American death triangle

In category theory in mathematics a generator (or separator) of a category $\mathcal{𝒞}$ is an object G of the category, such that for any two different morphisms $f,g:X\to Y$ in $\mathcal{𝒞}$, there is a morphism $h:G\to X$, such that the compositions $f\circ h\ne g\circ h$.

Generators are central to the definition of Grothendieck categories.

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 $f(x)\ne g(x)$. Hence the map $\mathbf{Z}\to X,$ $n\mapsto n\cdot x$ suffices.
• Similarly, the one-point set is a generator for the category of sets.

References

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External links

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