A while ago I asked *this* quetion: Canonical topology for big infinity topoi

and *this* question: How to resolve size issues with the regular epimorphism topology

Let me first summarize some of what I learned from these:

If $U$ is a fixed ambient Grothendieck universe, then one can define a $U$-site to be a (not necessarily $U$-small) Grothendieck site $(C,J)$ such that there exists a $U$-small set $G$ of objects, called topological generators, such that every object of $C$ admits a covering family all of whose sources are in $G$.

$U$-sites are useful to have around, because they allow you to deal with "large" Grothendieck sites whose topos of sheaves are equivalent to the topos of sheaves of a small site. For example, if $E$ is a $U$-topos, i.e. a category equivalent to sheaves of $U$-small sets on some $U$-small site, then it is certainly not $U$-small itself. However, it does carry a Grothendieck topology, the *canonical* Grothendieck topology which is generated by jointly surjective epimorphisms. If we choose a $U$-small site of definition for $E$ such that $K$ is subcanonical, so that $E\cong Sh_K\left(D\right)$, then the objects of $D$, considered as representable sheaves, form a $U$-small set of topological generators for $E$, showing that $E$ with the canonical topology is in fact a $U$-site. Now, the category of $U$-small presheaves on $E$ is not $U$-small, but it's locally $U$-small, and the inclusion of the full subcategory of sheaves admits a left-exact left-adjoint (expose ii, theorem 3.4 of SGA 4). Moreover, this category of sheaves, is equivalent to $E$ itself.

I'm pretty sure that I can prove that all of this goes through for $n$-topoi when $n$ is finite. The complication arises when $E$ is a genuine infinty topos. Indeed, one can still equip $E$ with the canonical topology, and by careful use of Grothendieck universes as above, construct the infinity topos $Sh_{\infty}\left(E,can\right)$. However, its possible that $E$ is not equivalent to infinity sheaves on some site (e.g. it could be hypersheaves), so in this case, $Sh_{\infty}\left(E,can\right)$ can not be equivalent to $E$, correct? Even worse, $E$ could land somewhere between sheaves and hypersheaves, so we can't just say $E$ is the hypercompletion of $Sh_{\infty}\left(E,can\right)$.

My quetsion is, what is the relationship between $E$ and $Sh_{\infty}\left(E,can\right)$ when $E$ is an infinity topos? When $E$ is equivalent to infinty sheaves on a site, are these the same?

cotopological localization(page 531 of Higher Topos Theory: xxx.lanl.gov/pdf/math/0608040v4.pdf) $\endgroup$ – André Henriques Apr 25 '12 at 19:03