---
tag: 3017J7P6JD
kind: concept
name: Pseudogroup
msc: [58H05, 57R30]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudogroup
same_as:
  wikidata: Q4382845
  wikipedia: Pseudogroup
---

## Summary

A pseudogroup on a [[topological space|0CQHFPGJ45]] is a collection of [[homeomorphisms|0QT00PB5G2]] between open subsets that is closed under composition, inversion, restriction and gluing.
It records the local symmetries of a space, which need not extend to global ones.
Pseudogroups are the transition rules of geometric structures: a smooth, complex or symplectic manifold is a space glued together by maps from the corresponding pseudogroup.
They also describe the transverse structure of foliations.

## Explanations

### L1 human

A group of symmetries acts on a whole space at once.
Many geometric questions only need symmetries that act on small pieces, such as a map from one disc to another.
A pseudogroup collects such local maps, with the rule that the collection is stable under the obvious operations: composing, inverting, shrinking the domain and patching pieces together.
For example, all smooth maps with smooth inverses between open subsets of $\R^n$ form a pseudogroup.
A smooth manifold is a space whose charts change by maps of this pseudogroup.