higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
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.
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:
Note that in particular the restriction of a map in $G$ remains in $G$.
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 |
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).
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
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
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
See also
Abstract pseudogroups are studied in the inverse semigroup literature, e.g.
Last revised on August 7, 2016 at 10:31:10. See the history of this page for a list of all contributions to it.