Properties of the category of smooth vector bundles over a smooth manifold

Question

I am wondering if there are any sources that discuss the properties of the category of vector bundles over a smooth manifold. It seems that most differential geometry texts I've looked at avoid explicitly discussing the properties of this category in a systematic fashion.

I know that the category is additive, but are there are any other useful things to know about this category? For instance:

1. Is it a closed monoidal category with respect to the tensor product?
2. Is there a sharp result that states the conditions under which kernels and quotients exist?
3. Under what conditions do functorial constructions on vector spaces induce corresponding constructions on smooth vector bundles?
4. How much of homological algebra applies to this category?

Answer 1

The reason why the categorical properties usually are not discussed in differential geometry texts is because they are not really needed (and most differential geometers probably don't have the categorical concepts in their standard repertoire). You will certainly find a categorical discussion of the category of concintuous vector bundles on a (say compact) space in text books on topological K-theory, and there is not much difference between continuous and smooth bundles in that respect.

As far as your concrete questions are concerned:

1. I think the question of being a closed monidal category reduces to point-wise questions so I should be true.

2. The condition for both is just constant (or to be formally correct, locally constant) rank.

3. Any functorial condition for vector spaces for which the induced maps on spaces of morphisms are smooth can be carried out for vector bundles. This is most transparent by describing vector bundles in terms of cocycles of transition functions, see section 8 of P. Michor's "Topics in Differential Geometry" (AMS Graduate Studies in Mathematics 93).

4. Quite a bit, but there always is the constant rank problem. In the end, most of it reduces to point-wise considerations.

Answer 2

The book Characteristic Classes (This may be behind a paywall) by Milnor & Stasheff answers some of your questions; in particular, they prove a theorem about "continuous functors": any functor on the category of vector spaces which is continuous on the space of isomorphisms between two vector spaces gives a functor on the category of vector bundles on some topological space.

One reason this category is not talked about very often is it is not that nice for exactly the reasons you mention. As Andreas Cap points out above, kernels and quotients do not always exist, and thus, homological algebra does not always work. The problem is you may have some morphisms of vector bundles whose rank changes on a submanifold of smaller dimension; the quotient or kernel of such a morphism won't be a vector bundle, since the dimension is not locally constant, so the local triviality condition is violated. (Some examples here).

Instead, one may work in the category of "coherent sheaves" - these were invented to be a category containing the category of vector bundles on a space, but including the missing kernels/quotients. This is common in algebraic geometry and related fields; the idea is to change your local picture from being some vector space over $\mathbb{R}$ or $\mathbb{C}$ to being a finitely generated module over the ring of smooth functions on your open sets. The main difference is that this allows things like "torsion" - maybe you have an element in your local module that is killed by the ideal of the ring of functions consisting of those functions that vanish on a submanifold of dimension one; this element can be thought of as being supported on that submanifold. This way, if the dimension of a kernel goes up near a point, you can model this behavior with such elements.

Unfortunately, coherent sheaves do not immediately give as nice of a geometric picture as vector bundles, and are fundamentally algebraic objects. The algebra is harder, and [to my knowledge - someone please correct me if I'm mistaken], things don't work so well over real manifolds - you need complex manifolds, where everything is basically algebraic already.