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The collection of all these cohesive -groupoids forms a cohesive (∞,1)-topos .
This is a cohesive site (…for the evident generalization of that definitions where Cech covers are generalized to hypercovers…). The key axiom to check is that for a hypercover of degreewise by a coproduct of contractibles, also the simplicial set obtained by sending each contractible to a point is contractible. This follows with ArtinMazur. See the proposition below
to be the (∞,1)-category of (∞,1)-sheaves on .
This is a cohesive (∞,1)-topos.
Proposition. For a locally contractible space (…maybe with some local size restriction, depending on the details of …), regarded as a 0-truncated locally contractible topological -groupoid, we have that the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor applied to coincides, up to equivalence, with the standard fundamental ∞-groupoid of .
Proof. By the analogous arguments as at ETop∞Grpd we may present by the left derived functor of the colimit functor on simplicial presheaves. This is the ordinary colimit applied to a cofibrant resolution of in . By Dugger’s cofibrant replacement theorem recalled at model structure on simplicial presheaves, such is given by a split hypercover degreewise a coproduct of objects in . By ArtinMazur there is a weak homotopy equivalence .