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By Deligne's theorem, each coherent topos has enough points. What would be an example of a Grothendieck topos with enough points which is not coherent?

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    $\begingroup$ The Zariski topos on a scheme (or locally spectral topological space) that is not quasi-separated should work, right? In algebraic geometry, the standard example is $\operatorname{Spec} k[x_1,x_2,\ldots]$ with the origin doubled, but topologically there are easier examples such as $S^\mathbf N$ with its unique closed point doubled, where $S$ is the Sierpiński space. $\endgroup$ Mar 9, 2022 at 17:31
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    $\begingroup$ A lot of presheaf topoi are not coherent and yet they always have enough points. $\endgroup$ Mar 10, 2022 at 14:27

1 Answer 1

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Here are some examples :

  1. For any topological space $X$, the topos of sheaf $\operatorname{Sh}(X)$ has enough points. In most cases this is not a coherent topos. If I remember correctly (for $X$ sober), this is only coherent if $X$ is a spectral space. In any case, spaces like $\mathbb{R}$ are definitely not coherent toposes.

  2. For any topos $\mathcal{T}$, and any set of points $X$ of $\mathcal{T}$, you get geometric morphism $X \to \mathcal{T}$ (where by $X$ I mean the topos $\operatorname{Set}/X$), you can take its image factorisation $X \twoheadrightarrow I \hookrightarrow \mathcal{T}$ and $I$ is a subtopos of $\mathcal{T}$ with enough points. Generally, non coherent toposes don't have a lot of coherent subtoposes (though of course this can happen), so this will often gives example of non-coherent topos with enough topos. There is also a way to do this for $X$ the class of all points of $\mathcal{T}$ despite the size issues.

  3. There is another completeness theorem like Deligne's which says that any "separable" Grothendieck topos has enough points. Separable essentially means that the topos can be defined by a site whose underlying category is countable and whose topology is generated by a countable family of basic covering Sieve. This is done in Makkai and Reyes's book "First order categorical logic" (theorem 6.2.4).

There are many separable toposes that are not coherent. In terms of classifying toposes, coherent means you only use finitary logic, while separable means you can use infinitary logic but the theory should have a countable signature and a countable set of axioms. So none of the two class is included in the other, but I would personally consider that in term of which classical topos they contains, separable toposes form a much larger class.

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  • $\begingroup$ #1 and #3 seem excellent examples. #2 seems a bit less clear: is it easy to show (in some class of examples) that the image is not coherent? $\endgroup$ Mar 9, 2022 at 22:04
  • $\begingroup$ @PeterLeFanuLumsdaine I agree with you. Looking at examples, It rarely is coherent, and it has no reason to be except accident, but I couldn't find a nice general condition in which it never is coherent. I guess the reason I included it is more to emphasize that it is easy to construct topos with enough points : they are coreflective amongst all topos, while nothing of the sort hold for coherent toposes... $\endgroup$ Mar 9, 2022 at 22:28

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