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Jun 27, 2017 at 8:53 vote accept Georg Lehner
Jun 27, 2017 at 7:38 comment added Jacob Lurie 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 $\mathcal{X}$).
Jun 27, 2017 at 7:12 history edited Jacob Lurie CC BY-SA 3.0
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Jun 27, 2017 at 4:56 history answered Jacob Lurie CC BY-SA 3.0