My question is: if I have a functor from the category of vector spaces Vect to the category of topological spaces Top (or differentiable manifolds Man) can I lift it to a functor from the category of vector bundles VectBun to the category of topological (or differentiable) bundles Bun?
Let $F:$ Vect $\to$ Top be a functor. I want to define $\tilde F:$ VectBun $\to$ Bun.
Objects. If $\pi: E \to B$ is a vector bundle, I can define $\tilde F(E) = \bigcup_{p\in B} F(E_p)$ and the obvious projection $\tilde \pi: \tilde F(E) \to B$. If $(U_i, \phi_i)_i$ is an atlas for $E$ I can define $\tilde \phi_i: \tilde \pi^{-1}(U_i) \to U_i \times F(\mathbb{R}^r)$ such that $\tilde \phi_i(\tilde e) = F(\phi_i|_{\{p\} \times E_p})(\tilde e)$ where $p = \tilde \pi(\tilde e)$. I can set a topology on $\tilde E$ saying that $A \subseteq \tilde E$ is open if and only if $\tilde \phi_i(A \cap \tilde\pi^{-1}(U_i))$ is open for every $i$.
Arrows. ...
My question are:
- Is it correct? To me, that seems correct, but I am afraid because functors "are not continuous in $p$" (I hope that it's clear what I am trying to say)
- Does this work also in the smooth category? If $F:$ Vect $\to$ Man , then $\tilde F(E)$ is a manifold? Again, my doubts come from the fact that functor are not "smooth in $p$"
In fact, for 2, I have that $(\tilde \phi_j \tilde \phi_i ^{-1})(p, x) = F(\phi_j \tilde \phi_i ^{-1}|_{\{p\}\times \mathbb{R}^r})(p, x)$ and $p$ is inside the functor. Then I don't know if $\tilde \phi_j \tilde \phi_i ^{-1}$.