Generator (category theory)
In mathematics, specifically category theory, a family of generators (or family of separators) of a category [math]\displaystyle{ \mathcal C }[/math] is a collection [math]\displaystyle{ \mathcal G \subseteq Ob(\mathcal C) }[/math] of objects in [math]\displaystyle{ \mathcal C }[/math], such that for any two distinct morphisms [math]\displaystyle{ f, g: X \to Y }[/math] in [math]\displaystyle{ \mathcal{C} }[/math], that is with [math]\displaystyle{ f \neq g }[/math], there is some [math]\displaystyle{ G }[/math] in [math]\displaystyle{ \mathcal G }[/math] and some morphism [math]\displaystyle{ h : G \to X }[/math] such that [math]\displaystyle{ f \circ h \neq g \circ h. }[/math] If the collection consists of a single object [math]\displaystyle{ G }[/math], 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 [math]\displaystyle{ \mathbf Z }[/math] is a generator: If f and g are different, then there is an element [math]\displaystyle{ x \in X }[/math], such that [math]\displaystyle{ f(x) \neq g(x) }[/math]. Hence the map [math]\displaystyle{ \mathbf Z \rightarrow X, }[/math] [math]\displaystyle{ n \mapsto n \cdot x }[/math] 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 elements 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
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