---
tag: XK933YCW13
kind: definition
name: Galois insertion
msc: [06A15]
defines: BPV58EPYCQ
uses: [MBVJ4JQXC5]
sources:
  - url: https://en.wikipedia.org/wiki/Galois_connection
---

## Definition

Let $(P, \le)$ and $(Q, \le)$ be [[partially ordered sets|011C29Q1S1]].
A *Galois insertion* of $Q$ into $P$ is a [[Galois connection|MBVJ4JQXC5]] with lower adjoint $f \colon P \to Q$ and upper adjoint $g \colon Q \to P$ such that $f(g(y)) = y$ for all $y \in Q$.

## Remarks

Let $G$ be a group, let $P$ be the set of subsets of $G$, and let $Q$ be the set of subgroups of $G$, both ordered by inclusion.
Let $f(S)$ be the [[subgroup generated by|0RG9SPQSWM]] $S$, and let $g(H) = H$ as a subset.
Then $f(S) \subseteq H$ if and only if $S \subseteq H$, and $f(g(H)) = H$, so this is a Galois insertion.
The pair of the inclusion $\Z \to \R$ and the floor function $\R \to \Z$ is a Galois connection but not a Galois insertion.
There the lower adjoint is the inclusion, and the inclusion of $\lfloor 1/2 \rfloor = 0$ is $0 \ne 1/2$.
For a Galois connection, these conditions are equivalent: $f \circ g$ is the identity of $Q$; $f$ is surjective; $g$ is injective.
The dual notion, with $g(f(x)) = x$ for all $x \in P$, is a Galois coinsertion.

## Explanations

### L1 human

A Galois insertion is a Galois connection in which the smaller side sits faithfully inside the larger side.
Going in and coming back out changes nothing.