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Let $E$ be a (Grothendieck) topos, e.g. $E = \text{Sh}(X)$ for a topological space $X$. And let $p = (p^*, p_*):\text{Set}\to E$ be a point of $E$, is there a way to "puncture" $E$ in some sense? By "puncture" I mean a way to remove the point $p$ and still get a topos.

e.g. one may expect to get $\text{Sh}(X\backslash\{p\})$ if $E = \text{Sh}(X)$ for a topological space $X$ and $p \in X$ is a usual point. But it's not obvious to see how to do this at the level of topos.

I'm asking this question because I want to know if topoi are "unpuncturable" in some sense.

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    $\begingroup$ The image of $p$ is a subtopos, and since subtoposes form a coHeyting algebra, you always can get the copseudocomplement of this subtopos. Note however that with toposes like $G$-sets (for some group $G$) you will get the trivial topos this way, which is probably not what you want. $\endgroup$ Oct 25, 2016 at 3:41
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    $\begingroup$ I can't think of anything else you could hope to get in the case of G-sets... $\endgroup$ Oct 27, 2016 at 3:05
  • $\begingroup$ @MikeShulman I can't think of anything else either. Another example is the presheaf topos $\text{Fun}(C^{\text{op}}, \text{Set})$, any object of $C$ defines a point, and there is no easy way to remove it from that category without destroying most of the structure (similar to the fact that puncture the complex plane destroys simply connectedness. E.g. if one "puncture" the category of simplicial sets at the point defined by [1], nothing good left). I'm asking this question just to confirm that puncture a topos is not a trivial thing. $\endgroup$
    – h__
    Oct 27, 2016 at 4:30
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    $\begingroup$ @h__: You can puncture $\mathbb{R}$ in type theory: $\Sigma_{x : \mathbb{R}} \mathrm{Id}_\mathbb{R}(x, 0) \to \mathrm{Empty}$. Even better is $\Sigma_{x : \mathbb{R}} (x < 0) + (0 < x)$. Your comment about algorithms is not correct since type theory (and constructive mathematics in greneral) can perfectly well speak about non-decidable predicates. There is no trouble, just the usual constructive difficulties. $\endgroup$ Oct 27, 2016 at 6:37
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    $\begingroup$ Note that even for topological spaces, it may well happen that the inclusion $X \setminus \{p\} \to X$ induces an equivalence of topoi. For example, this is the case if $p$ is the generic point of a variety. Thus, at the very least the topos doesn't know that the point is missing. $\endgroup$ Mar 26, 2022 at 10:59

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A more general question is how do you form the image of a geometric morphism f : E -> F, as a subtopos of F, and how do you form its complement? Recall that if U is a subobject of 1 in F then it defines an open subtopos (given by the operator U => ? on subobjects of 1) and its closed complement F-U is given by the operator of union with U on subobjects of 1. The closure of the image of f has to be the complement of the suprema of all the subobjects U of 1 in F for which f*(U) is 0. I have memories of a nice paper by Anders Kock about glueing which may yield more.

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    $\begingroup$ You probably mean Glueing Analysis for Complemented Subtoposes by Kock and Plewe? It contains characterization of some complemented subtoposes, but I wonder whether all complemented subtoposes have been described anywhere. Maybe for points it is easier... $\endgroup$ May 20, 2017 at 20:57
  • $\begingroup$ I believe Harry Simmonds wrote a paper on complementation, but I cannot remember the title. $\endgroup$ May 22, 2017 at 8:33
  • $\begingroup$ Hard to say - Simmons has almost all of his work, including the full publication list, displayed on his homepage but I could not find it there. Actually there also is a paper "Sublocale lattices" by Till Plewe which presents some further progress after his paper with Kock. However in any case all of the above could only help with localic toposes, whereas the question is about general ones. $\endgroup$ May 22, 2017 at 9:30

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