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