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  "body": "\n## Summary\n\nThe category of elements of a [[functor|28NQ0KRHD3]] $F \\colon \\mathcal{C} \\to \\mathbf{Set}$ has as objects the pairs $(A, x)$ with $x \\in F(A)$.\nIts morphisms are the morphisms of $\\mathcal{C}$ that carry one chosen element to the other.\nIt turns a set-valued functor into a category lying over $\\mathcal{C}$, and it is a special [[comma category|PC6HG1JEGM]].\nIt is used to write every [[presheaf|AND2WHZC8P]] as a colimit of representable functors, and it is the simplest case of the [[Grothendieck construction|272T16NWZ9]].\n\n## Explanations\n\n### L1 human\n\nA set-valued functor gives a set $F(A)$ for each object $A$, and functions between these sets for each arrow.\nThe category of elements spreads all these sets out: each element of each set becomes a point.\nThere is an arrow from $(A, x)$ to $(B, y)$ for each arrow $f \\colon A \\to B$ whose function sends $x$ to $y$.\nFor a group acting on a set, this gives a picture with one point per element of the set and one arrow per group element moving one point to another.\n",
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