Let $\mathsf C$ be a superextensive site with products.
A trivial fiber bundle is a bundle $\pi:E\rightarrow B$ which is isomorphic to $\pi_1:B\times F\rightarrow B$ in $\mathsf{C}/B$.
Let $\mathcal M$ be a class of arrows in $\mathsf C$. An arrow $\pi:E\rightarrow B$ is said to be locally in $\mathcal M$ if there's a covering $ \left\{u_i: U_i\rightarrow B \right\}$ such that $u_i^\ast(\pi)$ are all in $\mathcal M$.
A fiber bundle with fiber $F$ is an arrow which is locally a product projection of the form $X\times F\longrightarrow X$.
I'm trying to figure out whether the fact pullbacks exist for fiber bundles follows from these definitions in this generality. Let $\mathcal M$ be the class of product projections. My questions are:
- Is $\mathcal M$ stable under base change? (I think so because limits commute with limits, but I'm not sure whether this is actually right here.)
- Is the class of arrows locally in $\mathcal M$ stable under base change? (Again I'm tempted to just wave my hands and say everything here is just limits and they commute, so sure, but I'm not sure.)
- Is the class of arrows locally in $\mathcal M$ stable under base change assuming only $\mathcal M$ is? (Again, I'm guessing yes.)