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In Johnstone's book Topos Theory, he mentions an unresolved problem as to whether the 2-category $\mathbf{\text{Topos}}$ of Grothendieck toposes and geometric morphisms admits pseudo-colimits. $\mathbf{\text{Topos}}$ does admit lax colimits (including, famously, examples of Artin-Wraith gluing), and he remarks that insofar as

$$\mathbf{\text{Topos}}^{op} \to Cat$$

(which you can think of as the forgetful 2-functor from from Grothendieck toposes and left exact left adjoints to categories and functors) is represented by the object classifier $Set^{Fin}$, hence takes pseudo-colimits in $\mathbf{\text{Topos}}$ to pseudo-limits in $Cat$, what we are really asking is whether a pseudo-limit in $Cat$ of toposes and lex left adjoints gives back a topos.

Johnstone's book was written a long time ago, and I was wondering whether there has been progress on this problem since then.

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  • $\begingroup$ Usually (Top) denotes the category of topological spaces. So perhaps another notation would be appropriate. $\endgroup$ – Martin Brandenburg Mar 28 '11 at 21:35
  • $\begingroup$ Fair enough. I fixed it. $\endgroup$ – Todd Trimble Mar 28 '11 at 21:43
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The answer is yes; it seems to be due to Ieke Moerdijk in "The classifying topos of a continuous groupoid, I" (1988). There is also an exposition in section B3.4 of Johnstone's more recent book Sketches of an Elephant.

The idea is: a pseudo-limit in Cat of toposes and lex left adjoints "obviously" satisfies all the exactness conditions of Giraud's theorem, since lex left adjoints preserve all of that structure. Thus one "only" has to construct a small generating family in the limit category—but in general that is nontrivial!

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  • $\begingroup$ If the exactness conditions are obvious, then this really reduces to the fact (due to Bird) that locally presentable categories (and left adjoint functors) are closed under pseudo-limits in $Cat$. And indeed, in that case also most of the work goes into showing that one has a generating set. $\endgroup$ – Tim Campion May 19 '18 at 18:06

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