---
tag: K8YRDXE4WC
kind: definition
name: Pseudogroup
msc: [58H05, 57R30]
defines: 3017J7P6JD
uses: [0CQHFPGJ45, 0QT00PB5G2]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudogroup
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A *pseudogroup* on $X$ is a set $\Gamma$ of [[homeomorphisms|0QT00PB5G2]] $f \colon U \to V$ between open subsets $U, V$ of $X$ with these properties.

1. The identity map of $X$ is in $\Gamma$.
2. If $f \colon U \to V$ is in $\Gamma$, then $f^{-1} \colon V \to U$ is in $\Gamma$.
3. If $f \colon U \to V$ and $h \colon U' \to V'$ are in $\Gamma$, then the composite $h \circ f$, defined on the open set $f^{-1}(V \cap U')$, is in $\Gamma$.
4. If $f \colon U \to V$ is in $\Gamma$ and $W \subseteq U$ is open, then the restriction $f|_W \colon W \to f(W)$ is in $\Gamma$.
5. Let $f \colon U \to V$ be a homeomorphism between open subsets of $X$, and let $U = \bigcup_i U_i$ be a cover by open sets. If each restriction $f|_{U_i}$ is in $\Gamma$, then $f$ is in $\Gamma$.

## Remarks

Example: the set of all $C^\infty$ diffeomorphisms between open subsets of $\R^n$ is a pseudogroup on $\R^n$.
Each clause holds because being a diffeomorphism is preserved by inversion, composition and restriction, and it is a local property, which gives clause 5.
The same holds for real-analytic diffeomorphisms, for biholomorphic maps between open subsets of $\C^n$ and for symplectomorphisms between open subsets of $\R^{2n}$.

Non-example: on $\R$, let $\Gamma$ be the set of restrictions to open sets of the bijective affine maps $x \mapsto ax + b$, $a \ne 0$.
It satisfies clauses 1 to 4 but fails clause 5.
The map on $(0, 1) \cup (2, 3)$ that is the identity on $(0, 1)$ and $x \mapsto x + 10$ on $(2, 3)$ is affine on each piece but is not the restriction of a single affine map.

A pseudogroup generated by a set of local homeomorphisms is the smallest pseudogroup containing them.
A $\Gamma$-structure on a space, such as a smooth structure, is an atlas whose transition maps lie in $\Gamma$.

## Explanations

### L1 human

A pseudogroup is a set of local symmetries that can be composed, inverted, shrunk and patched together.