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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 as pointed out on MO here.1
to be the (∞,1)-category of (∞,1)-sheaves on .
This is an cohesive (∞,1)-topos.