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
Famous quotes containing the words group and/or case:
“Now, honestly: if a large group of ... demonstrators blocked the entrances to St. Patrick’s Cathedral every Sunday for years, making it impossible for worshipers to get inside the church without someone escorting them through screaming crowds, wouldn’t some judge rule that those protesters could keep protesting, but behind police lines and out of the doorways?”
—Anna Quindlen (b. 1953)
“It is often the case that the man who can’t tell a lie thinks he is the best judge of one.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)