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The notion of a coherent topos is a somewhat "refined" finiteness condition to put on a topos. One reason it is considered a "fruitful" notion is the Deligne completeness theorem, which tells us that every coherent topos has enough points.

A much "cruder" finiteness condition on a topos is to ask that it be locally finitely-presentable. I say that this is less "refined" because the notion of local finite presentability is not specifically adapted to topos theory -- it is a widely used notion in category theory more generally, whereas coherence is a notion which is not well-adapted to the study of categories like $Ab$ or $Grp$ which are very far from being toposes.

Despite the "crude" nature of this finiteness condition, it turns out to be specifically relevant to topos theory, due to the theorem that a topos is exponentiable if and only if it is a retract [EDIT: in a sense! see Marc Hoyois' comment below!] of a locally finitely presentable topos. This continues to be true $\infty$-categorically.

Moreover, every coherent topos is locally finitely presentable, motivating the following questions:

Question 1: Does every locally finitely presentable topos have enough points?

Equivalently More strongly, does every exponentiable topos have enough points?

Question 2: Does every compactly-generated $\infty$-topos have enough points? (Answer: no, as pointed out by Simon Henry below -- parameterized spectra are a counterexample)

Equivalently More strongly, does every exponentiable $\infty$-topos have enough points? (Answer: no, a fortiori)

Remarks:

  • There is reason to think that the answer should be affirmative. For instance, the answer is yes if we restrict attention to localic toposes. That is, every exponentiable locale is spatial.

  • If the answer is "yes", then we have a great strengthening of the Deligne completeness theorem. In fact, if the answer is "yes", then I might go so far as to say that the Deligne completeness theorem is a bit of a red herring, drawing as it does attention to the refined condition of coherence, when what is really signficant is the weaker and cruder condition of exponentiability.

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    $\begingroup$ For $\infty$-topos, you also have the hypercompleteness problem: $\infty$-topos with enough points are automatically hypercomplete, and while locally finitely presentable are not always (for e.g. the $\infty$-topos of parametrized spectra is locally finitely presentable). So a better question would be whether compactly generated hypercomplete $\infty$-topos have enough points (which is I believe the form that Deligne theorem takes for $\infty$-topos). $\endgroup$ Aug 10, 2021 at 17:29
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    $\begingroup$ @Ivan, that's not true, all locally compact hausdorff locales are exponentiable as toposes and most of them aren't scott topos. $\endgroup$ Aug 10, 2021 at 17:31
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    $\begingroup$ But indeed, theorem 2.58 of arxiv.org/pdf/2101.04015.pdf (the paper mention by Ivan) seem to claim that the answer to question 1 is yes. $\endgroup$ Aug 10, 2021 at 17:33
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    $\begingroup$ Author of said paper here. Indeed, "compactly generated" is weaker than "locally finitely presentable", but since finitely presentable objects are compact in the sense of my paper, Theorem 2.58 would indeed imply the 1-topos result that OP is after. @Marc Hoyois could you be more specific re the "gap"? My email address is in the paper if you would prefer to discuss that way. $\endgroup$ Aug 11, 2021 at 9:20
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    $\begingroup$ The sheaf condition is not a finite limit condition when the site does not have pullbacks, since it involves completions of the finite covering families to commutative squares (of which there may be infinitely many without a finite final subcategory); I constructed a counterexample demonstrating this for @Ivan earlier today, although the example needs some expansion in order to produce a compactly generated topos which is not locally finitely presented. However, this is off-topic! If I end up contributing an answer I will include such a counterexample in it. $\endgroup$ Aug 11, 2021 at 14:08

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