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My question is about the proof of Proposition A.6.1.6 in Lurie's Spectral Algebraic Geometry, which says the following:

Let $\mathcal{X}$ be any $\infty$-topos and denote by $\mathcal{X}^{coh}$ the full subcategory of the coherent objects. Then $\mathcal{X}^{coh}$ is a local $\infty$-pretopos.

In the proof, to show that $\mathcal{X}^{coh}$ is closed under geometric realizations of groupoid object, Lurie in a crucial way uses Proposition A.2.1.5 which states that if the pullback of a morphism along some effective epimorphism is relatively $n$-coherent, then the original morphism is already relatively $n$-coherent. This proposition, however, is only applicable if one assumes furthermore local $n$-coherence of $\mathcal{X}$, which again enters the proof in a very crucial way (no find an $n$-coherent cover of the object $U$ in the notation there).

Thus, it seems like the proof only works if $\mathcal{X}$ is already locally coherent, unless I just misunderstood the argument. Can it still somehow be salvaged for a general $\infty$-topos, as was claimed? I unfortunately also didn't find a proof of a classical analogon of this statement in the literature.

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  • $\begingroup$ I think the classical analogue you are looking for is theorem 3.3.7 / remark 3.3.9 in [Sketches of an elephant, Part D]: the full subcategory of coherent objects in a coherent topos is a pretopos. $\endgroup$
    – Zhen Lin
    Commented Mar 20, 2022 at 23:07
  • $\begingroup$ @ZhenLin Thanks a lot for the reference! As far as I can see coherence of the topos is still a requirement there, but I'll look at the proof more closely, maybe it yields some insights $\endgroup$
    – Markus Zetto
    Commented Mar 20, 2022 at 23:29
  • $\begingroup$ I haven't read that proof but have you checked that he does not apply A.2.1.5. to some slice topos over a coherent object ? Sometimes you reduce to something and are like "we can apply [X]" but you're applying X to a slice topos $\endgroup$
    – Maxime Ramzi
    Commented Mar 21, 2022 at 7:57
  • $\begingroup$ @MaximeRamzi That is definitely a good idea, but I thought trough it once more and if Lurie doesn't do a lot of steps between the lines that I don't see, he is definitely working in the topos here (and not a slice topos). One could think about working in $\mathcal{X}_X$ instead, but this is only locally $(n-1)$-coherent, so even there this argument wouldn't be applicable. $\endgroup$
    – Markus Zetto
    Commented Mar 21, 2022 at 9:05
  • $\begingroup$ I also forgot to mention that Prop. A.2.1.3, which is also used in the proof, depends on $\mathcal{X}$ being locally $n$-coherent as well, so there is even another point where this is needed. $\endgroup$
    – Markus Zetto
    Commented Mar 21, 2022 at 9:06

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If $X_0\to X$ is an effective epimorphism and $X_0$ is locally $n$-coherent, then $X$ is also locally $n$-coherent: every $Y$ over $X$ is covered by $Y\times_XX_0$, which is in turn covered by a coproduct of $n$-coherent objects. So in the proof of A.6.1.6 we know beforehand that $X$ is locally $n$-coherent for all $n$, hence all the results that assume local $n$-coherence apply.

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  • $\begingroup$ Thanks a lot, that is certainly easier than I thought it was going to be! Could one also use this argument in the proof of Theorem A.5.5.1, which states that in a locally $n$-coherent $\infty$-topos, the geometric realization of a semisimplicial object that is "pointwise" coherent and satisfies the Kan condition is still coherent? At the end of it, A.2.1.2 is used in a very similar way it is used here, so a similar argument might be able to completely get rid of the assumtion that $\mathcal{X}$ is locally $n$-coherent (note there's again an effective epimorphism from $X_0$ to $|X_\bullet|$) $\endgroup$
    – Markus Zetto
    Commented Mar 21, 2022 at 17:36
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    $\begingroup$ Yeah I guess so. You can just replace a general topos by the slice topos over the geometric realization, then apply A.5.5.1 as stated. $\endgroup$
    – Marc Hoyois
    Commented Mar 22, 2022 at 7:12

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