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I am struggling with Joyal-Tierney's paper Strong stacks and classifying spaces, (appeared in "Category Theory (Como, 1990)", volume 1488 of LNM, pp. 213–236, Springer 1991).

In particular one of the first sentences is totally out of my understanding:

Torsors solve the problem of finding all objects $T$ locally isomorphic to a given object $S$ of [a topos] $\mathcal E$. Recall that $T$ is locally isomorphic to $S$ iff there exists a covering $K$ (meaning $K$ is nonempty) and an isomorphism $K\times T\to K\times S$ over $K$. Equivalently, $T$ is locally isomorphic to $S$ iff $Iso(S,T)\neq\emptyset$.

So, it seems that $S,T$ are locally isomorphic iff... they are isomorphic! This obviously can't be, but I'm not able to interpret in any different way the notation "$Iso(S,T)$". Even the definition of covering is kind of smoky: I've always seen a covering defined as a collection of arrows $\{K_i\to T\}_{i\in I}$ such that $\amalg_{i\in I}K_i=K\to T$ is epi, hence I would rather say that $T$ is locally isomorphic to $S$ iff there exists a covering $\{K_i\}$ such that $K_i\times T\cong K_i\times S$ as objects in $\mathcal E/K_i$.

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1 Answer 1

1) If $K_i \times T \cong K_i \times S$ for all $i$, then $\sqcup_i K_i \times T \cong \sqcup_i K_i \times T$.

2) $\mathrm{Iso}(S,T) \neq \emptyset$ does not mean that this sheaf has a global section. It means that there is an open covering on which there are sections.

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