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

A smooth nn-manifold is a topological space MM equipped with an open cover U αU_\alpha with homeomorphisms ϕ α:U α n\phi_\alpha: U_\alpha \to \mathbb{R}^n such that the transition maps ϕ βϕ α 1:ϕ α(U αU β)ϕ β(U αU β)\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 n\mathbb{R}^n with some other topological space XX (eg. n\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 XX to specify what transition maps are allowed. The usage of pseudogroups to define the corresponding notion of manifolds is described at manifolds.



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

  • If g:UVg: U \to V is a homeomorphism and U αU_\alpha is a cover of UU, then gg is in GG if and only if each restriction g| U α:U αg(U α)g|_{U_\alpha}: U_\alpha \to g(U_\alpha) is in GG.

Note that in particular the restriction of a map in GG remains in GG.


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 n\mathbb{R}^nsmooth maps(n-dimensional) smooth manifolds
complex n-dimensional space n\mathbb{C}^nanalytic functions(n-dimensional) complex manifolds
n-dimensional upper half-space H n={(x 1,,x n) n:x 10}H^n = \{(x_1, \ldots, x_n) \in \mathbb{R}^n: x_1 \geq 0\}smooth mapsmanifolds with boundary
the nn-cube I n=[0,1] nI^n = [0, 1]^nsmooth mapsmanifold with corners

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).


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

  • 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

A textbook account is in

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

See also

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

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

Last revised on August 7, 2016 at 10:31:10. See the history of this page for a list of all contributions to it.