In reply to John Baez' questions on Nov. 26, I think that the general machinery which is relevant to the analysis of such categories of smooth spaces is the theory of concrete quasitoposes of Dubuc, cf. E. Dubuc, Concrete Quasitopoi, in "Applications of Sheaves", Proceedings Durham 1977 (Springer Lecture Notes 253). Note that the "differentiable spaces" of Chen (and the general machinery of Dubuc) deal with a COVARIANT determination of structure (i.e. the structure is given in terms of certain plots INTO a set/space), whereas the one considered by Mostow is CONTRAVARIANT (structure given in terms of certain functions OUT OF the set/space). You may have a "double" determination of mutually "balancing" plots and functions. This is the situation studied by Frölicher et al., aimed particularly at smooth spaces (cf. e.g. Frölicher and Kriegl: Linear Spaces and Differentiation Theory, Wiley 1988) (leading to the notion of Convenient Vector Space). A short survey of the Frölicher-Kriegl-Michor theory, and some references, are contained in Kock and Reyes: "Categorical Distribution Theory; Heat Equation" Aarhus Preprint 2004 no. 17; available electronically from the Aarhus Preprint Server, at http://www.imf.au.dk Anders Kock