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.
Read more about this topic: Injective Cogenerator
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