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

Consider the following: CC = open sets of RR^n, all n, and all smooth maps in between. Let DD be the category of sets X furnished with a notion of Admisible maps from any object U in CC. Add(U, X) subset Allmaps(U, X) For each X, Add(U, X) should be a presheaf on U, and a sheaf for the open coverings in CC. DD is not only complete, cocomplete and cartesian closed, but also it is a Quasitopos. Of course, it is the subcategory of separated sheaves of the topos of sheafs on CC for the canonical topology. Have a full and faithfull embedding Diff --> DD defined by: Given a manifold M: Add(U, M) = Diff(U, M). There is also a faithful functor F: DD ---> EGHaus (topological haussdorf spaces generated by open sets of euclidean spaces, like the manifolds) Inspect this category, probably it has the properties you mention in your msage. This category is very much related with the well adapted models I introduced for SDG. It is of no use for SDG since it lacks infinitesimals. This is due to the fact that bad limits that exists in Diff (the non transversal ones) are preserved by the embedding Diff --> DD. In well adapted model of SDG, Diff --> EE, only the transversal limits are preserved. best, eduardo dubuc