What's an example of an $\infty$-topos not equivalent to sheaves on a Grothendieck site?

Question

My question is as in the title:

Does anyone have an example (supposing one exists) of an $\infty$-topos which is known not to be equivalent to sheaves on a Grothendieck site?

An $\infty$-topos is as in Higher Topos Theory (HTT) 6.1.0.4: an $\infty$-category which is an accessible left-exact localization of presheaves on a small $\infty$-category.

A Grothendieck site is a small $\infty$-category $\mathcal{C}$ equipped with the $\infty$-categorical variant of the classical notion of a Grothendieck topology $\mathcal{T}$, as in HTT 6.2.2: a collection of sieves (subobjects $U\to j(C)$ of representable presheaves on $\mathcal{C}$) satisfying some axioms. Sheaves on $(\mathcal{C},\mathcal{T})$ are presheaves of $\infty$-groupoids on $\mathcal{C}$ which are local for the sieves in $\mathcal{T}$. Such form a full subcategory $\mathrm{Shv}(\mathcal{C},\mathcal{T})$ of the $\infty$-category of presheaves.

Note: the question Examples of $(\infty,1)$-topoi that are not given as sheaves on a Grothendieck topology appears superficially to be equivalent to this one. In practice it is not exactly the same. As answers to that question show, many interesting $\infty$-topoi exist which can be described without reference to any Grothendieck site. But it is still conceivable that a suitable site exists.

Also note: any $\infty$-topos $\mathcal{X}$ can be obtained as an accessible left-exact localization of some $\mathrm{Shv}(\mathcal{C},\mathcal{T})$ with respect to a suitable class of $\infty$-connected morphisms (HTT 6.2.2, 6.5.3.14), e.g., the class of hypercovers. However, this does not immediately preclude $\mathcal{X}$ being equivalent to $\mathrm{Shv}(\mathcal{C}',\mathcal{T}')$ for some other Grothendieck site $(\mathcal{C}',\mathcal{T}')$.

Added remark. I asked this question because I had, for a long time, tacitly assumed that such examples were plentiful, until I thought about it and realized I had no basis for thinking that. As no answers have yet been given, and I'm not aware of any tools which would likely lead to a resolution one way or the other, it looks to me that this should be regarded as an open question.

Answer 1

Not an answer -- the question is very much open! But I think it's worth compiling together some of the observations made in the comments (this answer is community wiki; feel free to add, correct, change it):

1. A fundamental difference between 1-topos theory and $\infty$-topos theory is that not every left exact localization of an $\infty$-topos $\mathcal E$ (even: of a presheaf $\infty$-topos) is localization with respect to a Grothendieck topology (a so-called topological localization). Rather, every left-exact localization $L$ of $\mathcal E$ factors as the topological localization at the Grothendieck topology $J$ generated by $L$, followed by a cotopological localization, so that $L\mathcal E$ lies somewhere between $J$-sheaves and the hypercompletion thereof.

2. Thus it's tempting to think, as Charles reports doing for some time, that almost any sheaf $\infty$-topos $\mathcal E$ which is not hypercomplete should yield examples of non-sheaf-$\infty$-toposes by taking cotopological localizations of $\mathcal E$. But of course, such $\infty$-toposes might admit sheaf presentations by changing the site.

3. Indeed, Charles gives an example of a sheaf $\infty$-topos which is not hypercomplete, but whose hypercompletion does turn out to be a sheaf $\infty$-topos (in fact a presheaf $\infty$-topos) over a different site. So it's unclear when the situation of (2) is likely to yield examples.

4. So far, we don't seem to have any candidate property enjoyed by sheaf $\infty$-toposes but not by $\infty$-toposes which are not-obviously-sheaf-$\infty$-toposes.

5. In 1-topos theory, we can do one better than stated in (1) above: every 1-topos is a sheaf topos over itself (or a suitable small subcategory thereof) via the canonical topology. This is known to fail for $\infty$-topoi. For example, let $C$ be a site such that representable sheaves are hypercomplete (e.g., a 1-site). If $Sh(C)$ is not hypercomplete (see below for examples), then the hypercompletion $Sh(C)^\mathrm{hyp}$ is not sheaves on itself with respect to the canonical topology.

6. Here are some examples of non-hypercomplete sheaf $\infty$-toposes, whose hypercompletions might be candidates for non-sheaf $\infty$-toposes. Maybe folks could add more:

• $Sh(Q)$, where $Q$ is the Hilbert cube (HTT 6.5.4.8)

• $\varprojlim_n Sh(B\mathbb Z/p^n)$ (HTT 7.2.2.31)

• parameterized spectra, or more generally $n$-excisive functors

Ironically, the classifying topos for $\infty$-connective objects is a sheaf topos.

Answer 2

Here is a conditional answer to the question. Consider the following

Hypothesis: If $\mathcal Y$ is a sheaf $\infty$-topos, then for any set of objects $Y_0 \subset \mathcal Y$, there exists a small, full subcategory $Y \subseteq \mathcal Y$ such that $Y_0 \subseteq Y$ and a topology $J$ on $Y$ such that $\mathcal Y \simeq Sh(Y,J)$ is canonically equivalent to sheaves on the site $(Y,J)$.

• Note that the analog of this Hypothesis is true for 1-topoi, because for sufficiently large full subcategories we can always take sheaves with respect to the canonical topology.

• This approach to proving the Hypothesis doesn't work for $\infty$-topoi.

• In fact, I don't even know if the hypothesis is true when $\mathcal Y = Spaces$.

Claim: If the Hypothesis is true, then every $\infty$-topos is a sheaf $\infty$-topos.

Proof: Note that the classifying topos $\mathcal C$ for $\infty$-connective morphisms is a sheaf $\infty$-topos (see here for the proof that the classifying topos for $\infty$-connective objects is a sheaf $\infty$-topos). So is the classifying topos $\mathcal O$ for objects. There is a geometric morphism $\mathcal O \to \mathcal C$ induced by a map of sites going the other direction.

If $\mathcal X$ is an $\infty$-topos, let $\mathcal X \to \mathcal Y$ be a geometric morphism exhibiting $\mathcal X$ as a cotopological localization of a sheaf $\infty$-topos $\mathcal Y$. So the localization is given by universally inverting some set $S$ of $\infty$-connective morphisms in $\mathcal Y$. By the Hypothesis, we may assume that these morphisms are between representables in the site presentation for $\mathcal Y$. So $\mathcal X$ is the pullback in the $\infty$-category of $\infty$-topoi $\mathcal X = \mathcal Y \times_{Psh(S) \otimes \mathcal C} (Psh(S) \otimes \mathcal O)$. The whole pullback diagram is induced by morphisms of sites, so the pullback $\mathcal X$ may be computed by taking a pushout in the $\infty$-category of sites and then passing to sheaves. Thus $\mathcal X$ is a sheaf $\infty$-topos.