In mathematics, specifically category theory, a family of generators (or family of separators) of a category \( {\mathcal {C}} \) is a collection \( {\displaystyle \{G_{i}\in Ob({\mathcal {C}})\mid i\in I\}} \) of objects, indexed by some set I, such that for any two morphisms\( f,g:X\to Y \) in \( {\displaystyle {\mathcal {C}},} \) if f\( {\displaystyle f\neq g} \) then there is some i in I and some morphism \( {\displaystyle h:G_{i}\to X} \) such that \( {\displaystyle f\circ h\neq g\circ h.} \) 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 \( {\mathbf Z} \) is a generator: If f and g are different, then there is an element \( x\in X \), such that \( f(x)\neq g(x) \). Hence the map \({\mathbf Z}\rightarrow X \), \( n\mapsto n\cdot x \)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.
In the category of modules over a ring R, a generator in a finite direct sum with itself contains an isomorphic copy of R as a direct summand. Consequently, a generator module is faithful, i.e. has zero annihilator.
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
generator in nLab
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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