categories: nice category of "smooth spaces"?

[baez@math.ucr.edu]

Dear Categorists -

I'm getting really annoyed at how the category Diff of smooth
manifolds and smooth maps isn't complete and cocomplete.  
Is there some category of "smooth spaces" that repairs these
defects?  Ideally I would like a category S with a bunch of
properties like:

1) Diff is a full subcategory of S
2) There is a faithful functor F: S -> CGHaus, so we can think
of smooth spaces as nice topological spaces (compactly generated
Hausdorff spaces) equipped with some extra structure.  
3) S has small limits and colimits
4) F preserves limits and colimits
5) The obvious functor from the category of simplices to CGHaus
factors through F, with the resulting smooth structure on a simplex
having reasonable properties (everyone knows what a smooth function
from a simplex to a manifold should be).

I can imagine asking much more, but this should give the idea.
I don't know much about schemes or synthetic differential geometry, 
so I don't know whether they achieve these goals.  I also don't 
know much about Pawel Gajer's "differential spaces".  Apparently
Gajer has made K(Z,n) into a "differential space" for all n; this
should be pretty easy if the category "differential spaces" has 
properties like 1)-5).  In case anyone wants to read his stuff,
here are the references:

Pawel Gajer, Geometry of Deligne cohomology, Invent. Math. 127 
(1997), 155-207.  

Pawel Gajer, Higher holonomies, geometric loop groups and smooth Deligne 
cohomology, Advances in Geometry, Birkhauser, Boston, 1999, pp. 195-235.

While I'm at it, has anyone formulated a good notion of a
"category internal to Diff"?  I.e. a gadget with a manifold
of objects, a manifold of morphisms, composition being a 
smooth map, and so on?  This would be a snap if Diff had 
finite limits, but it doesn't... which is one reason I'm 
getting annoyed!  

Should I discard Diff and work with something better instead?  

Best,
John Baez