The Stacks project

Remark 25.9.3. Note that the crux of the proof is to use Lemma 25.8.2. This lemma is completely general and does not care about the exact shape of the simplicial sets (as long as they have only finitely many nondegenerate simplices). It seems altogether reasonable to expect a result of the following kind: Given any morphism $a : K \times \partial \Delta [k] \to L$, with $K$ and $L$ hypercoverings, there exists a morphism of hypercoverings $c : K' \to K$ and a morphism $g : K' \times \Delta [k] \to L$ such that $g|_{K' \times \partial \Delta [k]} = a \circ (c \times \text{id}_{\partial \Delta [k]})$. In other words, the category of hypercoverings is in a suitable sense contractible.