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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nAn object $G$ of $\\mathcal{C}$ is a *generator* (or *separator*) if it has the following property.\nFor all [[morphisms|859F0TFVA3]] $f, g \\colon X \\to Y$ with $f \\neq g$, there is a morphism $h \\colon G \\to X$ with $f \\circ h \\neq g \\circ h$.\nMore generally, a set $\\mathcal{G}$ of objects is a *generating family* if for all $f \\neq g \\colon X \\to Y$ there are $G \\in \\mathcal{G}$ and $h \\colon G \\to X$ with $f \\circ h \\neq g \\circ h$.\n\n## Remarks\n\nWhen $\\mathcal{C}$ is locally small, $G$ is a generator exactly when the [[hom functor|VJXS8AK67E]] $\\Hom(G, -) \\colon \\mathcal{C} \\to \\mathbf{Set}$ is [[faithful|GRG8WEBSH5]].\nThe dual notion is a cogenerator (or coseparator): an object $C$ such that $f \\neq g \\colon X \\to Y$ implies $k \\circ f \\neq k \\circ g$ for some $k \\colon Y \\to C$.\n\nExample: in the category of [[abelian groups|082GHNPJ53]], $\\Z$ is a generator.\nIf $f \\neq g \\colon A \\to B$, pick $a \\in A$ with $f(a) \\neq g(a)$ and let $h \\colon \\Z \\to A$ be $n \\mapsto na$; then $(f \\circ h)(1) = f(a) \\neq g(a) = (g \\circ h)(1)$.\n\nNon-example: in the same category, the trivial group $0$ is not a generator.\nThe identity and the zero map of $\\Z/2\\Z$ differ, but the only morphism $h \\colon 0 \\to \\Z/2\\Z$ is zero, and both composites with $h$ are zero.\n",
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