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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.

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    $\begingroup$ Huh, so there's no theorem analogous to the one for 1-toposes, namely that (apart from size issues) they are equivalent to the category of sheaves on themselves for the canonical topology? (and having a small set of generators gives a full category that as a site has the same sheaves). Do we know where this argument breaks down? $\endgroup$ Nov 10 '19 at 7:15
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    $\begingroup$ @DavidRoberts I'm not sure. I think what happens is that if you try to generalize the proof, you learn that $\infty$-topoi are left-exact localizations of presheaves on themselves. The issue then is that left-exact localizations need not come from Grothendieck topologies in the $\infty$-case, i.e., need not be "topological localizations" in the sense of HTT 6.2.1.4, which are characterized as being determined by the class of monomorphisms which are inverted. $\endgroup$ Nov 10 '19 at 7:35
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    $\begingroup$ @DavidRoberts I just want to confirm what Charles said: there's no known classification of left-exact localizations of a given presheaf category. The argument breaks down because the class of arrows sent to effective epimorphisms ("covering maps") do not determine the ones that are sent to equivalences, only those sent to ∞-connected arrows. So the classical argument only tells you that every hypercomplete topos is the hypercompletion of a sheaf topos. $\endgroup$ Nov 10 '19 at 7:54
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    $\begingroup$ OK, now the question seems more plausible: the problem is with wanting to use only sieves to measure 'localness' $\endgroup$ Nov 10 '19 at 11:19
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    $\begingroup$ This doesn't seem related to the question. The OP means ∞-topos in the sense of (∞,1)-category theory, and the 2-category of categories is not an example of that. $\endgroup$ Nov 11 '19 at 7:21
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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.

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