Generator (category theory) - Alchetron, the free social encyclopedia

In category theory in mathematics a family of generators (or family of separators) of a category C is a collection { G i ∈ O b ( C ) | i ∈ I } of objects, indexed by some set I, such that for any two morphisms f , g : X → Y in C , if f ≠ g then there is some i∈I and morphism h : G i → X , such that the compositions f ∘ h ≠ g ∘ 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 Z is a generator: If f and g are different, then there is an element x ∈ X , such that f ( x ) ≠ g ( x ) . Hence the map Z → X , n ↦ n ⋅ 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.

References

Generator (category theory) Wikipedia

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