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Étale homotopy type of the moduli spaces of algebraic curves. (English) Zbl 0902.14019
Schneps, Leila (ed.) et al., Geometric Galois actions. 1. Around Grothendieck’s “Esquisse d’un programme”. Proceedings of the conference on geometry and arithmetic of moduli spaces, Luminy, France, August 1995. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 242, 85-95 (1997).
For an algebraic stack \(S\) one defines the étale homotopy type \((S)_{\text{et}}\) as the pro-simplicial set obtained via the connected component functor \(\pi_{0} : HR(S)^{0} \rightarrow\) simplicial sets. Here \(HR(S)^0\) is defined as the opposite of the category of hypercoverings of \(S\).
The article is devoted to a detailed proof of the statement that there is a weak equivalence of pro-simplicial sets between \(({\mathcal M}_{g,n} \otimes_{\mathbb Z} {\overline{\mathbb Q}})_{\text{et}} \) and \(K(\Gamma_{g,n} ,1)^\wedge\). Here \({\mathcal M}_{g,n}\), respectively \(\Gamma_{g,n}\), is the moduli space, respectively the mapping class group, of smooth \(n\)-pointed genus \(g\) curves; \(K(\Gamma_{g,n},1)\) is the latter’s associated Eilenberg-MacLane space and \(K(\Gamma_{g,n} ,1)^\wedge\;\) its profinite completion (in the sense of Artin-Mazur).
For the entire collection see [Zbl 0868.00041].

14H10 Families, moduli of curves (algebraic)
14F35 Homotopy theory and fundamental groups in algebraic geometry