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Stream: learning: questions

Topic: Constructing Bundle Categories and Base Changes

John Onstead (Feb 10 2025 at 12:40):

Let $\mathrm{VBund}(-)$ be the functor $\mathrm{Top}^{op} \to \mathrm{Cat}$ sending a space $X$ to the category of vector bundles over it and a morphism to the pullback functor/base change. Firstly, I want to know how to construct $\mathrm{VBund}(X)$ using universal properties (rather than explicitly, as has already been covered). For instance, I know that the pullback of $U: \mathrm{Top} \to \mathrm{Set}$ along $V: \mathrm{Vect} \to \mathrm{Set}$ is the category of topological vector spaces. Ideally then, the category of vector bundles over $X$ is some pullback between $V$ and some functor $F: \mathrm{Bund}(X) \to \mathrm{Set}$ from the category of locally trivial "plain" fiber bundles over $X$ to $\mathrm{Set}$. The obvious choice is just the composition $U \circ dom$ with $dom$ sending a bundle to the domain space, since this seems to capture the intuition of "vector space object internal to bundles" we get from the explicit definition of $\mathrm{VBund}(X)$, but I'm not sure how to confirm this.

John Onstead (Feb 10 2025 at 12:41):

So that's my first question- how can one express the category $\mathrm{VBund}(X)$ as a pullback in $\mathrm{Cat}$? Thanks!

John Onstead (Feb 10 2025 at 22:14):

Also, here's another possibility I just thought of. Let $F$ be the functor that sends a bundle with isomorphic fibers to the underlying set of one of its fibers. An object in the pullback category $\mathrm{Bund}(X) \times_{\mathrm{Set}} \mathrm{Vect}$ with $F$ as this functor is a triple $(U(p^{-1}), p: E \to X, Y)$ where $U(p^{-1})$ is the underlying set of a fiber of $p$ and where $Y$ is a vector space such that $V(Y) = U(p^{-1})$. Thus, an object is a bundle such that every fiber is associated with the additional structure of a particular vector space, which seems a logical definition for a vector bundle.

John Onstead (Feb 10 2025 at 22:16):

(If some context helps, the reason I ask is because I'm ultimately trying to determine when a particular thing can be "pulled back" along a morphism so I can understand which structures are amenable to effective descent. For instance, I recently learned a manifold structure cannot be pulled back along an arbitrary continuous function. The above question is sort of a warmup to help better understand the more general situation)

John Baez (Feb 10 2025 at 22:24):

John Onstead said:

Also, here's another possibility I just thought of. Let $F$ be the functor that sends a bundle with isomorphic fibers to the underlying set of one of its fibers.

Which one? If you work with bundles over pointed spaces you can get a functor sending a bundle to the underlying set over the basepoint. But you won't get a functor if you randomly choose a point in the base and take the underlying set of the fiber over that point.

John Baez (Feb 10 2025 at 22:27):

Oftwn people work with a category of bundles whose fibers are all isomorphic to a specified space $F$, and then you have $F$ at your disposal and the problem I raised goes away.