Timeline for What is the exact definition of the $\infty$-topos of sheaves on a localic $\infty$-groupoid?
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| Nov 18, 2023 at 18:57 | comment | added | Arshak Aivazian | I meant with a suitable localization, yes (the phrase "with a suitable model structure" in the text implied that). Okay, but for the rest $1 < n < \infty$ is this theme developed? I mean is there a definition of a $\infty$-category of local $n$-groupoids that is equivalent to a $\infty$-category of $n$-topos? | |
| Nov 18, 2023 at 11:56 | comment | added | Marc Hoyois | It is definitely not fully faithful. The generalization of the classical statement would be that it is a localization onto its essential image. I do not know if this question has been considered before. | |
| Nov 18, 2023 at 9:13 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Nov 18, 2023 at 9:08 | comment | added | Arshak Aivazian | @MarcHoyois Thank you! Is it unknown whether this functor is a full embedding (and for $n < \infty$ is this an equivalence)? | |
| Nov 18, 2023 at 7:35 | comment | added | Marc Hoyois | The natural functor is the embedding $[\Delta^\mathrm{op},\mathrm{Top}_0]\hookrightarrow [\Delta^\mathrm{op},\mathrm{Top}_\infty]$ followed by the colimit over $\Delta^\mathrm{op}$. One knows that every $\infty$-topos is in the essential image up to hypercompletion (Prop. A.4.3.1 in SAG). I think it is generally expected, but not known, that this functor fails to be essentially surjective, in contrast to the classical case (or the case of $n$-topoi for finite $n$). | |
| Nov 18, 2023 at 0:08 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Nov 17, 2023 at 21:58 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Nov 17, 2023 at 21:53 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |