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)?

notsubmanifolds 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"? – Andrew Stacey Aug 2 '12 at 14:45polyfold(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? – Mariano Suárez-Alvarez♦ Aug 3 '12 at 6:15