Injective cogenerator

When working with unfamiliar algebraic objects, one would like to find some approximation using more familiar objects. There are two main types of objects that one should familiarize one's self with in order to do these approximations.

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 with a zero object is some object G so that every other nonzero object H has some nonzero morphism f:G -> H.

• A cogenerator is some object C such that every other nonzero object H has some nonzero morphism f:H -> C. (Note the change in order).

Discussion of the abelian group case

Assuming one has a nice category, say of abelian groups, then one can form direct sums of copies of G until the morphism f:Sum(G) -> H is onto, and one can form direct products of C until the morphism f:H-> Prod(C) is one to one.

As an example, the integers are a generator of the category of abelian groups (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.

General theory

One is often interested in projective generators (even finitely generated projective generators, called progenerators) and minimal injective cogenerators. Both examples so far have these extra properties.

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. In topological language, we try to find covers of unfamiliar objects. Finding a cogenerator allows one to express every object as a subobject of a direct product of copies of the cogenerator.

The cogenerator Q/Z is quite useful in the study of modules over general rings. One forms the (algebraic) character module H* of homomorphisms from H to Q/Z. Being a cogenerator says precisely that H* is 0 if and only if H is zero. 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 zero. Similar statements apply to the topological (or Pontryagin) character module of continuous homomorphisms from H to R/Z (the circle group).

Every H* is very special, it is pure-injective (also called algebraically compact), which says more or less that solving equations in H* is relatively straightforward. Since * is such a pleasant operation (that neither creates nor destroys zero modules, morphisms, etc.) one can often consider a problem after applying the * to simplify matters.

The topological dual turned out to be quite useful since the dual of a discrete (untopologized) group was compact. This more topological field of study is loosely known as abstract harmonic analysis, or fourier analysis on groups[?].