On the homotopy theory of Grothendieck ∞-groupoids

We present a slight variation on a notion of weak ∞-groupoid introduced by Grothendieck in Pursuing Stacks and we study the homotopy theory of these ∞-groupoids. We prove that the obvious definition for homotopy groups of Grothendieck ∞-groupoids does not depend on any choice. This allows us to give equivalent characterizations of weak equivalences of Grothendieck ∞-groupoids, generalizing a well-known result for strict ∞-groupoids. On the other hand, given a model category M in which every object is fibrant, we construct, following Grothendieck, a fundamental ∞-groupoid functor Π∞ from M to the category of Grothendieck ∞-groupoids. We show that if X is an object of M, then the homotopy groups of Π∞(X) and of X are canonically isomorphic. We deduce that the functor Π∞ respects weak equivalences.