Generator (category theory)

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In category theory in mathematics a family of generators (or family of separators) of a category \mathcal C is a collection \{G_i\in Ob(\mathcal C)|i\in I\} of objects, indexed by some set I, such that for any two morphisms f, g: X \rightarrow Y in \mathcal C, if f\neq g then there is some i∈I and morphism h : G_i \rightarrow X, such that the compositions f \circ h \neq g \circ h. If the family consists of a single object G, we say it is a generator.

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 \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.


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