It is a classical theorem of Freyd that if a small category is complete (has all small limits—in fact, having small products suffices), then it is a preorder (has at most one morphism between any two objects). The proof of this theorem (which can be found [here](http://ncatlab.org/nlab/show/adjoint+functor+theorem) or in CWM) is non-constructive, i.e. it uses the Law of Excluded Middle. Therefore, it can potentially fail in the internal logic of an elementary topos. And in fact, it *does* fail in the effective topos, and more generally in realizability topoi, where there do exist small complete categories that are not preorders.


However, I have heard it said that Freyd's theorem cannot fail in a *Grothendieck* topos; i.e. that a small complete category in a Grothendieck topos must still be a preorder—despite the fact that the internal logic is still in general intuitionistic, so that Freyd's proof cannot work. Can someone explain why this is, or (even better) give a reference containing a proof?



Small complete categories in a Grothendieck topos
Hi. I mentioned that I had thought about this on nForum a while back - sorry I didn't get back to you sooner. The following sketch of a proof is mainly due to Colin McLarty.


Two features which distinguish a Grothendieck topos from a more general topos are


1. That it has a geometric morphism to Sets, namely the global sections functor.
2. That it has an object of generators (i.e. there is an object G such that if $f,g: A \to B$ are not equal then there exists an arrow $h: G \to A$ with $fh \neq gh$)


Let $C$ be a small complete category object in a Grothendieck topos $T$ which is not a preorder. Then $C^G$ is also a small complete category in this topos essentially because exponentials commute. The global sections functor applied to $C^G$ gives a small complete category in the category of sets which is not a preorder (the property of being a small complete category is preserved by geometric morphisms, and the special property of G allows the property of being "not a preorder" to carry through), which is a contradiction.


It is a little easier to think about in the case of sheaves on some topological space. There a small complete category object which is not a preorder would have to fail to be a preorder on some open set, and the sections on that open set would be a small complete category which is not a preorder. $G$ takes the place of this open set above.


If you have any questions about this let me know. In particular I can write out all of the adjunctions showing various properties are preserved, but I don't want to get too nitty gritty if it isn't useful to you.


Kind regards,
Steven Gubkin


