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).

4$\begingroup$ The case of sheaves valued in $\infty$groupoids is the content of a recent preprint of (BerwickEvans)[Boavida de Brito]Pavlov: arxiv.org/pdf/1912.10544.pdf $\endgroup$ – Dylan Wilson Jan 7 '20 at 19:41

$\begingroup$ 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! $\endgroup$ – domenico fiorenza Jan 7 '20 at 20:22

$\begingroup$ @domenicofiorenza There's no reason to close the question. $\endgroup$ – Kevin Arlin Jan 8 '20 at 1:15

1$\begingroup$ 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. $\endgroup$ – domenico fiorenza Jan 8 '20 at 6:24
As requested, the comment as an answer:
The case of sheaves valued in ∞groupoids is the content of a recent preprint of (BerwickEvans)[Boavida de Brito]Pavlov: arxiv.org/pdf/1912.10544.pdf.