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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.)

Morgan Rogers (he/him) (Nov 05 2024 at 13:11):

I've been replaced! ;) Nice one @Ryuya Hora

Ivan Di Liberti (Nov 05 2024 at 21:34):

We are all replaceable, my friend.

Ryuya Hora (Nov 08 2024 at 01:33):

If we consider toposes of profinite topological group actions, we need to consider strongly non-complete one. (If G is strongly complete prof.top.grp., then the topos $\mathrm{Cont}(G)$ is equivalent to $[\mathrm{Disc}(G), \mathbf{FinSet}]$.)

So, what I've been thinking as a candidate is, the topos of continuous actions of $(\mathbb{Z}/2\mathbb{Z})^{{\mathbb{N}} }$. (Although (Z/2Z)^ℕ has only countably many open subgroups, it has uncountably many index-2 subgroups.)

This topos can be written down as follows:

Its object is a set $X$ equipped with a countable family of endofunctions $\{\tau_i :X \to X\}_{i\in \mathbb{N}}$ such that
・Each endofunction is an involution $\tau_i \circ \tau_i = \mathrm{id}_X$.
・The involutions are commutative $\tau_i \circ \tau_j =\tau_j \circ \tau_i$
・Only finitely many involutions are non-trivial $∃n\in \mathbb{N}, ∀i>n, \tau_i= \mathrm{id}_X$
Morphisms are functions preserving all involutions.

This defines a boolean, connected, essentially small, and locally finite elementary topos.

David Michael Roberts (Nov 08 2024 at 04:57):

This seems like a finitist analogue of an unbounded Set-topos I considered in a paper, where instead of the product of |N copies of Z/2. I had ORD-many, and with careful encoding considered the category of continuous actions on sets, for a "topology" on that large group. I didn't encode this using involutions, though.

Ryuya Hora (Nov 08 2024 at 05:33):

David Michael Roberts said:

This seems like a finitist analogue of an unbounded Set-topos I considered in a paper, where instead of the product of |N copies of Z/2. I had ORD-many, and with careful encoding considered the category of continuous actions on sets, for a "topology" on that large group. I didn't encode this using involutions, though.

Thank you for your comment. I also think this is a kind of finitist analogue. (I think this would work for an arbitrary strong limit cardinal $\kappa$, like $\aleph_0$ and the inaccessible cardinal corresponding to the fixed Grothendieck universe. In that case, we would consider completion with respect to $\kappa$-small quotient groups.)

For others’ reference, let me include a link to the paper here.

David Michael Roberts (Nov 08 2024 at 09:39):

Heh, I see now why I didn't use involutions: it wasn't a class-product of Z/2's!

That was in a thesis I supervised where the student used not a product of Z's like I did, but Z/2's and the project was on proving a (probably) stronger independence result, namely that the groupoid of anafunctors from the discrete groupoid with objects |N to the delooping of Z/2 could be not essentially small (WISC just for surjections to |N implies it is ess. small, but there is possibly a sliver of daylight between these two statements). This stronger independence statement is known to be true using material set theory methods, but the point was to do it in stack semantics.

Ideally one day this will turn into a paper.