This question centers around section 6.5 of Borceux and Janelidze's Galois Theories.
Definition 1. Let $\mathcal M$ be a class of arrows in a category (in our case $\mathsf{Top}$). Say an arrow $f:X\rightarrow Y$ is locally in $\mathcal M$ if there's an open cover of $Y$ such that the base change along each open inclusion is in $\mathcal M$.
Question 1. (How) Can this be rephrased in the internal language of the sheaf topos $\mathsf{Sh}(Y)$? (if that's even the right topos).
Definition 2. Let $\Sigma \mathcal M$ be the class of all arrows isomorphic to ones of the form $$\coprod_{\lambda\in \Lambda}f_\lambda:\coprod _\lambda X_\lambda \longrightarrow \coprod _\lambda Y_\lambda$$ where each $f_\lambda$ is in $\mathcal M$.
Question 2. What's the geometric intuition behind this enlargement of $\mathcal M$? Does it have anything to do with suspension?
Then we have the following proposition. The authors write below the proof that "the notion "is locally in", which is an instance of the general idea of localization, can be described using the pullbacks along surjective étale maps instead of the inverse images of open neighborhoods."
Proposition 6.5.2. For an arrow in $\mathsf{Top}$ TFAE:
- it is locally in $\mathcal M$
- it is locally in $\Sigma \mathcal M$
- there exists an étale surjection that pulls it back to an arrow in $\Sigma \mathcal M$.
Question 3. Again, why does this make sense geometrically? Inclusions of opens are very far from surjections...
Question 4. How, formally, is "locally in $\mathcal M$" an instance of 'localization'?