Subject: Chen spaces vs diffeological spaces From: John Baez Date: Wed, 12 Sep 2007 02:39:16 -0700 Andrew Stacey wrote: > > I think that Frolicher spaces and diffeological spaces (Chen's spaces) > > are one and the same. I'm afraid that diffeological spaces and Chen's spaces are not one and the same! At least in Iglesias-Zemmour's book, the "plots" used to define a diffeology on a space X are maps from R^n's to X. In Chen spaces, the "plots" are maps from convex subsets of R^n's to X. In both cases, a function f: X -> R is defined to be smooth iff its composites with all plots are smooth. Since [0,1] is a convex subset of R, it's easy to make X = [0,1] into a Chen space such that the smooth functions f: X -> R are precisely those that are smooth in the usual sense (meaning: even at the endpoints). On the other hand, I don't know if there's a diffeology on X = [0,1] that accomplishes this goal! For various - so far futile - attempts to settle this question, see: http://golem.ph.utexas.edu/category/2007/04/quantization_and_cohomology_we_19.html#c009006 I recently emailed Iglesias-Zemmour and he said he'd try to settle it. I hope he does! To me, diffeological spaces will be almost useless if I can't put a diffeology on [0,1] that captures the usual notion of smooth function f: [0,1] -> R. So for now, I'm using Chen spaces. Best, jb