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Stream: learning: questions

Topic: Giraud's theorem in weaker categories than elementary topoi

Haruki Toyota (Nov 26 2025 at 01:56):

I'm studying the elementary topos-theoretic generalization of Giraud's theorem reading Johnstone's Elephant B.1 and I noticed that many of lemmas used in the proof of it hold for weaker categories such as regular or Barr-exact categories.

Is there a known generalization of Giraud's theorem in the setting of such weaker categories?

Mike Shulman (Nov 26 2025 at 17:43):

Yes, there is, if you're willing to omit or fiddle with the size conditions. An analogue of a "category of sheaves" is the $\kappa$-ary exact completion of a $\kappa$-ary site, which I described in my paper Exact completions and small sheaves.

Then there's a theorem that a category is $\kappa$-ary exact if and only if it is the $\kappa$-ary exact completion of some $\kappa$-ary site. If you specialize this to $\kappa=\infty$, the "size of the universe", you get that a category is the category of small sheaves on some site if and only if it satisfies all the conditions of Giraud's theorem except for the existence of a small generator.

You can further refine the general theorem to say that given a $\kappa$-ary exact category, it is the $\kappa$-ary exact completion of any subcategory that is "$\kappa$-ary dense". In particular, that recovers the full Giraud theorem since a "small generator" is the same as being $\infty$-ary dense, and the $\infty$-ary exact completion of a small site is its category of sheaves (that is, all sheaves on a small category are small).

Haruki Toyota (Nov 27 2025 at 05:12):

It looks interesting, but it seems that everything in your paper is formulated in the category of sets. I think one of the beauties of Giraud’s theorem described in the Elephant is that it holds relative to any elementary topos. Is there a generalization from $\mathbf{Set}$ to elementary topoi or weaker categories? (Subquestion: Can we define the notion of “arity class” in weaker categories?)

David Michael Roberts (Nov 27 2025 at 06:15):

Is the question about fibred categories over a base elementary topos S? As in, when is such a fibred category really a bounded S-topos?

I think that people like Vickers and Maietti etc who work on arithmetic universes might have relevant things to say in that case: what does it mean to have a topos over a base "list-arithmetic pretopos"?

Mike Shulman (Nov 27 2025 at 20:01):

I haven't written it out, but I think nearly everything in that paper is constructive, so it should hold relative to any base topos with a NNO. I was careful to give a definition of arity class that makes sense constructively (although constructively it won't be true any more that every arity class is either a cardinal number or $\{1\}$), and to use subsingletons (Remark 2.5) in place of LEM when possible. If you want to rephrase it as an external definition about some elementary topos (rather than simply interpreting it in the internal logic of that topos) it would take a little work, but the result ought to be nice -- it should consist of some of the axioms of a [[class of small maps]].

Haruki Toyota (Nov 29 2025 at 00:39):

@David Michael Roberts
Yes, my question is about fibred categories over a base elementary topos S. I will send them an e-mail. Thank you!
@Mike Shulman
Thanks for your reply! Let me process this for a bit.