category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a branch of mathematics, the concept of an injective cogenerator is drawn from examples such as

Pontryagin duality In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...

. Generators are objects which cover other objects as an approximation, and (dually) cogenerators are objects which envelope other objects as an approximation. More precisely: * A generator of a

category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

with a

zero object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

is an object ''G'' such that for every nonzero object H there exists a non

zero morphism In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object. Definitions Suppose C is a category, and ''f'' : ''X'' → ''Y'' is a morphism in C. The ...

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 group In mathematics, 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 the order in which they are written. That is, the group operation is commut ...

s, one can in fact form direct sums of copies of ''G'' until the morphism :''f'': Sum(''G'') →''H'' is

surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...

; and one can form direct products of ''C'' until the morphism :''f'':''H''→ Prod(''C'') is

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...

. 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 In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...

). 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 In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

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

Finding a generator of an

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 ab ...

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 useful in the study of

modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...

over general rings. If ''H'' is a left module over the ring ''R'', one forms the (algebraic)

character module In mathematics, especially in the area of abstract algebra, every module has an associated character module. Using the associated character module it is possible to investigate the properties of the original module. One of the main results discovere ...

''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 In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

from left ''R''-modules to right ''R''-modules. Every ''H''* is pure-injective (also called algebraically compact). 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 In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \ ...

R/Z.

In general topology

The

Tietze extension theorem In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness ...

can be used to show that an interval is an injective cogenerator in a category of

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

s subject to

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. These are sometime ...

References

{{reflist Category theory