Dear Categorists - Algebraic topologists like to use not Top but a "convenient category" of topological spaces which has the advantage of being cartesian closed. The most popular seems to be CGHaus, the category of compactly generated Hausdorff spaces, or "Kelly spaces". It's cartesian closed, complete and cocomplete. I'll take these as the requirements for a category being "convenient". Lawvere and others have considered various topoi of "smooth spaces" in their work on synthetic differential geometry. I'm looking for something similar, but apparently a bit different. I want a convenient category of smooth spaces equipped with a forgetful functor to CGHaus: U: Smooth -> CGHaus so I can do differential geometry and the apply it to algebraic topology with the greatest of ease. I want U to be faithful, so that smoothness is just a *property* of continuous maps between smooth spaces. And, I want U to preserve limits and colimits. The topoi used in synthetic differential geometry don't seem to meet these requirements, because the all-important "infinitesimal arrow" D doesn't seem to have a good underlying Hausdorff space. You could say its underlying Hausdorff space is the one-point space, but this would not give a *faithful* functor U. Chen and Mostow have given definitions of "smooth space" that might meet my requirements, and I'm wondering what people think of them: K.T. Chen, Iterated paths integrals of differential forms and loop space homology, Ann. Maths. 97 (1973) 213-237. M. A. Mostow, The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations, J. Diff. Geom. 14 (1979), 255-293. I only have the energy to describe Chen's definition. I'm hoping someone can give me an elegant proof that it gives a category meeting my requirements. I think it does, but I don't know an elegant proof. I'll use "space" to mean an object of CGHaus, though Chen actually uses it to mean an object of Top. Definition: A "smooth space" X is a space equipped with, for each convex open subset U of some R^n, a collection P(U,X) of continuous maps f: U -> X, called "plots". These need to satisfy various properties: 1) If f: U -> X is a plot and g: V -> U is a smooth map with V convex open in some R^m, then fg is a plot. 2) If U is covered by convex open sets U_a and the restriction of f: U -> X to each U_a is a plot, then f is a plot. 3) If U = R^0, every map f: U -> X is a plot. Definition: If X and Y are smooth spaces and g: X -> Y is continuous, g is defined to be "smooth" if for every plot f: U -> X, gf: U -> Y is a plot. We can try to psychoanalyze this definition: Instead of talking about "convex open sets of R^n's" and smooth maps between these, I'd prefer to talk about R^n's and smooth maps between these, because I believe these form an equivalent category. Let's call this category OpenBall. If we do this and then drop all but condition 1), Chen's definition would say that a smooth space is a space X equipped with a presheaf P on OpenBall assigning to each R^n a set of "plots" P(R^n,X). He also requires that P be a subpresheaf of the presheaf that assigns to each R^n the set of all continuous functions from R^n to X. Then, 2) says this presheaf P is actually a sheaf with respect to some Grothendieck topology on OpenBall. Then, 3) is some extra condition that guarantees smooth maps between smooth spaces are determined by what they do to global points. Or if you prefer, a condition that guarantees constant maps are smooth. So, his definition seems like a way of starting with OpenBall equipped with some Grothendieck topology, taking the category of sheaves on this, and then forming a subcategory with the help of the functor OpenBall -> CGHaus. I'm wondering if anyone recognizes this construction as a standard trick for building topoi... or the composite of a couple standard tricks. Or, if there's some similar but better way to meet my requirements! Happy Thanksgiving, jb 27-Nov-2004 13:23:18 -0400,1354;000000000000-00000000