6
$\begingroup$

So if $C$ is a 1-site and $D$ is a subsite (with the induced coverage), there are some conditions that ensure that the pre-composition and right Kan extension functors yield an equivalence of categories $$Sh(C)\simeq Sh(D)$$

Such a subsite $D$ is called $\textit{dense}$. Now, if C is an $(\infty,1)$-site and D is a sub-$(\infty,1)$-site of C (with the induced covering families), then are there analogous conditions than ensure that the usual pre-composition and Kan extension functors yield an equivalence of $(\infty,1)$-categories?

Improve this question
$\endgroup$
4
  • $\begingroup$ I will be extremely surprised if the same condition doesn't work for ∞-sites, but I cannot find a reference. $\endgroup$
    – Denis Nardin
    Commented Mar 4, 2017 at 23:23
  • 2
    $\begingroup$ @Denis It doesn't actually. The typical example is there's a basis of the topology on the Hilbert cube with a different ∞-category of sheaves. @ Karthik There are some sufficient conditions in Lemma C.3 in my paper math.mit.edu/~hoyois/papers/lefschetz.pdf. They're not very satisfying because they require existence of some pullbacks, but they suffice in many situations. $\endgroup$
    – Marc Hoyois
    Commented Mar 5, 2017 at 1:16
  • $\begingroup$ @MarcHoyois Great, I'd accept this as an answer. $\endgroup$
    – Karthik Yegnesh
    Commented Mar 5, 2017 at 1:38
  • $\begingroup$ @MarcHoyois Augh, I think I knew that counterexample once... I did not realize that a basis of a topology is an example of this, although it is obvious. $\endgroup$
    – Denis Nardin
    Commented Mar 5, 2017 at 2:45

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .