# Does the 2-category of toposes admit pseudo-colimits?

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.

• Usually (Top) denotes the category of topological spaces. So perhaps another notation would be appropriate. – Martin Brandenburg Mar 28 '11 at 21:35
• Fair enough. I fixed it. – Todd Trimble Mar 28 '11 at 21:43

• 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. – Tim Campion May 19 '18 at 18:06