Generator (category theory)
In category theory in mathematics a generator (or separator) of a category is an object G of the category, such that for any two different morphisms in , there is a morphism , such that the compositions .
Generators are central to the definition of Grothendieck categories.
- 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.
- 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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