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  "body": "\n## Summary\n\nA Galois insertion is a [[Galois connection|MBVJ4JQXC5]] $(f, g)$ in which the lower adjoint undoes the upper adjoint: $f(g(y)) = y$ for every $y$.\nThen $g$ embeds $Q$ into $P$ as the set of closed elements of the [[closure operator|17FVAN6HXX]] $g \\circ f$.\nGalois insertions describe how a structure such as \"subgroups\" sits inside a larger ordered set such as \"all subsets\".\n\n## Explanations\n\n### L1 human\n\nIn a Galois insertion one ordered set sits inside the other, and each element of the big set has a best approximation from above in the small set.\nFor a group, the subgroups sit inside the collection of all subsets.\nEach subset $S$ has a smallest subgroup containing it, the subgroup generated by $S$.\nGoing from a subgroup to its underlying set and back gives the same subgroup.\n",
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