An $(\infty,1)$-topos according to Lurie is defined as (accessible) left exact localization of a presheaf $(\infty,1)$-category $\text{P}(\mathcal C)$. Those $(\infty,1)$-topoi $\text{Sh}(\mathcal C)$ arrising from a site $C$ correspond precisely to topological left exact localizations of $\text{P}(\mathcal C)$.

What is an example of an $(\infty,1)$-topos not given as $\text{Sh}(\mathcal C)$ - i.e. an $(\infty,1)$-topos arrising from a non-topological localization? Should I think of them as pathological or as useful to have?

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    $\begingroup$ They are useful to have! There are many natural examples of ∞-topoi that have a priori nothing to do with Grothendieck topologies, such as ∞-categories of n-excisive or finitary functors with values in an ∞-topos, or ∞-categories of coalgebras in an ∞-topos. These may happen to be ∞-category of sheaves "by accident", and I don't know how one could rule that out, but knowing they are sheaves would not be useful. $\endgroup$ Jun 27, 2017 at 3:17
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    $\begingroup$ Of related interest, just as Grothendieck toposes were generalized to elementary toposes, Mike Shulman has proposed a notion of elementary ∞-toposes here golem.ph.utexas.edu/category/2017/04/elementary_1topoi.html, written up as this nLab entry ncatlab.org/nlab/show/elementary+%28infinity%2C1%29-topos. $\endgroup$ Jul 4, 2017 at 6:44

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Marc's examples are good ones, but let me add two more (which are closely related to each other):

1) Let $\mathcal{C}$ be an accessible $\infty$-category which admits small filtered colimits, and let $\mathcal{X}$ be the $\infty$-category of functors from $\mathcal{C}$ to $\mathcal{S}$ which preserve small filtered colimits. Then $\mathcal{X}$ is an $\infty$-topos, but I do not know how to realize $\mathcal{X}$ as an $\infty$-category of sheaves on a site.

2) Let $X$ be a locally compact topological space, and let $\mathcal{X}$ be the $\infty$-topos of sheaves on $X$. Then $\mathcal{X}$ is exponentiable in the setting of $\infty$-topoi: that is, for every $\infty$-topos $\mathcal{Y}$, there exists another $\infty$-topos $\mathcal{Y}^{\mathcal{X}}$ such that geometric morphisms from $\mathcal{Z}$ into $\mathcal{Y}^{\mathcal{X}}$ are the same as geometric morphisms from $\mathcal{Z} \times \mathcal{X}$ into $\mathcal{Y}$ (where the product is formed in the $\infty$-category of $\infty$-topoi). I do not know if $\mathcal{Y}^{\mathcal{X}}$ can be realized as an $\infty$-topos of sheaves on a site, even if it is assumed that such a description is known for $\mathcal{Y}$.

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    $\begingroup$ Oh, and there's an even more obvious source of examples: if $\mathcal{X}$ is an $\infty$-topos which does arise as sheaves on a site, then you can form the hypercompletion and Postnikov completion of $\mathcal{X}$. These are $\infty$-topoi which aren't obviously sheaves on a site (my guess would be that they are not, except in cases where they coincide with $\mathcal{X}$). $\endgroup$ Jun 27, 2017 at 7:38

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