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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.

  1. 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 $i$.

  2. Arrows. ...

My question are:

  1. 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)
  2. 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 $\tilde \phi_j \tilde \phi_i ^{-1}$.

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  • $\begingroup$ I think this can be understood better in terms of classifying spaces. Basically $F$ induces a map of the nerve of the category of isomorphismes of vector spaces to that of the homeomorphisms of spaces, so that it induces a functor between the classifying spaces of the nerves of these categories. As these classifying spaces classify vector bundles and bundles over spaces, from this you get your $\tilde{F}$. What is scary for me here is that usually we work with the group completion of these monoids, so we would $F$ to be monoidal, but somehow in your case, we don't seem to need this... $\endgroup$
    – user43326
    Dec 3, 2019 at 15:44
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    $\begingroup$ Your categories are topologically enriched and you need your functor to be continuous. Then your construction will provide continuous cocycles i.e bundles. $\endgroup$
    – ThorbenK
    Dec 3, 2019 at 18:04
  • $\begingroup$ @ThorbenK To me seems that the "just topological" construction works, even if my functors are not continuous. Am I wrong? $\endgroup$
    – CNS709
    Dec 5, 2019 at 16:50
  • $\begingroup$ For your transition functions to be continuous (smooth) you need your functor to be continuous (smooth) $\endgroup$
    – ThorbenK
    Dec 5, 2019 at 17:51
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    $\begingroup$ Perhaps it's helpful to break it by restricting to a single object: at the end of the day, you have a homomorphism $GL_n \to Homeo(F)$ for some $n$ and some $F$, and you need this to be continuous. (At a high level, we are simply taking "B" of this map, but of course it's helpful to write down what's really happening at the level of transition functioms etc.) $\endgroup$ Dec 6, 2019 at 8:24

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