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I was wondering about whether there is a natural category-theoretic framework that encapsulates the idea of vector bundle maps between different manifolds. Of course, we have the category $\mathbf{Diff}$ of smooth manifolds, and for any $M\in\mathbf{Diff}$, we have the category $\mathbf{Vec}(M)$ of $\mathbb R$-vector bundles over $M$. But is there a natural way of considering bundle maps $E\to F$ given $f:M\to N$ in $\mathbf{Diff}$ and with $E\in\mathbf{Vec}(M)$ and $F\in\mathbf{Vec}(N)$? Possibly we could let $\mathbf{Vec}(\mathbf{Diff})$ have as objects the categories $\mathbf{Vec}(M)$ though this does leave the question of what the natural morphisms are.

I'm not very familiar with higher category theory, but would that be the appropriate framework for this be that of a $2$-category?

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Given a vector bundle $\pi : E \to M$ over $M$, a vector bundle $\rho : F \to N$ over $N$, and a map $f : M \to N$, a natural notion of a morphism from $E$ to $F$ is a map $g : E \to F$ such that $\rho \circ g = \pi \circ f$, as in the following diagram: $$\begin{matrix} E & \overset{g}{\to} & F \\ {\scriptsize \pi} \downarrow && \downarrow {\scriptsize \nu} \\ M & \underset{f}{\to} & N \end{matrix}$$ By the universal property of pullbacks, this is equivalent to saying that $g$ is a map $E \to f^*F$ over $M$, where $f^*F$ is the pullback bundle of $F$ along $f$.

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  • $\begingroup$ Is tops theory remotely useful in differential geometry? $\endgroup$ – Monstrous Moonshine Nov 16 '17 at 20:01

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