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  "body": "\n## Summary\n\nThe pullback, or fibre product, of two morphisms $f \\colon X \\to Z$ and $g \\colon Y \\to Z$ with a common target is a universal commutative square over them.\nIt is the [[limit|CHS5AFJGWE]] of the diagram $X \\to Z \\leftarrow Y$.\nIn sets it is the set of pairs $(x, y)$ with $f(x) = g(y)$, and it includes preimages and intersections as special cases.\nPullbacks are used to restrict, intersect and change the base of objects throughout geometry and topology.\n\n## Explanations\n\n### L1 human\n\nTwo maps land in the same place $Z$.\nThe pullback collects the pairs of points, one from $X$ and one from $Y$, that land on the same point of $Z$.\nIf $Y$ is a subset of $Z$ and $g$ is the inclusion, the pairs are just the points $x$ with $f(x) \\in Y$, so the pullback is the preimage $f^{-1}(Y)$.\nIf both $X$ and $Y$ are subsets of $Z$ with their inclusions, the pullback is the intersection $X \\cap Y$.\n",
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