Generator (category theory)

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

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