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Let $X$ be a topological space. We may naturally associate to $X$ a simplicial set which is a Kan complex. This in turn gives rise to the fundamental $\infty$-groupoid associated to $X$. From this object we can recover the usual fundamental groupoid associated to $X$. In this way we can view the fundamental groupoid as a higher dimensional decategorification of an $\infty$-groupoid.

Now assume that $X$ is a scheme. There is a natural concept of the the etale fundamental groupoid associated to $X$. More precisely the objects are geometric points of $X$ and morphisms are isomorphisms of the associated etale fibre functors.

My question is the following: Is there a concept of the etale fundamental $\infty$-groupoid associated to $X$, from which we can recover the usual etale fundamental groupoid as in the topological setting?

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Yes: it is due to Artin and Mazur. –  Moosbrugger Nov 8 '11 at 0:12
    
@Moosbrugger: care to add a reference? –  Yosemite Sam Nov 8 '11 at 8:13
    
Artin and Mazur, "Etale homotopy type", Lecture Notes in Mathematics 100. See also Friedlander, "Étale homotopy of simplicial schemes", Annals of Mathematics Studies 104. –  Alex Nov 8 '11 at 10:23
    
thanks ${ }$ –  Yosemite Sam Nov 8 '11 at 23:30

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