Re: KT Chen's smooth CCC

[John Baez <baez@math.ucr.edu>]

Bill Lawvere wrote:

>By urging the study of the good geometrical ideas and constructions of
>Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, Spanier,
>Steenrod, I am of course not advocating the preferential resurrection of
>the particular categories they tentatively devised to contain the
>constructions.

I chose Chen's framework when Urs Schreiber and I were doing some work
in mathematical physics and we needed a "convenient category" of smooth
spaces.  I decided to choose one that was easy to explain to people
brainwashed by the "default paradigm", in which spaces are sets equipped
with extra structure.  Later I realized I needed to write a paper
establishing some properties of Chen's framework.  By doing that I guess
I'm guilty of reinforcing the default paradigm, and for that I apologize.

If I understand correctly, one can actually separate the objections
to continuing to develop Chen's theory of "differentiable spaces"
into two layers.

Let me remind everyone of Chen's 1977 definition.  He didn't state
it this way, but it's equivalent:

There's a category S whose objects are convex subsets C of R^n
(n = 0,1,2,...) and whose maps are smooth maps between these.
This category admits a Grothendieck pretopology where a cover
is an open cover in the usual sense.

A differentiable spaces is then a sheaf X on S.   We think of
X as a smooth space, and X(C) as the set of smooth maps from C to X.

But the way Chen sets it up, differentiable spaces are not all
the sheaves on S: just the "concrete" ones.

These are defined using the terminal object 1 in S.  Any convex set
C has an underlying set of points hom(1,C).  Any sheaf X on S has an
underlying set of points X(1).  Thanks to these, any element of X(C)
has an underlying function from hom(1,C) to X(1).  We say X is "concrete"
if for all C, the map sending elements of X(C) to their underlying
functions is 1-1.

The supposed advantage of concrete sheaves is that the underlying
set functor X |-> X(1) is faithful on these.  So, we can think of
them as sets with extra structure.

But this advantage is largely illusory.  The concreteness condition
is not very important in practice, and the concrete sheaves form not
a topos, but only a quasitopos.

That's one layer of objections.  Of course, *these* objections
can be answered by working with the topos of *all* sheaves on S.
This topos contains some useful non-concrete objects: for example,
an object F such that F^X is the 1-forms on X.

But now comes a second layer of objections.  This topos of sheaves
still lacks other key features of synthetic differential geometry.
Most importantly, it lacks the "infinitesimal arrow" object D such
that X^D is the tangent bundle of X.

The problem is that all the objects of S are ordinary "non-infinitesimal"
spaces.  There should only be one smooth map from any such space to D.
So as a sheaf on S, D would be indistinguishable from the 1-point space.

So I guess the real problem is that the site S is concrete: that is,
the functor assigning to any convex set C its set of points hom(1,C)
is faithful. I could be jumping to conclusions, but it seems to me
that that sheaves on a concrete site can never serve as a framework
for differential geometry with infinitesimals.

Best,
jb