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Proposition 4.1 of Vistoli's notes says $\mathsf{Top}^2$ is a stack over $\mathsf{Top}$. The proof starts by looking at the fibered product $\coprod_iU_i \times _U\coprod_iU_i$, however this object never comes up again in the whole proof. Why is it introduced at all and why is it relevant to descent?

The same pullback is also mentioned here...

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popo
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I think you are right, it does not seem to be relevant. You have to construct a map $X→U$ . The way of doing this is to consider first the disjoint union of the $U_i$ (call it $U'$), then you have a map from the disjoint union of the $X_i$ (call it $X'$) to $U'$. Now you have $X'\to U'\to U$,and if you quotient $X'$ by the relation "to correspond through the relevant transition map" then you get a map $X\to U$ (where $X$ is the quotient).

Now as to why $U'\times_U U'$ is relevant in descent theory. In fact, it is crucial even in the definition of a stack! Given a site $F$ and a covering $U'\to U$, recall that the corresponding category of local data is given by those $\alpha\in F(U)$ such that the two possible pullback along the projections $U'\times_UU'\to U$ "coincide". Better: they are pairs $(\alpha, \sigma)$ where $\alpha\in F(U)$ and $\sigma$ is an isomorphism between the two pullbacks satisfying the cocycle condition.

So $U'\times_U U'$ in descent theory has the role of the pairwise intersections $U_{ij}:= U_i\cap U_j$ in a classical open covering of a topological space. If you define continuous maps (vector bundles, quasi-coherent sheaves, other geometric objects defined on spaces) on a covering $U'=\coprod U_i$ of $U$ such that on $U'\times_UU'=\coprod U_{ij}$ your constructions coincide (better: you can give isomorphisms that satisfy the cocycle conditions, a condition on $U'\times_UU'\times_UU'=\coprod U_{ijk}$), then your constructions yield a global one on $U$. To show that a certain fibered category is a stack, this is what you have to verify (as you know, you also need to show that "being isomorphic" is a local property).

So no relevance to the proof (at least the way I see it) but high relevance to the theory.

Sonner
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