KT Chen's smooth CCC

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

Hi -

Bill Lawvere mentioned that KT Chen had a cartesian closed category
of smooth spaces.  I've found this very useful in my work on geometry.
I kept wanting more properties of this category, so finally my student
Alex Hoffnung and I wrote a paper about it:

Convenient Categories of Smooth Spaces
http://arxiv.org/abs/0807.1704

Abstract: A "Chen space" is a set X equipped with a collection of
"plots" - maps from convex sets to X - satisfying three simple
axioms. While an individual Chen space can be much worse than a
smooth manifold, the category of all Chen spaces is much better
behaved than the category of smooth manifolds.  For example, any
subspace or quotient space of a Chen space is a Chen space, and
the space of smooth maps between Chen spaces is again a Chen space.
Souriau's "diffeological spaces" share these convenient properties.
Here we give a unified treatment of both formalisms. Following ideas
of Dubuc, we show that Chen spaces, diffeological spaces, and even
simplicial complexes are examples of "concrete sheaves on a concrete
site". As a result, the categories of such spaces are locally
cartesian closed, with all limits, all colimits, and a weak subobject
classifier. For the benefit of differential geometers, our treatment
explains most of the category theory we use.

In particular, at some point we break down and admit we're dealing
with a "quasitopos".

Best,
jb