Injective cogenerator
Encyclopedia
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 (dually) cogenerators are objects which envelope other objects as an approximation. When working with unfamiliar algebraic objects, one can use these to approximate with the more familiar.
More precisely:
s, one can in fact form direct sums of copies of G until the morphism
is surjective; and one can form direct products of C until the morphism
is injective.
For example, the integers are a generator of the category of abelian groups (since every abelian group is a quotient of a free abelian group
). This is the origin of the term generator. The approximation here is normally described as generators and relations.
As an example of a cogenerator in the same category, we have Q/Z, the rationals modulo the integers, which is a divisible
abelian group. Given any abelian group A, there is an isomorphic copy of A contained inside the product of |A| copies of Q/Z. This approximation is close to what is called the divisible envelope - the true envelope is subject to a minimality condition.
Finding a generator of an abelian category
allows one to express every object as a quotient of a direct sum of copies of the generator. Finding a cogenerator allows one to express every object as a subobject of a direct product of copies of the cogenerator. One is often interested in projective generators (even finitely generated projective generators, called progenerators) and minimal injective cogenerators. Both examples above have these extra properties.
The cogenerator Q/Z is quite useful in the study of modules
over general rings. If H is a left module over the ring R, one forms the (algebraic) character module H* consisting of all abelian group homomorphisms from H to Q/Z. H* is then a right R-module. Q/Z being a cogenerator says precisely that H* is 0 if and only if H is 0. Even more is true: the * operation takes a homomorphism
to a homomorphism
and f* is 0 if and only if f is 0. It is thus a faithful contravariant functor
from left R-modules to right R-modules.
Every H* is very special in structure: it is pure-injective (also called algebraically compact), which says more or less that solving equations in H* is relatively straightforward. One can often consider a problem after applying the * to simplify matters.
All of this can also be done for continuous modules H: one forms the topological character module of continuous group homomorphisms from H to the circle group R/Z.
is an injective cogenerator in a category of topological space
s subject to separation axiom
s.
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...
, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality
Pontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...
. Generators are objects which cover other objects as an approximation, and (dually) cogenerators are objects which envelope other objects as an approximation. When working with unfamiliar algebraic objects, one can use these to approximate with the more familiar.
More precisely:
- A generator of a categoryCategory theoryCategory 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...
with a zero object is an object G such that for every nonzero object H there exists a nonzero morphismZero morphismIn category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...
f:G → H.
- A cogenerator is an object C such that for every nonzero object H there exists a nonzero morphism f:H → C. (Note the reversed order).
The abelian group case
Assuming one has a category like that of abelian groupAbelian 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, one can in fact form direct sums of copies of G until the morphism
- f: Sum(G) →H
is surjective; and one can form direct products of C until the morphism
- f:H→ Prod(C)
is injective.
For example, the integers are a generator of the category of abelian groups (since every abelian group is a quotient of a 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...
). This is the origin of the term generator. The approximation here is normally described as generators and relations.
As an example of a cogenerator in the same category, we have Q/Z, the rationals modulo the integers, which is a 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...
abelian group. Given any abelian group A, there is an isomorphic copy of A contained inside the product of |A| copies of Q/Z. This approximation is close to what is called the divisible envelope - the true envelope is subject to a minimality condition.
General theory
In topological language, we try to find covers of unfamiliar objects.Finding a generator 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...
allows one to express every object as a quotient of a direct sum of copies of the generator. Finding a cogenerator allows one to express every object as a subobject of a direct product of copies of the cogenerator. One is often interested in projective generators (even finitely generated projective generators, called progenerators) and minimal injective cogenerators. Both examples above have these extra properties.
The cogenerator Q/Z is quite useful in the study of modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over general rings. If H is a left module over the ring R, one forms the (algebraic) character module H* consisting of all abelian group homomorphisms from H to Q/Z. H* is then a right R-module. Q/Z being a cogenerator says precisely that H* is 0 if and only if H is 0. Even more is true: the * operation takes a homomorphism
- f:H → K
to a homomorphism
- f*:K* → H*,
and f* is 0 if and only if f is 0. It is thus a faithful contravariant 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 when in the category of small categories....
from left R-modules to right R-modules.
Every H* is very special in structure: it is pure-injective (also called algebraically compact), which says more or less that solving equations in H* is relatively straightforward. One can often consider a problem after applying the * to simplify matters.
All of this can also be done for continuous modules H: one forms the topological character module of continuous group homomorphisms from H to the circle group R/Z.
In general topology
The Tietze extension theorem can be used to show that an intervalInterval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
is an injective cogenerator in a category of 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...
s subject to separation axiom
Separation axiom
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms...
s.