Fiber bundles with varying fibers via pullbacks along étale surjections

Suppose $\mathsf C$ is a complete extensive category.

I managed to prove that a bundle $\alpha$ is a fiber bundle with fiber $F$ if there exists an associated cover $p:\coprod_iU_i\rightarrow B$ such that $p^\ast\alpha$ is a trivial fiber bundle with fiber $F$. For this, I used the fact extensive categories with products are distributive.

$$\require{AMScd} \begin{CD} U_i\times F @>>> A\\ @VVV @VV{\alpha}V\\ U_i @>>> B \end{CD}\forall i\iff\require{AMScd} \begin{CD} \coprod_iU_i\times F @>>> A\\ @V{p^\ast\alpha}VV @VV{\alpha}V\\ \coprod_iU_i @>>{p}> B \end{CD}$$

Now, I read that:

It is not really important to require all $F_b$ above to be homeomorphic to each other. Without that requirement there is still a reasonable description of locally trivial bundles, namely as those which split over some $(E,p)$, where $p$ is a surjective local homeomorphism, with trivial bundles replaced by their coproducts in the arrow category of $\mathsf{Top}$.

But I don't understand what this means. Suppose we restrict only to $p:\coprod _i U_i\rightarrow B$ the associated cover of a covering family. Is it saying the right square is a pullback iff all left ones are?

$$\require{AMScd} \begin{CD} U_i\times F_i @>>> A\\ @VVV @VV{\alpha}V\\ U_i @>>> B \end{CD}\forall i\iff\require{AMScd} \begin{CD} \coprod_iU_i\times F_i @>>> A\\ @V{p^\ast\alpha}VV @VV{\alpha}V\\ \coprod_iU_i @>>{p}> B \end{CD}$$

What do I do here?

Added. I have answered my original question below, but am clueless about the case of $p:E\rightarrow B$ a general étale surjection.

1 Answer

The for associated covers proof is exactly the same, and rests on the following characterization of universal coproducts.

Let $\mathsf{C}$ be a category with pullbacks and coproducts. TFAE.

• Coproducts are universal.
• The square on the right is a pullback if the square on the left is a pullback for each $i$.$$\require{AMScd} \begin{CD} P_i @>>> X_i\\ @VVV @VVV\\ A @>>> B \end{CD}\forall i\implies\require{AMScd} \begin{CD} \coprod_iP_i @>>> \coprod_iX_i\\ @VVV @VVV\\ A @>>> B \end{CD}$$
• The square on the right is a pullback if and only if the square on the left is a pullback for each $i$.$$\require{AMScd} \begin{CD} P_i @>>> X_i\\ @VVV @VVV\\ A @>>> B \end{CD}\forall i\iff\require{AMScd} \begin{CD} \coprod_iP_i @>>> \coprod_iX_i\\ @VVV @VVV\\ A @>>> B \end{CD}$$

I don't know what to do about the case of general étale surjections.