Small complete categories in a Grothendieck topos

2010-10-24T21:16:07Z

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 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?

Answer by Steven Gubkin for Small complete categories in a Grothendieck topos

2010-12-31T22:17:22Z

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

That it has a geometric morphism to Sets, namely the global sections functor.
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

Small complete categories in a Grothendieck topos - MathOverflow

Small complete categories in a Grothendieck topos

Answer by Steven Gubkin for Small complete categories in a Grothendieck topos