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Stream: deprecated: our papers

Topic: The representing localic groupoid for a geometric theory

Graham Manuell (May 25 2023 at 13:37):

@Joshua Wrigley and I have just put a new paper on the arXiv: https://arxiv.org/abs/2305.15209

This is an attempt to make the Joyal-Tierney result that every topos is the topos of sheaves on a localic groupoid more accessible to people who are interested in topos theory, but who are not experts. In particular, we provide an explicit description of a localic groupoid that represents the classifying topos of a given geometric theory.

Matteo Capucci (he/him) (May 26 2023 at 07:51):

Nice!

Matteo Capucci (he/him) (May 26 2023 at 07:52):

I had to remind myself this is true for Grothendieck topoi, not all elementary topoi. Am I right?

Graham Manuell (May 26 2023 at 10:48):

Yes. That's right.

John Baez (May 26 2023 at 20:48):

For example the category of finite sets is an elementary topos, but it's not equivalent to the topos of sheaves on any site, much less on a localic groupoid, since it's not cocomplete.

David Michael Roberts (May 31 2023 at 05:28):

So then here's a question, which I posed as a reaction elsewhere to the paper coming out: is every $Set$-topos the category of (small?) sheaves on a large localic groupoid? Since locales and groupoids are dual, a large locale is merely a large frame, and is makes sense to talk about small (co)sheaves on such a thing (and then talk about coequivariant things, using a cogroupoid object).

David Michael Roberts (May 31 2023 at 09:52):

Since left adjoints preserve colimits, any elementary topos $\mathcal{E}$ with a geometric morphism $f\colon \mathcal{E} \to Set$ automatically has arbitrary small coproducts of objects in the image of $f^*$. I haven't thought about whether it would have arbitrary coproducts, though.

Graham Manuell (May 31 2023 at 13:27):

This question is a little above my pay grade, since I'm not really a topos theorist myself and all my intuition comes from the analogy with pointfree topology. It is not even clear to me (at least without thinking about it more seriously) whether sheaves on a large frame necessarily form a topos.

I know the category of uniformly continuous G-sets for a topological group G is an elementary topos that does not generally have small coproducts, but I don't know if it has a geometric morphism to Set.

David Michael Roberts (May 31 2023 at 13:45):

You should takes small sheaves, those that are a small colimit of representables. And even then, in general it might not work. But sometimes it does!

Morgan Rogers (he/him) (May 31 2023 at 13:49):

The result about uniformly continuous $G$-sets that @Graham Manuell stated comes from Example A2.1.7 in Sketches of an Elephant. The discrete $G$-sets (those for which the action is trivial) are always uniformly continuous, and the inclusion of these is left adjoint to the global sections functor. (For future readers, this discussion is happening in parallel).