derived smooth geometry
(also called or ) is the category whose
For the second statement one needs that every paracompact manifold admits a differentially good open cover : an open cover by open balls that are diffeomorphic to a Cartesian spaces. The proof for this is spelled out at good open cover.
For the second statement observe that the Joyal-Jardine model structure on simplicial sheaves is a presentation for the hypercompletion of the (∞,1)-category of (∞,1)-sheaves (see presentations of (∞,1)-sheaf (∞,1)-toposes). By the above result it follows that there is an equivalence of (∞,1)-categories between the hypercompletions
Now CartSp is even an ∞-cohesive site. By the discussion there it follows that (before hypercompletion) is a cohesive (∞,1)-topos. This means that it is in particular a local (∞,1)-topos. But this implies (as discussed there), that the (∞,1)-category of (∞,1)-sheaves already is the hypercomplete (∞,1)-topos. Therefore finally