Contents

# Contents

## Idea

The definition of a smooth manifold is, as usual, given by

A smooth $n$-manifold is a topological space $M$ equipped with an open cover $U_\alpha$ with homeomorphisms $\phi_\alpha: U_\alpha \to \mathbb{R}^n$ such that the transition maps $\phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \to \phi_\beta(U_\alpha \cap U_\beta)$ are smooth.

If we replace $\mathbb{R}^n$ with some other topological space $X$ (eg. $\mathbb{C}^n$), and “smooth” with some other suitable restriction (eg. analytic), then we get a different kind of manifold. A pseudogroup is a structure we put on $X$ to specify what transition maps are allowed. The usage of pseudogroups to define the corresponding notion of manifolds is described at manifolds.

## Definition

###### Definition

Let $X$ be a topological space (or locale) and let $H$ be the groupoid whose objects are the open subsets of $X$ and morphisms are the homeomorphisms between them. A pseudogroup (also called transformation pseudogroup) on $X$ is a wide sub-groupoid $G$ of $H$ (ie. $G$ contains all open subsets of $X$) satisfying the sheaf property:

• If $g: U \to V$ is a homeomorphism and $U_\alpha$ is a cover of $U$, then $g$ is in $G$ if and only if each restriction $g|_{U_\alpha}: U_\alpha \to g(U_\alpha)$ is in $G$.

Note that in particular the restriction of a map in $G$ remains in $G$.

###### Example

We have the following correspondence between common pseudogroups and kinds of manifolds (the precise correspondence is defined in manifold):

Topological spaceMorphismsManifold
real n-dimensional space $\mathbb{R}^n$smooth maps(n-dimensional) smooth manifolds
complex n-dimensional space $\mathbb{C}^n$analytic functions(n-dimensional) complex manifolds
n-dimensional upper half-space $H^n = \{(x_1, \ldots, x_n) \in \mathbb{R}^n: x_1 \geq 0\}$smooth mapsmanifolds with boundary
the $n$-cube $I^n = [0, 1]^n$smooth mapsmanifold with corners
###### Definition

More generally, an abstract pseudogroup is a sub-inverse semigroup of the inverse semigroup of partial bijections of a set (or, even more generally, local automorphisms of an object in more general context).

## Properties

### Relation to effective étale stacks

In the language of topological groupoids/Lie groupoids and their associated geometric stacks, pseudogroups correspond to effective étale stacks. (Lawson-Lenz 11, Carchedi 12, section 2).

The concept was introduced in

• Élie Cartan, Sur la structure des groupes infinis de transformation (suite) Ann. Sci. École Norm. Sup. (3), 22:219–308, 1905 (Numdam)

• Élie Cartan, Les groupes de transformations continus, infinis, simples. Annales Scientifiques de l’École Normale Supérieure 26: 93–161, 1909 (Numdam)

• St. Golab, Über den Begriff der “Pseudogruppe von Transformationen” Mathematische Annalen 116: 768–780 (1939)

The history of the concept is reviewed in

• B. L. Reinhart

who attributes the term “pseudogroup” to

The relation of pseudogroups to Lie groupoids originates with Haefliger, see at Haefliger groupoid.

Further discussion of the relation of psuedogroups to étale groupoids/étale stacks includes

A textbook account is in

• Alan Paterson, Groupoids, Inverse Semigroups, and their Operator Algebras