Show that a functor which preserves colimits has a right adjoint

In Moerdijk, Classifying spaces and classifying topoi, page 22, we find the following statement: a functor between topoi which preserves colimits must have a right adjoint, necessarily unique up to isomorphism (MacLane, Categories for the Working Mathematician, page 83).

Despite the reference, I actually fail to see the motivation for this. Can someone give me a hint for a proof (or an indication on how the reference is related to the problem)? It seems to me that the Theorem in CWM only covers the "uniqueness" part...

I don't know a reference for elementary topoi, but one for presheaf topoi can be found around pages 41-43 of MacLane-Moerdijk Sheaves in Geometry and Logic: they prove (Corollary 4) that a functor $A:\mathbb{C}\rightarrow\mathcal{E}$ where $\mathcal{E}$ is cocomplete and $\mathbb{C}$ is small extends along the Yoneda embedding $y$ to a unique (up to iso) $L_A:\widehat{\mathbb{C}}\rightarrow\mathcal{E}$ which preserves colimits, and such a functor has a right adjoint by construction (Theorem 2).

If now you consider some colimit preserving functor $F:\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{D}}$, the composition $A=Fy$ must extend in such a way: so it extends to $L_A=F$, but then $F$ must have a right adjoint.

• The theorem is false for elementary topoi, basically since they can be small: consider the inclusion of finite sets into sets. – Kevin Carlson Jan 15 '18 at 18:44
• Just a question: I understand that by $\hat{\mathbb C}$ you mean presheaves. What if I have sheaves? It is not always true that $\mathbb C$ embeds in $Sh(\mathbb C,J)$, i.e. not every topology is subcanonical. – W. Rether Nov 9 '18 at 9:24
• Update: using the adjoint pairs (inclusion, sheafification) between the topos of sheaves and the category of presheaves solves the problem. Thanks! – W. Rether Nov 18 '18 at 12:14
• @W.Rether also, if you're still interested you can check slide 19 at oliviacaramello.com/Teaching/Lectures15_to_18.pdf for a reference to a special adjoint functor theorem which holds in this case – TheMadcapLaughs Dec 30 '18 at 22:43

This is false in general without further hypotheses; see adjoint functor theorem. The reference to CWM, as you say, is only a reference for the uniqueness. The various adjoint functor theorems do imply this statement for a functor between Grothendieck topoi.

• Thank you. Indeed, the context I'm interested in is that of topoi. Edited! – W. Rether Jan 14 '18 at 21:13
• In particular, the nLab mentions (at the bottom of the Statements section) a fairly general result about functors between locally presentable categories which states any functor between locally presentable categories is a left adjoint if and only if it preserves colimits. This includes sheaf toposes but also many other handy things like categories of algebras. – Derek Elkins Jan 14 '18 at 23:11