# Representing spaces of $\infty$-stacks

In The stable moduli space of Riemannsurfaces: Mumford’s conjecture, Madsen and Weiss introduce the representing space $$|\mathcal{F}|$$ of a sheaf of sets $$\mathcal{F}$$ on the site $$\mathscr{X}$$ of smooth manifolds as the geometric realization of the simplicial set $$[n]\mapsto \mathcal{F}(\Delta^n)$$ and prove that concordance classes of elements in $$\mathcal{F}(X)$$ bijectively correspond to homotopy classes of maps from $$X$$ to $$|\mathcal{F}|$$. As also the classifying space of a topological group $$G$$ admits a description as geometric realization (but now of a simplicial topological space rather than a simplicial set) and concordance classes of principal $$G$$-bundles on are the same thing as isomorphism classes, this suggests that more generally one should have a representing space for not only sheaves, but also for stacks and higher stacks on $$\mathscr{X}$$, similarly defined by a topological realization construction, and representing concordance classes. Unfortunately, I have not been able to precisely locate such a statement in the literature, so I'm asking for a reference here (in case such a result has some hope to be correct).

• The case of sheaves valued in $\infty$-groupoids is the content of a recent preprint of (Berwick-Evans)-[Boavida de Brito]-Pavlov: arxiv.org/pdf/1912.10544.pdf – Dylan Wilson Jan 7 '20 at 19:41
• Thanks, Dylan! If you can promote your comment to an answer (just the same text) I'll accept it and consequently close the question: that's precisely what I was looking for! – domenico fiorenza Jan 7 '20 at 20:22
• @domenicofiorenza There's no reason to close the question. – Kevin Arlin Jan 8 '20 at 1:15
• Right, "close" was not the correct term here, as it means a precise thing on MO. What I meant was: I'll make the question appear as "answered" so that MO users won't think I'm not satisfied with Dylan comment and struggle to find a more satisfying answer. – domenico fiorenza Jan 8 '20 at 6:24