The Stacks project

Remark 25.11.1. One feature of this description is that if one of the multiple intersections $U_{i_0} \cap \ldots \cap U_{i_{n + 1}}$ is empty then the covering on the right hand side may be the empty covering. Thus it is not automatically the case that the maps $I_{n + 1} \to (\text{cosk}_ n\text{sk}_ n I)_{n + 1}$ are surjective. This means that the geometric realization of $I$ may be an interesting (non-contractible) space.

In fact, let $I'_ n \subset I_ n$ be the subset consisting of those simplices $i \in I_ n$ such that $U_ i \not= \emptyset $. It is easy to see that $I' \subset I$ is a subsimplicial set, and that $(I', \{ U_ i\} )$ is a hypercovering. Hence we can always refine a hypercovering to a hypercovering where none of the opens $U_ i$ is empty.

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