Subject: Chen spaces vs diffeological spaces
From: John Baez <baez@math.ucr.edu>
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