a ring
  Ring (mathematics)
  In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element...
  
  , then the category of all left (or right) modules
  Module (mathematics)
  In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may lie in an arbitrary ring...
  
   over R is an abelian category. In fact, it can be shown that any small abelian category is equivalent to a full subcategory of such a category of modules (Mitchell's embedding theorem
  Mitchell's embedding theorem
  Mitchell's embedding theorem, also known as the Freyd-Mitchell theorem, is a mathematical result about abelian categories; it states that these categories, while rather abstractly defined, are all quite concrete categories of modules...
  
  ).
• If R is a left-noetherian ring
  Noetherian ring
  In abstract algebra, a Noetherian ring, named after Emmy Noether, is a ring that satisfies the ascending chain condition on ideals. Explicitly this means: given an increasing sequence of idealsthere exists an for which...
  
  , then the category of finitely generated left modules over R is abelian. In particular, the category of finitely generated modules over a noetherian commutative ring
  Commutative ring
  In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
  
   is abelian; in this way, abelian categories show up in commutative algebra
  Commutative algebra
  Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
  
  .
• As special cases of the two previous examples: the category of vector space
  Vector space
  A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
  
  s over a fixed field
  Field (mathematics)
  In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
  
   k is abelian, as is the category of finite-dimensional vector spaces over k.
• If X is a topological space
  Topological space
  Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
  
  , 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 a topological space
  Topological space
  Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
  
  , then the category of all sheaves
  Sheaf (mathematics)
  In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
  
   of abelian groups on X is an abelian category. More generally, the category of sheaves of abelian groups on a Grothendieck site
  Grothendieck topology
  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. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
  
   is an abelian category. In this way, abelian categories show up in algebraic topology
  Algebraic topology
  Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...
  
   and algebraic geometry
  Algebraic geometry
  Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
  
  .
• If C is a small category
  Category theory
  In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
  
   and A is an abelian category, then the category of all functor
  Functor
  In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms in the category of small categories....
  
  s from C to A forms an abelian category (the morphisms of this category are the natural transformation
  Natural transformation
  In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
  
  s between functors). If C is small and preadditive
  Preadditive category
  In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
  
  , 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.

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, the coproduct
  Coproduct
  In category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
  
   ∐A_i exists in A (i.e. A is cocomplete).
• AB4) A satisfies AB3), and the coproduct of a family of monomorphisms is a monomorphism.
• AB5) A satisfies AB3), and filtered colimits of exact sequence
  Exact sequence
  In mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory, an exact sequence is a sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the...
  
  s are exact.

and their duals

• AB3*) For every set {A_i} of objects of A, the product
  Product (category theory)
  In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
  
   ΠA_i exists in A (i.e. A is complete
  Complete category
  In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F : J → C where J is small has a limit in C...
  
  ).
• AB4*) A satisfies AB3*), and the product of a family of epimorphisms is an epimorphism.
• AB5*) A satisfies AB3*), and filtered limits 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

Zero morphism

In category theory, a zero morphism is a special kind of "trivial" morphism. Suppose C is a category, and for any two objects X and Y in C we are given a morphism 0_XY : X → Y with the following property: for any two morphism f : R → S and g : U → V we obtain a commutative...

 from A to B.
This can be defined as the zero

0 (number)

0 is both a number and the numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, zero is used as a placeholder in place value systems...

 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

Coimage

In mathematics, particularly in algebra, the coimage of a homomorphismis the quotientof domain and kernel.The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies....

of f, while the monomorphism is called the image

Image (category theory)

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....

of f.

Subobject

Subobject

In category theory, a branch of mathematics, a subobject is, roughly speaking, an object which sits inside another object in the same category. The notion is a generalization of the older concepts of subset from set theory and subgroup from group theory...

s and quotient objects are well-behaved

Well-behaved

Mathematicians very frequently speak of whether a mathematical object — a function, a set, a space of one sort or another — is "well-behaved" or not. The term has no fixed formal definition, and is dependent on mathematical interests, fashion, and taste...

 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

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may lie in an arbitrary ring...

 over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product

Tensor product

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. In each case the significance of the symbol is the same: the most general bilinear operation. In some contexts, this...

 of a finitely generated abelian group G and any object A of A.
The abelian category is also a comodule

Comodule

In mathematics, a comodule is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.- Formal definition :...

; Hom(G,A) can be interpreted as an object of A.
If A is complete

Complete category

In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F : J → C where J is small has a limit in C...

, then we can remove the requirement that G be finitely generated; most generally, we can form finitary

Finitary

In mathematics or logic, a finitary operation is one, like those of arithmetic, that takes a finite number of input values to produce an output. An operation such as taking an integral of a function, in calculus, is defined in such a way as to depend on all the values of the function , and is so...

 enriched limits in A.

Related concepts

Abelian categories are the most general setting for homological algebra

Homological algebra

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...

.
All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequences, and derived functor

Derived functor

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :...

s.
Important theorems that apply in all abelian categories include the five lemma

Five lemma

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....

 (and the short five lemma

Short five lemma

In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma....

 as a special case), as well as the snake lemma

Snake 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...

 (and the nine lemma

Nine lemma

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. It states: ifis a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well...

 as a special case).

History

Abelian categories were introduced by Alexander Grothendieck in his famous Tôhoku paper in the middle of the 1950s in order to unify various cohomology theories. At the time, there was a cohomology theory for sheaves

Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...

, and a cohomology theory for group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

s. The two were defined completely differently, but they had formally almost identical properties. In fact, much of category theory

Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....

 was developed as a language to study these similarities. Grothendieck managed to unify the two theories: they both arise as derived functor

Derived functor

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :...

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.

To do

There are still several facts listed in Preadditive category

Preadditive category

In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...

, Additive category

Additive category

In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A₁,...,A_n of C have a biproduct A₁ ⊕ ⋯ ⊕ A_n in C.In mathematics, specifically in category theory, an additive category is...

, and Preabelian category that should be repeated here when this is the most common context in which they're used.

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