higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
$Diff$ (also called $Man$ or $Mfd$) is the category whose
morphisms are differentiable functions (smooth functions) between these.
The category $Diff$ becomes a large site by equipping it with the coverage consisting of open covers.
This is an essentially small site: a dense sub-site for $Diff$ is given by CartSp${}_{smooth}$.
The first statement follows trivially as for Top: the preimage of an open subset under a continuous function is again open (by definition of continuouss function).
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.
The sheaf topos over $Diff$ is a cohesive topos.
The hypercompletion of the (∞,1)-sheaf (∞,1)-topos over $Diff$ is a cohesive (∞,1)-topos.
For the first statement, use that by the comparison lemma discussed at dense sub-site we have an equivalence of categories
By the discussion at CartSp we have that $CartSp_{smooth}$ is a cohesive site. By the discussion there the claim follows.
For the second statement observe that the Joyal-Jardine model structure on simplicial sheaves $Sh(Diff)^{\Delta^{op}}_{loc}$ is a presentation for the hypercompletion of the (∞,1)-category of (∞,1)-sheaves $\hat Sh_{(\infty,1)}(Diff)$ (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${}_{smooth}$ is even an ∞-cohesive site. By the discussion there it follows that $Sh_{(\infty,1)}(CartSp_{smooth})$ (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
The cohesive topos $Sh(Diff) \simeq Sh(CartSp_{smooth})$ is in particular the home of diffeological spaces. See there for more details.
is that of smooth ∞-groupoids. Discussed at Smooth∞Grpd.
The theory of differentiable stacks is that of geometric stacks in the (2,1)-sheaf (2,1)-topos