nLab pseudogroup

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 space	Morphisms	Manifold
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 maps	manifolds with boundary
the $n$-cube $I^n = [0, 1]^n$	smooth maps	manifold 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).

Related concepts

• manifold
• inverse semigroup

References

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

• Veblen, Whitehead 1932

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

• Mark Lawson, Daniel Lenz, Pseudogroups and their étale groupoids (arXiv:1107.5511)

• David Carchedi, section 2 of Étale Stacks as Prolongations (arXiv:1212.2282)

A textbook account is in

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

See also

• Wikipedia, Pseudogroup

Abstract pseudogroups are studied in the inverse semigroup literature, e.g.

• Mark V. Lawson, Inverse semigroups: the theory of partial symmetries, World Scientific, 1998.