Category of abelian groups
Encyclopedia
In mathematics
, the category
Ab has the abelian group
s as objects and group homomorphism
s as morphism
s. This is the prototype of an abelian category
.
The monomorphism
s in Ab are the injective group homomorphisms, the epimorphism
s are the surjective group homomorphisms, and the isomorphism
s are the bijective group homomorphisms.
The zero object of Ab is the trivial group {0} which consists only of its neutral element.
Note that Ab is a full subcategory of Grp, the category of all groups
. The main difference between Ab and Grp is that the sum of two homomorphisms f and g between abelian groups is again a group homomorphism:
(x+y) = f(x+y) + g(x+y) = f(x) + f(y) + g(x) + g(y)
The third equality requires the group to be abelian. This addition of morphism turns Ab into a preadditive category
, and because the direct sum of finitely many abelian groups yields a biproduct
, we indeed have an additive category
.
In Ab, the notion of kernel in the category theory sense
coincides with kernel in the algebraic sense
, i.e.: the kernel of the morphism f : A → B is the subgroup K of A defined by K = {x in A : f(x) = 0}, together with the inclusion homomorphism i : K → A. The same is true for cokernel
s: the cokernel of f is the quotient group
C = B/f(A) together with the natural projection p : B → C. (Note a further crucial difference between Ab and Grp: in Grp it can happen that f(A) is not a normal subgroup
of B, and that therefore the quotient group B/f(A) cannot be formed.) With these concrete descriptions of kernels and cokernels, it is quite easy to check that Ab is indeed an abelian category
.
The product
in Ab is given by the product of groups
, formed by taking the cartesian product
of the underlying sets and performing the group operation componentwise. Because Ab has kernels, one can then show that Ab is a complete category
. The coproduct
in Ab is given by the direct sum; since Ab has cokernels, it follows that Ab is also cocomplete.
Taking direct limit
s in Ab is an exact functor
, which turns Ab into an abelian category
.
We have a forgetful functor
Ab → Set
which assigns to each abelian group the underlying set, and to each group homomorphism the underlying function
. This functor is faithful, and therefore Ab is a concrete category
. The forgetful functor has a left adjoint
(which associates to a given set the free abelian group
with that set as basis) but does not have a right adjoint.
An object in Ab is injective
if and only if it is divisible
; it is projective
if and only if it is a free abelian group. The category has a projective generator (Z) and an injective cogenerator
(Q/Z). This implies that Ab is an example of a Grothendieck category.
Given two abelian groups A and B, their tensor product
A⊗B is defined; it is again an abelian group. With this notion of product, Ab is a symmetric monoidal category
.
Ab is not cartesian closed
(and therefore also not a topos
) since it lacks exponential object
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
Ab has the abelian group
Abelian group
In abstract algebra, 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 as objects and group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
s as morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s. This is the prototype of an abelian category
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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
.
The 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 X \hookrightarrow Y....
s in Ab are the injective group homomorphisms, the 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 the surjective group homomorphisms, and the isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
s are the bijective group homomorphisms.
The zero object of Ab is the trivial group {0} which consists only of its neutral element.
Note that Ab is a full subcategory of Grp, the category of all groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
. The main difference between Ab and Grp is that the sum of two homomorphisms f and g between abelian groups is again a group homomorphism:
(x+y) = f(x+y) + g(x+y) = f(x) + f(y) + g(x) + g(y)
- = f(x) + g(x) + f(y) + g(y) = (f+g)(x) + (f+g)(y)
The third equality requires the group to be abelian. This addition of morphism turns Ab into a 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...
, and because the direct sum of finitely many abelian groups yields a biproduct
Biproduct
In category theory and its applications to mathematics, a biproduct of a finite collection of objects in a category with zero object is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects...
, we indeed have an additive category
Additive category
In mathematics, specifically in category theory, an additive category is a preadditive category C such that all finite collections of objects A1,...,An of C have a biproduct A1 ⊕ ⋯ ⊕ An in C....
.
In Ab, the notion of kernel in the category theory sense
Kernel (category theory)
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra...
coincides with kernel in the algebraic sense
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
, i.e.: the kernel of the morphism f : A → B is the subgroup K of A defined by K = {x in A : f(x) = 0}, together with the inclusion homomorphism i : K → A. The same is true for 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: the cokernel of f is the quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
C = B/f(A) together with the natural projection p : B → C. (Note a further crucial difference between Ab and Grp: in Grp it can happen that f(A) is not a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of B, and that therefore the quotient group B/f(A) cannot be formed.) With these concrete descriptions of kernels and cokernels, it is quite easy to check that Ab is indeed an abelian category
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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
.
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...
in Ab is given by the product of groups
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
, formed by taking the cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of the underlying sets and performing the group operation componentwise. Because Ab has kernels, one can then show that Ab is a complete category
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. Dually, a cocomplete category is one in which all small colimits exist...
. The coproduct
Coproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes 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...
in Ab is given by the direct sum; since Ab has cokernels, it follows that Ab is also cocomplete.
Taking direct limit
Direct limit
In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...
s in Ab is an 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...
, which turns Ab into an abelian category
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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
.
We have a forgetful functor
Forgetful functor
In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...
Ab → Set
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
which assigns to each abelian group the underlying set, and to each group homomorphism the underlying function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
. This functor is faithful, and therefore Ab is a concrete category
Concrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...
. The forgetful functor has a left adjoint
Adjoint functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
(which associates to a given set the free abelian group
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
with that set as basis) but does not have a right adjoint.
An object in Ab is injective
Injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
if and only if it is divisible
Divisible group
In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n...
; it is projective
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
if and only if it is a free abelian group. The category has a projective generator (Z) and an injective cogenerator
Injective cogenerator
In category theory, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality. Generators are objects which cover other objects as an approximation, and cogenerators are objects which envelope other objects as an approximation...
(Q/Z). This implies that Ab is an example of a Grothendieck category.
Given two abelian groups A and B, their 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, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
A⊗B is defined; it is again an abelian group. With this notion of product, Ab is a symmetric monoidal category
Monoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
.
Ab is not cartesian closed
Cartesian closed category
In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in...
(and therefore also not a topos
Topos
In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
) since it lacks exponential object
Exponential object
In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories...
s.