Injective Cogenerator - The Abelian Group Case

The Abelian Group Case

Assuming one has a category like that of abelian groups, 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). 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.