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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?

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  • $\begingroup$ Can you be more specific about what kind of subcategory $C$ is allowed to be? Does it contain at least one $n$-manifold for all $n$ for example? $\endgroup$
    – David Carchedi
    Commented Nov 13, 2013 at 18:37
  • $\begingroup$ @DavidCarchedi: In the examples I have in mind C is supposed to be some category of infinite-dimensional manifolds, so in particular it contains all finite-dimensional manifolds. $\endgroup$
    – Dmitri Pavlov
    Commented Nov 14, 2013 at 12:09
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    $\begingroup$ In that case, as long as your infinite-dimensional manifolds are locally modeled on "convenient vector spaces" (e.g. Frechet spaces), then $f$ is infinity dense. I'm a bit busy this moment, but I will try to type up the proof soon. $\endgroup$
    – David Carchedi
    Commented Nov 14, 2013 at 17:12
  • $\begingroup$ @DavidCarchedi: Ok, thanks a lot for letting me know this. I'm eagerly waiting for the details. $\endgroup$
    – Dmitri Pavlov
    Commented Nov 14, 2013 at 17:38
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    $\begingroup$ @DmitriPavlov : I don't quite see how the counterexample you quote from HTT is a counter example of this. To me it is the typical counter example of a non-Hypercomplete infinity topos of sheaves. I don't see the connection with what you are talking about. $\endgroup$
    – Simon Henry
    Commented Nov 19, 2018 at 9:37

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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.

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