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Let $f : \mathbf{X}\to \mathbf{Y}$ be a morphism of topoi; in his 1977 monograph, Johnstone describes the open mapping cylinder of $f$ as the following pushout of topoi: $\require{AMScd}$ \begin{CD} \mathbf{X} @>{f}>> \mathbf{Y}\\ @V{\textit{closed point}}VV {\scriptsize\textit{(cocart)}}@VVV\\ \mathbf{X}\times\mathbb{S} @>>> \mathbf{M}_f^\circ \end{CD}

Because $\mathbf{M}_f^\circ$ is defined by a pushout, it is easy to view it as a figure shape --- and in particular, it is easy to see by mapping into the "topos of sets" / object classifier $\mathbb{O}$ that the topos, viewed as a category of sheaves or "Giraud frame", is the Artin gluing of the direct image functor $f_*$. On the other hand, it is not so clear how to understand the (generalized) points of $\mathbf{M}_f^\circ$, since points are described by mapping in. What exactly does $\mathbf{M}_f^\circ$ classify?

A hint is suggested by the fact that $\mathbf{M}_f^\circ$ is a kind of generalized Sierpiński cone / scone, where instead of pushing out along the terminal map $\mathbf{X} : \mathbf{X}\to \mathbf{1}$, we push out along $f : \mathbf{X}\to\mathbf{Y}$. In the case of the actual scone $\hat{\mathbf{X}} = \mathbf{M}^\circ_{\mathbf{X}}$, the points are easy to describe using the fact that the scone can be reconstructed as the restriction of the Vickers-Johnstone "lower bagdomain topos" along the inclusion $\mathbb{S}\hookrightarrow\mathbb{O}$: \begin{CD} \hat{\mathbf{X}} @>>> \mathbf{B}_{\mathsf{L}}(\mathbf{X})\\ @VVV {\scriptsize\textit{(cart)}} @VVV\\ \mathbb{S} @>>> \mathbb{O} \end{CD}

Under this identification, a point $\mathbf{W}\to \hat{\mathbf{X}}$ is determined by the data of an open $U\in \mathscr{O}_{\mathbf{W}}$ together with a point $\mathbf{W}_{/U}\to \mathbf{X}$. This universal property, by the way, makes perfect sense if you think of the scone as a "toposification" of the lift monad from domain theory: a point of the scone $\hat{\mathbf{X}}$ is just a "partial point" of $\mathbf{X}$.


I would like to find out whether there is a similar way to understand the open mapping cylinder for an arbitrary morphism of topoi $f$, where I have a crisp description of what it classifies in the style of the above. And if this makes sense, I am also curious about the closed mapping cylinder obtained by flipping the orientation of $\mathbb{S}$ by gluing with the open point $\mathbf{X}\to\mathbf{X}\times\mathbb{S}$ instead of the closed point.

It seems like there may be a way to do it by viewing $\mathbf{X}$ in a relative style as a $\mathbf{Y}$-topos and considering the ordinary scone, but I have to admit that I am not yet skilled with using the relative point of view in the category of topoi.

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    $\begingroup$ It's kind of shocking -- people have been studying toposes in general and Artin glueing in particular for decades -- I'm surprised I can't find a description of what the Artin glueing classifies anywhere! Do you know the answer when, say $f$ is etale? $\endgroup$
    – Tim Campion
    Feb 13, 2021 at 19:07
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    $\begingroup$ Following your suggestion about relativization, one would expect that it classifies something like commutative squares$$\begin{array}{ccc}\mathbf{W}_{/U}&\to&\mathbf{X}\\\downarrow&&\downarrow\\\mathbf{W}&\to&\mathbf{Y}\end{array}$$except that one has to take isomorphisms witnessing commutativity of these squares as part of the structure... $\endgroup$ Feb 13, 2021 at 21:42
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    $\begingroup$ In other words, it should classify isomorphisms $y|_U\cong f(x)$ (for various $U\in\mathcal O_{\mathbf W}$) where $y$ is a thing in $\mathbf W$ classified by $\mathbf Y$ and $x$ is a thing in $\mathbf W_{/U}$ classified by $\mathbf X$ $\endgroup$ Feb 13, 2021 at 21:48
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    $\begingroup$ Matthias Hutzler has work in progress on something related: gitlab.com/MatthiasHu/gluing-classifying-toposes $\endgroup$
    – David Roberts
    Feb 14, 2021 at 5:37
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    $\begingroup$ This seems really related to what I have been working on lately! :-) I think @მამუკაჯიბლაძე 's guess is right, but I can't really see it from the definition in the question yet. In terms of geometric theories, this is what I call "conditional extension" of the theory extension classified by the X --> Y. (After adding a new proposition symbol to the theory for Y, which corresponds to the product with sierpinski topos.) I will try to answer as soon as I understand a bit more. $\endgroup$ Feb 14, 2021 at 13:10

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