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I have given pseudogroup an entry of its own, for the moment just copying there the definition from manifold. This is so as to be able to add references for the concepts, which I did.
It seems that you meant a pseudogroup of transformations, the entry wanted in inverse semigroup, rather than more general notions (when more general partial/local automorphisms Iso(X) replace the selfhomeomorphisms $Homeo(X)\subset Iso(X)$ or diffeomorphism of manifolds). The latter case is the classical case, the semigroup specialists now consider analogues in formal sense…
general pseudogroup of transformations is a sub-inverse-semigroup of $Homeo(X)$.
General pseudogroup in modern sense is just a sub-inverse semigroup of some group of “partial automorphisms” (in some concrete geometries often itself sub-inverse semigroups of partial bijections of underlying sets).
This is now quite standard among specialists on pseudogroup theory, wikipedia being mostly written by readers of old books. Very like sheaves are not nowdays necessarily on topological spaces but on sites etc.
You know where the edit button is.
I updated the page.
Golab whom you cite was a Polish mathematician, born in Bosnia, with quite an interesting biography which can be found online.
Glad to hear Zoran’s remarks on inverse semigroups.
In my unpublished web there was an article on “cartologies” which is based on Carboni’s notion of bicategories of partial maps and has to do with abstract notions of manifold, especially categories of such (as opposed to groupoids of manifolds). It’s still a bit of a mess, but I decided to put a copy on my published web for what interest it might have. (Putting it there might also spur me to do a little more on this.)
Interesting context, Todd! BTW some dollar signs starting formulas missing just above your Proposition 2 – in points labelled 2 and 3.
Added a short ideas section, simplified the definition and reformatted the example to give more information. Some clean-up of pseudogroup and manifold can/should be done to reduce redundancy.
A table in pseudogroup has strange terminology like complex plane for $\mathbb{C}^n$ and half-plane for upper half-space in the real $n$-dimensional space $\mathbb{R}^n$. I suggest using more standard complex $n$-dimensional space $\mathbb{C}^n$, real $n$-dimensional space $\mathbb{R}^n$, (real) $n$-dimensional upper half-space (notation discuttable as $\mathbb{H}^n$ is as a rule used for the hyperbolic $n$-dimensional space and the upper half-space can be for $n=2$ used as a model for the hyperbolic space).
The terminology was copied from the original passage, but I agree it is rather werid. Urs’ comment at #5 still applies.
Re #13: clearly, yes.
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