The prime motivation to introduce the category of manifolds with corners is to have a convenient theory for the analysis on simplices that is as powerful as for smooth manifolds (with boundaries).

As far as I understand, polygonally bounded subdomains of $\mathbb R^n$ are not submanifolds with corners in general, at least for two reasons: (i) The category of manifolds with corners does not include "inward corners" (ii) The number of polygonal boundary pieces that meet at a common point can be arbitrarly large.

Is there a category of manifolds which contains arbitrary polygonal domains (where the boundary pieces may be curved)?

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    $\begingroup$ Do you mean "are not submanifolds with corners in general"? As for the actual question, there are plenty of categories of generalised smooth spaces that would include these but I guess you want something a bit smaller. What properties do you want for your category of "manifolds"? $\endgroup$ – Andrew Stacey Aug 2 '12 at 14:45
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    $\begingroup$ You can also use the category of PL manifolds with boundary or topological submanifolds with boundary. It all depends on the equivalence relation you want to have on your domains (or sheaves of functions that you would like to consider). If you allow that a (closed) quadrant is isomorphic to a half-plane, then PL or TOP suffice. If not, you have to use stratified spaces as in Rafe's answer. Also, by "polygon" you probably mean "polyhedron" (otherwise $n=2$). $\endgroup$ – Misha Aug 2 '12 at 15:11
  • $\begingroup$ yes, I meant not manifolds with corners. --- It seems that even manifolds with corners are only a very small subbranch that has barely been investigated and canonicalized. $\endgroup$ – shuhalo Aug 2 '12 at 17:19
  • $\begingroup$ What goes wrong if one defines a polyfold (excuse the silly neologism) as, say, a Hausdorff 2nd countable space which is locally modelled on closed polyhedral cones just as manifolds with corners are locally modeled on orthants? $\endgroup$ – Mariano Suárez-Álvarez Aug 3 '12 at 6:15

There is a class of what I call "smoothly stratified spaces". This is a bit less general than the larger class of stratified spaces (satisfying Thom-Mather axioms) in that it doesn't allow cusps. These spaces come up in many settings, but definitely include all polygonal and polyhedral domains. They are the setting for analysis in Cheeger's old paper `Spectral geometry of cone-like spaces' JDG 1983 (?), and are also discussed in detail in a recent paper of mine with Albin, Leichtnam and Piazza (The signature package on Witt spaces, just came out in Ann ENS). One point is that if you have a polyhedron, e.g. in $\mathbb{R}^3$, with vertices which are more than trivalent, then it is simply not a manifold with corners, but you can think of the local structure near each vertex as a cone over a spherical polygon.


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