Lacking properties of the category of smooth manifolds

Question

According to Wikipedia "the category of smooth manifolds with smooth maps lacks certain desirable properties"(http://en.wikipedia.org/wiki/Differentiable_manifold#Generalizations). What are these desirable properties and why should a geometer care?

Answer 1

One is that the category of smooth manifolds does not have pullbacks; in particular, the intersection of two smooth submanifolds of a smooth manifold is not necessarily a smooth manifold, and even when it is it doesn't necessarily have the correct behavior (e.g. additivity of codimension). In general you can only take intersections like this under transversality hypotheses, so whenever you want to take such intersections (e.g. to relate them to the cup product) you need to perturb the manifolds involved slightly. Transversality issues can be annoying to deal with; I'm not familiar with the details, but I've been told they crop up in Floer theory and Gromov-Witten theory, for example. One way to motivate certain flavors of derived differential geometry is that in those contexts we can always take intersections having the correct properties (e.g. codimensions behaving the correct way) without dealing with transversality issues. See, for example, the work of Dominic Joyce.

Another is that the category of smooth manifolds does not have exponential objects; that is, given two (finite-dimensional) smooth manifolds $M$ and $N$ there's no natural way to turn the space of smooth maps $M \to N$ into a (finite-dimensional) smooth manifold itself. This would be a great thing to do if you could do it for a number of reasons; already the case that $M = S^1$ (so the construction of free loop spaces $LN$) is of great interest in connection with e.g. string topology, index theory, loop groups, elliptic cohomology...