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Stream: learning: questions

Topic: $$\aleph_0$$-condensed sets

Madeleine Birchfield (Dec 07 2024 at 00:18):

So $\kappa$-condensed sets can be defined for a regular cardinal $\kappa$. For example, the $\kappa$-pyknotic sets are defined to be $\kappa$-condensed sets for an inaccessible cardinal $\kappa$. And the light condensed sets are the $\aleph_1$-condensed sets.

What are the $\aleph_0$-condensed sets?

David Michael Roberts (Dec 07 2024 at 10:36):

Sheaves on the category of finite sets with the coherent coverage, no? But this implies that a sheaf is determined by its value on a point, by preservation of finite coproducts. And then one needs that surjections are sent to equalisers.

David Michael Roberts (Dec 07 2024 at 11:28):

At least, the value of the sheaf on objects is determined by where it sends the point. The arrow component of the functor is determined by where it sends the map {1,2}->*. I'm guessing the final category you're after is just equivalent to Set, but I'm checking it in a low-brow way.

David Michael Roberts (Dec 07 2024 at 11:35):

Ok, and the value of the functor on {a,b}->* is the diagonal map, so, yes, the sheaf is specified completely by the choice of a single set.
I know there's abstract characterisations of the sheaf cat as a cocompletion, so the answer is going to end up being Set. But it's sometimes fun to get one's hands dirty...

David Michael Roberts (Dec 07 2024 at 16:49):

To be concrete, every aleph0-condensed set is Set(—,X):FinSet^op-->Set for some arbitrary set X.

Madeleine Birchfield (Dec 07 2024 at 17:40):

I had a hunch there was something degenerate about $\aleph_0$-condensed sets but wasn't exactly sure what the degeneracy was going to be.

Madeleine Birchfield (Dec 07 2024 at 17:44):

Next thing I wonder is, given two regular cardinals $\kappa \leq \lambda$, do the $\kappa$-condensed sets embed into the $\lambda$-condensed sets? Because if so than the embedding of $\aleph_0$-condensed sets into other condensed sets provides a way of representing the "uncondensed" sets in condensed sets. (If not then never mind.)

Madeleine Birchfield (Dec 07 2024 at 18:01):

Actually, I think that $\aleph_0$-condensed sets already embed into $\kappa$-condensed sets for $\aleph_0 \leq \kappa$, since I believe $\kappa$-condensed sets form a local topos over Set.

Madeleine Birchfield (Dec 07 2024 at 18:02):

At least $\kappa$-condensed $\infty$-groupoids form a local $(\infty,1)$-topos over $\infty\mathrm{Grpd}$ modulo size issues, and I think everything still holds when we 0-truncate everything.

David Michael Roberts (Dec 07 2024 at 21:21):

I think you need strong limit or something, not regular, to get a ff functor.

David Michael Roberts (Dec 07 2024 at 21:24):

Hmm, Proposition 2.9 in https://www.math.uni-bonn.de/people/scholze/Condensed.pdf does the case where both cardinals are uncountable strong limit. So you just need to consider kappa=aleph_0 and lambda a strong limit.

Madeleine Birchfield (Dec 07 2024 at 21:26):

so essentially, it is the pyknotic sets that embed in larger cardinal pyknotic sets.

David Michael Roberts (Dec 07 2024 at 21:27):

Strong limit≠inaccessible, though.

David Michael Roberts (Dec 07 2024 at 21:28):

Aleph_0 is still strong limit

Madeleine Birchfield (Dec 07 2024 at 21:28):

Yes but Scholze used uncountable strong limits, so excluding $\aleph_0$.

Madeleine Birchfield (Dec 07 2024 at 21:29):

unless there are other uncountable strong limits that aren't inaccessible that I'm forgetting off the top of my head

Madeleine Birchfield (Dec 07 2024 at 21:30):

On the other hand, I don't think there is any barrier to adaping Scholze's proof to the case of $\kappa = \aleph_0$.

David Michael Roberts (Dec 07 2024 at 21:32):

There is a cofinal sequence of uncountable strong limit cardinals in ZFC

David Michael Roberts (Dec 07 2024 at 21:32):

Scholze deliberately avoided assuming the existence of inaccessibles

Madeleine Birchfield (Dec 07 2024 at 21:34):

Ah okay.

David Michael Roberts (Dec 07 2024 at 21:34):

Regarding adapting Scholze's proof, the aleph_0-small extremally disconnected sets are the finite discrete spaces, and the claims in the first paragraph should be immediate, no? I don't quite see instantly why the unit of the adjunction should be an iso, but I assume that's not a problem at all.

David Michael Roberts (Dec 07 2024 at 21:36):

Finite discrete spaces are a full subcat of lambda-small extremally disconnected spaces for all (strong limit) lambda, so it should be easy.

David Michael Roberts (Dec 07 2024 at 21:37):

BTW, beth_{alpha+omega} is always strong limit, for all ordinals alpha, just so you have a concrete example!

Madeleine Birchfield (Dec 07 2024 at 21:39):

Thanks

David Michael Roberts (Dec 07 2024 at 21:39):

The point is that one _can_ use regular cardinals instead of strong limit, but the characterisations as different kinds of sheaves might not work.

Madeleine Birchfield (Dec 07 2024 at 21:40):

Unfortunately the example I was really interested in was the relationship between $\aleph_0$-condensed sets and light condensed sets, the latter which I believe are not from a strong limit cardinal.

David Michael Roberts (Dec 07 2024 at 21:41):

That is, freely passing between the sites of kappa-small profinite sets, kappa-small extremally disconnected spaces, and compact hausdorff spaces is not so cheap.

David Michael Roberts (Dec 07 2024 at 21:41):

One needs to use the 'real' definition, not a characterisation.

David Michael Roberts (Dec 07 2024 at 21:42):

Possibly using the opposite of countable boolean algebras as the site of definition for light condensed sets, and the opposite of finite boolean algebras for aleph_0-small condensed sets...

David Michael Roberts (Dec 07 2024 at 21:43):

Though the latter feels weird! Since those are all very specific sizes.
And yes, aleph_1 is not strong limit.

Madeleine Birchfield (Dec 07 2024 at 21:44):

isn't the opposite of finite boolean algebras just finite sets?

Madeleine Birchfield (Dec 07 2024 at 21:45):

no, I think it's the other way around - the opposite of finite sets is finite boolean algebras

David Michael Roberts (Dec 07 2024 at 21:47):

The equivalence is via the contravariant powerset functor, I believe.

Madeleine Birchfield (Dec 07 2024 at 21:48):

but the op functor in Cat is an involution so maybe the opposite of finite boolean algebras is finite sets

David Michael Roberts (Dec 07 2024 at 21:49):

In particular, Scholze uses only the countable boolean alg site, the profinite sets that are countable _sequential_ limits, and the profinite sets that are countable limits, not extremally disconnected stuff.

David Michael Roberts (Dec 07 2024 at 21:54):

From memory. It was a long time ago I watched the lectures!

Madeleine Birchfield (Dec 07 2024 at 21:57):

In this paper the author uses all countable limits, not just the sequential limits, in defining light profinite sets:

Definition 1. A light profinite set is a topological space that arises as a countable limit of finite discrete sets.

Madeleine Birchfield (Dec 07 2024 at 21:57):

but I've seen it defined using countable sequential limits, countable cofiltered limits, etc

Madeleine Birchfield (Dec 07 2024 at 22:00):

but every finite set is a light profinite set because it is the sequential limit of the constant sequential diagram involving the identity function on the finite set.

Madeleine Birchfield (Dec 07 2024 at 22:01):

so it might be possible to show that finite sets embed fully faithfully in light profinite sets and then maybe use that to show that $\aleph_0$-condensed sets embed fully faithfully in light condensed sets.

David Michael Roberts (Dec 07 2024 at 22:18):

Scholze proved the equivalence of the sequential/non-sequential in his lectures, I recall. Or at least stated it...

David Michael Roberts (Dec 07 2024 at 22:21):

https://youtu.be/_4G582SIo28?feature=shared at 14:40 is where the relevant material starts

David Michael Roberts (Dec 07 2024 at 22:23):

Finite sets manifestly embed fully faithfully in (light) profinite sets, as the constant sequential diagrams, for example.

David Michael Roberts (Dec 07 2024 at 22:26):

In fact at 25:00 he defines a light profinite set in such a way it includes finite sets. Namely via countable, not just countable infinite boolean algebras.

David Michael Roberts (Dec 07 2024 at 22:33):

The equivalence with sequential profinite is about 39:10

David Michael Roberts (Dec 07 2024 at 22:50):

The definition of light condensed set is around 1:00:40