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.
Examples
- 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.
References
- 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
External links
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