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  "body": "\n## Summary\n\nA generator, also called a separator, of a [[category|A2DE66937D]] is an object $G$ with the following property.\nIts morphisms into other objects are enough to tell any two different parallel morphisms apart.\nEquivalently, the [[hom functor|VJXS8AK67E]] $\\Hom(G, -)$ is faithful.\nGenerators and the dual cogenerators appear as hypotheses in the special adjoint functor theorem, and generators are part of the definition of a Grothendieck category.\n\n## Explanations\n\n### L1 human\n\nTwo functions $f, g \\colon X \\to Y$ are different exactly when they differ at some point $x$ of $X$.\nIn the category of sets, a point of $X$ is the same as a map from the one-point set $\\{*\\}$ to $X$.\nSo $\\{*\\}$ can \"probe\" every difference between maps: it is a generator.\nIn the category of abelian groups, the group $\\Z$ plays the same role, since a homomorphism $\\Z \\to A$ is the same as an element of $A$.\nA generator is an object whose probes see everything.\n",
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