# Is Set a finitely presentable object in Topoi?

The setting for this question is the (2,1) category Topoi.

• 0-cells in Topoi are Grothendieck topoi.
• 1-cells are geometric morphisms and have the direction of the right adjoint.
• 2-cells are invertible natural transformation (between the left adjoints).

Johnstone (and Lurie) proves that this 2-category has pseudo colimits that can be computed in Cat. To be precise, given a diagram in Topoi its colimit is the limit of the corresponding diagram in Cat associated to the inverse images.

Q. Does the representable functor $$\mathsf{pt}:= \text{Topoi(Set, )}$$ preserve directed colimits?

After some naive optimism, I have the feeling that the answer is no, but I suspect that it might be known.

• A candidate counterexample would be a topos $$\mathcal{E}$$ whose subtopoi lattice is not infinitary distributive, in fact a point is an atom in this coHeyting algebra.
• Another idea is to build a topos with with some points as a directed colimit of topoi without points, but I am not sure that this is possible at all.
• If a counterexample exist, I expect that it's possible to exhibit a localic counterexample.
• Is your functor $\mathrm{pt}$ valued in sets or in categories (which I feel is the most natural choice)? – Denis Nardin Feb 4 at 10:35
• It takes value in categories! Thanks for the question! – Ivan Di Liberti Feb 4 at 10:39
• I believe I can see that $Set$ is $\kappa$-presentable in the category of locally $\kappa$-presentable topoi, but I don't think it's any more presentable than that. However, I don't have a counterexample... – Tim Campion Feb 4 at 14:52

If $$\mathcal{C}$$ is a small category with finite limits then geometric morphisms from $${\rm Set}$$ to the presheaf topos $${\rm PSh}(\mathcal{C})$$ are in bijection with left exact functors $$\mathcal{C} \to {\rm Set}$$, or, equivalently, with objects in the pro-category $${\rm Pro}({\cal C})$$. More explicitly, if $${\bf X} = \{X_i\}_{i \in {\cal I}}$$ is a pro-object in $${\cal C}$$ then the corresponding point $${\bf X}^*:{\rm PSh}(\mathcal{C}) \to {\rm Set}$$ sends a presheaf $$F :\mathcal{C}^{\rm op} \to {\rm Set}$$ to $${\rm colim}_{i \in {\cal I}} F(X_i)$$. If $${\cal C}_1,{\cal C}_2$$ are two categories with finite limits and $$f: {\cal C}_1 \to {\cal C}_2$$ is a functor (which does not necessarily preserve finite limits) then we have a geometric morphism $$f_*: {\rm PSh}(\mathcal{C}_1) \leftrightarrows {\rm PSh}(\mathcal{C}_2): f^*$$ given by restriction and right Kan extension. We can then check that the functor on points $${\rm Pro}({\cal C}_1) \to {\rm Pro}({\cal C}_2)$$ induced by $$f_*$$ sends $$\{X_i\}_{i \in {\cal I}}$$ to $$\{f(X_i)\}_{i \in {\cal I}}$$.
If we now take a sequence $${\cal C_1} \to {\cal C_2} \to ... \to {\cal C_n} \to ... \to$$ of categories with finite limits (where the functors $${\cal C_i} \to {\cal C_{i+1}}$$ are not assumed to preserve finite limits) then $${\rm colim}^{\rm Topoi}_i {\rm PSh}({\cal C}_i) \simeq {\rm lim}^{\rm Cat}_i {\rm PSh}({\cal C}_i) \simeq {\rm PSh}({\rm colim}_i{\cal C_i}),$$ but in general the map $${\rm colim}_i{\rm Pro}({\cal C_i}) \to {\rm Pro}({\rm colim}_i{\cal C_i})$$ is not an equivalence. For example, it is often not essentially surjective: take $${\cal C_n} = [n]$$ with each consecutive map $$[n] \to [n+1]$$ being the inclusion as $$[n] \cong \{1,...,n+1\} \subseteq [n+1]$$ (these categories have finite limits because they are posets in which every finite subset has a minimal element).
• Thanks for your answer! I completely believe it, because I dot expect the Ind-completion to be an accessible KZ-doctrine over Cat, but is there an evident argument that proves $\text{colim} \text{Pro}([n]) \not \cong \text{Pro}([\omega])$? – Ivan Di Liberti Feb 7 at 10:01
• @IvanDiLiberti, Note that for every $n$ we have ${\rm Pro}([n]) \simeq [n]$, but there is a non-constant object in ${\rm Pro}([\omega])$, namely, the inverse system given by the identity diagram $\omega \to \omega$ (and which corresponds to the terminal left exact functor sending everything to the point). – Yonatan Harpaz Feb 7 at 10:52