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Stream: learning: questions

Topic: Locally finite elementary topos

Vincent Moreau (Oct 30 2024 at 10:27):

Hello everyone! I am looking for examples of essentially small elementary toposes which are locally finite, which by that I mean that every hom-set is a finite set.

If $\mathbf{J}$ is a small category, then the functor category $[\mathbf{J}, \mathbf{FinSet}]$ may or may not be an elementary topos, and may or may not be locally finite. I am interested in the case where both properties hold.

My question is the following: is every essentially small locally finite elementary topos of the form $[\mathbf{J}, \mathbf{FinSet}]$ for some small $\mathbf{J}$?

Ryuya Hora (Oct 30 2024 at 11:12):

Vincent Moreau said:

My question is the following: is every essentially small locally finite elementary topos of the form [J,FinSet] for some small J?

For a topological group G, its continuous actions on finite sets form an essentially small locally finite elementary topos. But i don't know whether it is (accidentally?) of the form of [J, FinSet]. (For example, the topos of finite continuous actions of the profinite integers is equivalent to the topos [Z, FinSet].

Vincent Moreau (Oct 30 2024 at 11:20):

I see thanks!

Ivan Di Liberti (Oct 30 2024 at 12:00):

Check out: Stack Representation of Finitely Presented Heyting Pretoposes by Lingyuan Ye. I am not sure it contains the answer, but I think it is relevant and it could be the state of art on the topic.

Vincent Moreau (Oct 30 2024 at 13:37):

Great, thanks!

Mike Shulman (Oct 30 2024 at 15:12):

Vincent Moreau said:

My question is the following: is every essentially small locally finite elementary topos of the form $[\mathbf{J}, \mathbf{FinSet}]$ for some small $\mathbf{J}$?

Note that being locally finite, for an elementary topos, is equivalent to admitting a geometric morphism to FinSet. I thought I remembered reading somewhere in Sketches of an Elephant that the answer to this question was yes, perhaps even without the "essentially small" qualifier, but I can't find it right now.

Ivan Di Liberti (Oct 30 2024 at 20:57):

Mike Shulman said:

Vincent Moreau said:

My question is the following: is every essentially small locally finite elementary topos of the form $[\mathbf{J}, \mathbf{FinSet}]$ for some small $\mathbf{J}$?

Note that being locally finite, for an elementary topos, is equivalent to admitting a geometric morphism to FinSet. I thought I remembered reading somewhere in Sketches of an Elephant that the answer to this question was yes, perhaps even without the "essentially small" qualifier, but I can't find it right now.

I remember a comment along these lines in Aspects of topoi by Freyd.

Ryuya Hora (Oct 31 2024 at 04:03):

Mike Shulman said:

Vincent Moreau said:

My question is the following: is every essentially small locally finite elementary topos of the form $[\mathbf{J}, \mathbf{FinSet}]$ for some small $\mathbf{J}$?

Note that being locally finite, for an elementary topos, is equivalent to admitting a geometric morphism to FinSet. I thought I remembered reading somewhere in Sketches of an Elephant that the answer to this question was yes, perhaps even without the "essentially small" qualifier, but I can't find it right now.

What I've read in Sketches of an Elephant is Cor 2.2.22* which states that a topos $E$ is of the form of $[\mathbf{J},\mathbf{FinSet}]$ for a finite category $\mathbf{J}$, if and only if $E$ is bounded over $\mathbf{FinSet}$.

If the essential smallness assumption (relative to a fixed Grothendieck universe U) is omitted, I think the topos of (U-)large group actions on FinSet provides a counter-example. For example, let $E$ be a category whose objects are finite set $X$ equipped with a (U-)large family of automorphisms $\{\rho_{\kappa}\colon X\to X\}_{\kappa \text{: U-small cardinal}}$. This category $E$ is locally finite, boolean, elementary topos, but not essentially small.

Mike Shulman (Oct 31 2024 at 05:21):

Right, there it is, thanks: C2.2.22. So a topos that's bounded over FinSet is always a topos of finite presheaves on a finite category, and thus a fortiori essentially small, whereas your example is an unbounded FinSet-topos that's not essentially small or presheaves on any small category. But can an essentially small FinSet-topos be unbounded?

Ryuya Hora (Oct 31 2024 at 06:01):

Mike Shulman said:

But can an essentially small FinSet-topos be unbounded?

Yes, for example, the topos of finite $\mathbb{Z}$-actions $[\mathbb{Z},\mathbf{FinSet} ]$ is essentially small and unbounded $\mathbf{FinSet}$-topos.

This topos is equivalent to the topos of finite continuous $\hat{\mathbb{Z}}$-actions. For other profinite group, the situation is similar. But I don't know whether they can always be presented as $[J, \mathbf{FinSet}]$.
(For example, how about finite continuous $\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-actions (instead of $\mathbf{Gal}(\overline{\mathbb{F}_p}/\mathbb{F}_p) \cong \hat{\mathbb{Z}}$)? I don't feel this is a finite presheaf topos, but I don't know how to prove it.)