---
tag: BPV58EPYCQ
kind: concept
name: Galois insertion
msc: [06A15]
specializes: [Q5JHEPRYQ6]
sources:
  - url: https://en.wikipedia.org/wiki/Galois_connection
---

## Summary

A Galois insertion is a [[Galois connection|MBVJ4JQXC5]] $(f, g)$ in which the lower adjoint undoes the upper adjoint: $f(g(y)) = y$ for every $y$.
Then $g$ embeds $Q$ into $P$ as the set of closed elements of the [[closure operator|17FVAN6HXX]] $g \circ f$.
Galois insertions describe how a structure such as "subgroups" sits inside a larger ordered set such as "all subsets".

## Explanations

### L1 human

In 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.
For a group, the subgroups sit inside the collection of all subsets.
Each subset $S$ has a smallest subgroup containing it, the subgroup generated by $S$.
Going from a subgroup to its underlying set and back gives the same subgroup.