Generator (category theory)
In category theory in mathematics a family of generators (or family of separators) of a category is a collection of objects, indexed by some set I, such that for any two morphisms in , if then there is some i∈I and morphism , such that the compositions . If the family consists of a single object G, we say it is a generator (or separator).
Generators are central to the definition of Grothendieck categories.
The dual concept is called a cogenerator or coseparator.
- In the category of abelian groups, the group of integers is a generator: If f and g are different, then there is an element , such that . Hence the map suffices.
- Similarly, the one-point set is a generator for the category of sets. In fact, any nonempty set is a generator.
- In the category of sets, any set with at least two objects is a cogenerator.
- Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2, p. 123, section V.7
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