Rovelli has introduced an extended idea of diffeomorphism which is smooth except at a finite number of points. The group of extended (or 'almost smooth') diffeomorphisms plays a role in the Fairbairn/Rovelli paper that just came out and also in the key chapter 6 of the new book "Quantum Gravity". Someone I exchange emails with has been wondering what to call these morphisms and one possibility that really ought to be considered, in my view, is to follow the example of the Skippy Peanut Butter people. For them, peanut butter is either Smooth or else it is smooth except at a finite number of points in which case it is Chunky almost smooth peanut butter is chunky so almost smooth homeomorphisms are chunkymorphisms ---------------- Einstein's original vintage 1915 General Relativity was, it seems, "diffeomorphism invariant" meaning that if you had some matter and a geometry which was a solution to the equation then you could skootch the matter and the metric around by a smooth map and it would still be a solution. As far as I know the original GR is not however invariant under chunkymorphisms. [correction: it seems it may be after all! one can extend the idea of a solution of the einstein equation to almost smooth metrics! see a later post in this thread] One thing about Rovelli is his audacity. I dont know how to spell it in Yiddish but there is a word for it. It is high risk to contemplate extending the diffeomorphism group. Maybe someone will have some thoughts about this. or maybe I will have some to add later. If you want background on chunkymorphisms (called by the correct name that Rovelli uses) its all thru Chapter 6 of the book and especially around pages 170-173 and 192 The book is online here http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html or else look at Fairbairn/Rovelli http://arxiv.org/gr-qc/0403047 here is an earlier thread on the Fairbairn/Rovelli paper https://www.physicsforums.com/showthread.php?t=16144 I think there probably is a differentiable-with-finite-number-of-singularities category, maybe someone who does category theory can look at it and see what it would be like. Fairbairn and Rovelli make a start on this. It seems potentially interesting.