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?
Best Answer
I will write what I think is a proof that in fact every infinity topos is equivalent to sheaves over itself. Please let me know if I am making any errors. I am basically adapting a proof from SGA4 of the classical statement.
Let $U$ a Grothendieck universe and suppose that $E$ is a left-exact localization of presheaves of $U$-small infinity groupoids on some $U$-small site. Then $E$ possesses a $U$-small set of generators, $X_\alpha$, $\alpha \in A$. By HTT 6.3.5.17, the Yoneda embedding of $E$ into sheaves of $V$-small infinity groupoids on $E$, with $U \in V$ a larger Grothendieck universe, preserves $U$-small colimits.
Lemma: If $i:F \hookrightarrow G$ is a mono with $F$ and $G$ infinity sheaves on $E$, with $G$ representable, then $F$ is representable.
Pf: Consider the family $\left(f:X_\alpha \to F, f \in F\left(X_\alpha\right)_0\right), \alpha \in A$, which is jointly epimorphic. Hence, the corresponding Cech-nerve is effective. Each iterative fiber product in this nerve, say $$X_{\alpha_1} \times_F ...\times_{F} X_{\alpha_n}\simeq X_{\alpha_1} \times_G ...\times_{G} X_{\alpha_n},$$ since $i$ is mono. But the right-hand side is representable. Hence, this is actually an effective groupoid object in $E$, so it has a colimit $C$ in $E$. Since the inclusion of $E$ into $V$-sheaves (the Yoneda embedding) preserves colimits, we conclude that $C$ represents $F$.
Now consider $H$ to be an arbitrary sheaf of $U$-small infinity groupoids on $E$. Consider the family $\left(g:X_\beta \to H, f \in H\left(X_\beta\right)_0\right), \beta \in B$. Each iterative fiber product in the associated nerve, say $X_{\alpha_1} \times_H ...\times_{H} X_{\alpha_n}$ is a subobject of the representable sheaf $X_{\alpha_1} \times...\times X_{\alpha_n}$, but also an infinity sheaf, hence it is representable by an object of $E$ by the lemma. So we again have an effective groupoid object which actually lies entirely in $E$, and as in the lemma, we conclude $H$ is representable.