TITLE: Which dense inclusions of sites are ∞-dense?
QUESTION [8 upvotes]: An inclusion of sites f: D→C is dense if it induces an equivalence between the categories of sheaves on C and D.
Likewise, f is ∞-dense is it induces an equivalence between the ∞-categories of ∞-sheaves on C and D.
An ∞-dense inclusion is dense but the opposite is false in general, see Counterexample 6.5.4.2 in Lurie's Higher Topos Theory.
A nontrivial example of an ∞-dense inclusion of sites is given by the inclusion Cart→Man
of cartesian spaces into all smooth manifolds, as explained in
Is the site of (smooth) manifolds hypercomplete?
As it turns out, in the 1-categorical case we have an ample supply of dense inclusions of sites,
as explained by Proposition C2.2.16 of Johnstone's Sketches of an Elephant:
essentially small sites of definition of a given Grothendieck topos E that are subcanonical
can be identified with full subcategories of E whose objects form a separating family,
with the Grothedieck topology induced from the canonical topology on E.
Every inclusion of such sites is dense, but not necessarily ∞-dense.
I wonder if there is a (practical) criterion that allows us to check whether a dense inclusion is also ∞-dense.
We can assume the source site D to be hypercomplete, if necessary.
The example that I have in mind takes D to be the site of smooth manifolds (or cartesian spaces)
and C a subcategory of the category of sheaves of sets on smooth manifolds
that contains some infinite-dimensional smooth manifolds (e.g., smooth mapping spaces Map(M,N) between finite-dimensional manifolds).
What additional conditions are needed to ensure that a dense inclusion of sites is ∞-dense?

REPLY [2 votes]: If you content yourself with the hypercomplete case, a sufficient practical criterion is given in my paper joint with Tony Yue Yu, that you can find here: http://arxiv.org/abs/1412.5166. The result we are concerned with is lemma 2.35.
It is the analogue of the well known result in algebraic geometry, that can be found here: http://stacks.math.columbia.edu/tag/03A0.