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  "body": "\n## Summary\n\nThe functor of points of a [[scheme|5A35TYZD02]] $X$ sends each scheme $T$ to the set of morphisms $T \\to X$, called the $T$-points of $X$.\nBy the [[Yoneda lemma|D6DF1ARPVD]], the scheme $X$ is determined by this [[functor|28NQ0KRHD3]].\nThis lets one define schemes by the data they parametrize, as is done for moduli spaces and Hilbert schemes.\n\n## Explanations\n\n### L1 human\n\nInstead of asking what a space is made of, one can ask what all the maps into it look like.\nFor the affine plane over $\\Z$, a map from $\\operatorname{Spec} A$ into it is just a pair $(a, b)$ of elements of the ring $A$.\nSo the plane \"is\" the rule that sends each ring $A$ to the set $A^2$.\nSolutions of equations work the same way: the scheme of $x^2 + y^2 = 1$ sends $A$ to the set of solutions of $x^2 + y^2 = 1$ in $A$.\n",
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