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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...

Thank you in advance.

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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.

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    $\begingroup$ The theorem is false for elementary topoi, basically since they can be small: consider the inclusion of finite sets into sets. $\endgroup$ – Kevin Carlson Jan 15 '18 at 18:44
  • $\begingroup$ 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. $\endgroup$ – W. Rether Nov 9 '18 at 9:24
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    $\begingroup$ Update: using the adjoint pairs (inclusion, sheafification) between the topos of sheaves and the category of presheaves solves the problem. Thanks! $\endgroup$ – W. Rether Nov 18 '18 at 12:14
  • $\begingroup$ @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 $\endgroup$ – TheMadcapLaughs Dec 30 '18 at 22:43
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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.

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  • $\begingroup$ Thank you. Indeed, the context I'm interested in is that of topoi. Edited! $\endgroup$ – W. Rether Jan 14 '18 at 21:13
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    $\begingroup$ 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. $\endgroup$ – Derek Elkins Jan 14 '18 at 23:11

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