Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: community: our work

Topic: Chris Grossack (they/them)

Chris Grossack (they/them) (Jul 16 2023 at 20:22):

I don't want to say what I'm working on JUST yet (though lots of people who were at CT2023 already know), but here's an interesting question along the way

Chris Grossack (they/them) (Jul 16 2023 at 20:32):

Every finite product category is automatically symmetric monoidal, where the cartesian product becomes a tensor product.

Now say we have a (strong) symmetric monoidal functor $F$ between two finite product categories. Is $F$ is automatically a finite product functor? That is, if we have a structure map $\alpha_{A,B} : F(A \times B) \overset{\sim}{\to} FA \times FB$ is $\alpha_{A,B}$ automatically the map $\langle F \pi_A, F \pi_B \rangle$? I'm going back and forth on whether I believe it or not, but I'm not seeing it immediately.

Oh, and be kind if this is obvious, haha. I haven't spent much time trying to prove it at all, since I thought it would be fun to start a conversation here ^_^

Chris Grossack (they/them) (Jul 16 2023 at 20:53):

Ah, it looks like the answer is "yes". See the baby example here:
https://mathoverflow.net/questions/313600/explicit-left-adjoint-to-forgetful-functor-from-cartesian-to-symmetric-monoidal

John Baez (Jul 16 2023 at 21:54):

Sounds right. So "being cartesian" is just a property of a symmetric monoidal category, not a structure - the forgetful 2-functor from cartesian categories to symmetric monoidal categories is 1-faithful.

John Baez (Jul 16 2023 at 21:55):

So great to see you've joined the #practice: our work team!

Chris Grossack (they/them) (Oct 05 2023 at 17:40):

Here's a cute question that's come up for me a handful of times. My guess is that there's no good answer, but it would be REALLY convenient if someone does know a good answer (even a good answer that only works in certain circumstances would be interesting)

Chris Grossack (they/them) (Oct 05 2023 at 17:41):

Fix a monad $T$ on $\mathcal{C}$. Is there a way to recognize when an arrow in the kleisli category $x \to Ty$ is a monomorphism, ideally only referencing facts about $\mathcal{C}$?

Chris Grossack (they/them) (Oct 05 2023 at 17:44):

In case we work with the familiar group monad on $\mathsf{Set}$, this would give a categorical way of understanding embeddings between free groups. This is already a nontrivial problem, which makes me doubt any kind of simple answer... But maybe there's a less-than-simple answer, haha

David Egolf (Oct 05 2023 at 18:12):

I'm just starting to learn about monads, but I would be curious about the (hopefully?) simpler case where one considers an arrow $x \to Tx$.

dusko (Oct 06 2023 at 00:56):

Chris Grossack (they/them) said:

Fix a monad $T$ on $\mathcal{C}$. Is there a way to recognize when an arrow in the kleisli category $x \to Ty$ is a monomorphism, ideally only referencing facts about $\mathcal{C}$?

the forgetful functor to $\cal C$ is a right adjoint, so it preserves pullbacks, so it also preserves the monics. so every monomorphism between free algebras must be a monomorphism between the carriers. the fact that it is sufficient, i.e. that a monomorphism between the carriers must also be a monomorphism between the algebras as soon as it is an algebra homomorphism follows from the definition of algebra homomorphisms in the category of all algebras (usually in the eilenberg moore form) and the fact that the category of free algebras (say in the kleisli form) is fully embedded.

the difficult parts of the question are what is simple and what is nontrivial. it usually helps to abstract away the parts that cannot play a role in the answer, e.g. the presentation of free algebras in the kleisli form. the implementation details often hide general answers.

dusko (Oct 06 2023 at 01:01):

ps note that the kleisli morphism $x\to Ty$ is not a morphism between the underlying algebra carriers. it is its lifting $Tx\to Ty$. so $x\to Ty$ is monic if and only if the underlying function of its lifting $Tx\to Ty$ is a monic in $\cal C$...

which is how we would present it to define its composition with whichever two kleisli morphisms would be testing whether this is a monic anyway. so i guess we could answer this question within the kleisli category itself, if we wanted to ignore my prosletizing about abstracting away the presentation :))

Chris Grossack (they/them) (Oct 06 2023 at 01:41):

@dusko -- Thanks for looking into this! Unfortunately the presentation is the important thing for my application. I have an adjunction $A \dashv C$ between graphs and groups, and I want a combinatorial condition on graphs $\Gamma$ and $\Delta$ that is satisfied if and only if we have an embedding $A \Gamma \to A \Delta$ at the level of groups.

We know that all maps $A \Gamma \to A \Delta$ are classified by graph homs $\Gamma \to CA \Delta$, which brings us to the question of recognizing monos in the kleisli category.

dusko (Oct 06 2023 at 05:33):

if we have the adjunction that induces the monad $T=CA$, isn't it easier to look at the category of free $T$-algebras as the category where the objects are graphs and a morphism from $\Gamma$ to $\Delta$ is the group homomorphism $A\Gamma\to A\Delta$?

Chris Grossack (they/them) (Oct 06 2023 at 06:01):

Sure, but that's equivalent to the kleisli category, haha. And it doesn't help us find a graph theoretic condition for monomorphism-ness, at least as far as I can see. The reason I find the kleisli category appealing is because everything is happening in the category of graphs. We've just changed the target from $\Delta$ to $CA\Delta$, but that's ok because it's still just some graph built from the data in $\Delta$.

dusko (Oct 06 2023 at 21:26):

Chris Grossack (they/them) said:

Sure, but that's equivalent to the kleisli category, haha. And it doesn't help us find a graph theoretic condition for monomorphism-ness, at least as far as I can see.

i imagine it must cost some effort to not see the answer to your original question from the 3 versions that i just gave:
1) forgetful functor from to $\cal C$ preserves and reflects the kleisli monomorphisms
2) direct test in the kleisli itself: $f\circ x=f\circ y\implies x=y$ means that you lift $f$ and then compose
3) if you are given a resolution of your monad, then the kleisli category is isomorphic with the category of morphisms between the images along the left adjoint from the resolution.
in all cases, your kleisli morpism is a monic if its transpose in the category of groups is injective.

now you say that you want a "graph theoretic condition", but what is the graph theoretic property of $\Gamma\to CA\Delta$ corresponding to a group embedding $A\Gamma\hookrightarrow A\Delta$ depends on what $C$ and $A$ are, which you didn't give.

Chris Grossack (they/them) (Oct 06 2023 at 22:12):

@dusko Thanks for the help, then! You're probably right that a satisfying answer to my question will likely involve more specifics of $A \dashv C$ than I gave.

Also, for future reference, your last post comes off as extremely rude -- particularly the tone of your first sentence. I imagine most people would prefer not to be talked to that way ^_^

Morgan Rogers (he/him) (Oct 07 2023 at 09:08):

I agree with @Chris Grossack (they/them) on that last point, especially since there's surely plenty more to be said about monomorphisms in Kleisli categories (both generally and in Chris' specific case).

@Chris Grossack (they/them) how much do you know about the interaction of your monad with monomorphisms? If $T$ (or $A$) preserves them, your life should be a lot easier. How explicitly can you compute these functors? In my experience, it should be easy enough to figure out the answer in the restricted case of finite graphs where you can leverage finiteness, and then to figure out how the result extends (or fails to extend) to the infinite case.

Notification Bot (Oct 14 2023 at 11:35):

9 messages were moved from this topic to #general: off-topic > don't blatantly insult people by Morgan Rogers (he/him).

Chris Grossack (they/them) (Nov 08 2023 at 20:14):

Not categorical, but extremely cool: It turns out that 37 is the "median second prime factor".

By this I mean if you look at the second smallest prime factor of a large number $N$, it will be $< 37$ about half the time, and $> 37$ about half the time!

Chris Grossack (they/them) (Nov 08 2023 at 20:15):

I just wrote up a blog post about this: https://grossack.site/2023/11/08/37-median

It turns out to not be very hard to prove ^_^

Ralph Sarkis (Nov 08 2023 at 20:51):

Great read Chris. I had to think a little bit about why you can multiply the probabilities of numbers having (or not) a certain factor, maybe it is worth explaining why they are "independent".

Matteo Capucci (he/him) (Nov 10 2023 at 11:25):

Super interesting Chris!

Matteo Capucci (he/him) (Nov 10 2023 at 11:25):

Analytic number theory is wild

Nathanael Arkor (Nov 13 2023 at 17:35):

@Chris Grossack (they/them): not sure if you had spotted this already, but your post has become pretty popular on Hacker News (https://news.ycombinator.com/item?id=38242946).

Chris Grossack (they/them) (Nov 13 2023 at 17:37):

@Nathanael Arkor Some of my other friends told me this yesterday! Apparently I was number 1 for a while?? It's weird that my blog went from 10-20 views per day (which is pretty normal) up to 10 THOUSAND views yesterday. I'm glad people are enjoying it ^_^

Chris Grossack (they/them) (Nov 13 2023 at 17:40):

I'm almost done with another post, but it's much more technical (it's about 2-categories and the problems they solve). I'm trying to decide if I want to write another elementary blog post next instead, or if I want to finish this one.

John Baez (Nov 13 2023 at 22:25):

Congrats, Chris. Hacker News is a big amplifier, so you are now famous - and rightfully so.

Recently I tried to visit Greg Egan's website and got this message:

Bandwidth Limit Exceeded

The server is temporarily unable to service your request due to the site owner reaching his/her bandwidth limit. Please try again later.

That seemed odd. He's occasionally had trouble with his internet service provider, so I emailed him, and he said

A page of mine with a lot of big GIF files briefly got linked on the site Hacker News (where people post links to things they like, and other people can vote on them, and the top 100 or so are listed on the site's main page) ... and enough people viewed the page to exceed my web site's 500Gb data transfer limit for the month. So my site will be unavailable until December.

Morgan Rogers (he/him) (Nov 14 2023 at 08:57):

Oof so Hacker news can either make you famous or break your stuff...

Chris Grossack (they/them) (Jul 04 2024 at 06:08):

It's finally done!

Have a 3 part series on the topological topos

First, let's talk about some fundamentals. How do we think of objects in the topos? How do we compute with them? How do they relate to objects in "the real world"?
https://grossack.site/2024/07/03/life-in-johnstones-topological-topos

Then, since post 1 is loooong, a little palate cleanser. It's a short post about how we can use the topological topos to more effectively study algebraic structures equipped with a topology.
https://grossack.site/2024/07/03/topological-topos-2-algebras

Lastly, another long one. Let's talk seriously about the internal logic of the topological topos. What's true and why? And what does this truth mean in the real world? This one isn't for the faint of heart!
https://grossack.site/2024/07/03/topological-topos-3-bonus-axioms

Thanks for all the encouragement, everyone! This was easily the most effort I've ever put into... maybe any math project I've ever done, haha. But I think it turned out really well ^_^.

If you have comments, I would love to hear what you think!

Emilio Minichiello (Jul 04 2024 at 11:42):

Started reading the first post and will come back to it later, just wanted to say its really excellently written and detailed! I really like the care you took to motivate and flesh out the concepts.

Nathan Corbyn (Jul 04 2024 at 12:42):

These posts are so lovely to read and very instructive--thank you Chris!

Madeleine Birchfield (Jul 05 2024 at 13:18):

"Dependent Choice implies Countable Choice, which itself implies Weak Countable Choice. But WCC implies that the dedekind reals and the cauchy reals agree."

You linked to the wrong weak version of countable choice in this sentence in the third blog post. It is $\mathrm{AC}_{\mathbb{N}, 2}$ rather than Bridges and Schuster's weak countable choice ($\mathrm{WCC}$) which proves that the Dedekind and Cauchy reals coincide. As far as I'm aware, whether $\mathrm{WCC}$ proves that the Dedekind and Cauchy reals coincide with each other is still an open question.

Chris Grossack (they/them) (Jul 05 2024 at 15:57):

Madeleine Birchfield said:

"Dependent Choice implies Countable Choice, which itself implies Weak Countable Choice. But WCC implies that the dedekind reals and the cauchy reals agree."

You linked to the wrong weak version of countable choice in this sentence in the third blog post. It is $\mathrm{AC}_{\mathbb{N}, 2}$ rather than Bridges and Schuster's weak countable choice ($\mathrm{WCC}$) which proves that the Dedekind and Cauchy reals coincide. As far as I'm aware, whether $\mathrm{WCC}$ proves that the Dedekind and Cauchy reals coincide with each other is still an open question.

Oh wow, I had no idea there were TWO things called WCC, both of which are implied by either CC or LEM, haha. Thanks for the correction, I just fixed it!

Morgan Rogers (he/him) (Jul 05 2024 at 17:07):

"It’s possible that the presence of nonsequential spaces will render a mono in Seq no longer monic in Top!"
Does that actually happen? ;)

Chris Grossack (they/them) (Jul 05 2024 at 18:08):

Morgan Rogers (he/him) said:

"It’s possible that the presence of nonsequential spaces will render a mono in Seq no longer monic in Top!"
Does that actually happen? ;)

Hm... Are you saying it doesn't? :P. I would believe it... the monos in top are the continuous injections, and I guess monos in seq are ALSO continuous injections, since the forgetful functor to set preserves limits.

If you agree, I can go ahead and update that

Chris Grossack (they/them) (Jul 05 2024 at 18:27):

ooh, I think I see how to characterize Seq inside $\mathcal{T}$ type-theoretically. I'll figure that out sometime soon and add it to the post

Madeleine Birchfield (Jul 06 2024 at 00:08):

"Actually, I think I remember reading somewhere that the analytic omniscience principles are always statements about the cauchy reals. The reason countable choice makes them properties of the dedekind reals is because under CC the dedekind and cauchy reals agree."

There are multiple possible notions of analytic omniscience principles, since one can construct multiple notions of real numbers in a topos which in general are not equivalent to each other. Examples include the Cauchy real numbers $\mathbb{R}_C$, the Escardo-Simpson real numbers $\mathbb{R}_E$ which are the initial Cauchy complete Archimedean ordered field, the Dedekind real numbers $\mathbb{R}_D$, etc.

Each notion of real number has its own analytic omniscience principles - it makes sense to talk about the analytic LPO/WLPO/LLPO/Markov's principle for the Cauchy real numbers, Escardo-Simpson real numbers, Dedekind real numbers etc, and they are not equivalent to each other unless some axiom collapses the entire hierarchy of constructive real numbers like excluded middle or countable choice. For example, any topos which satisfies Brouwer's continuity principle for Dedekind real numbers contradicts analytic LPO for Dedekind real numbers, but is consistent with the analytic LPO for Cauchy real numbers so long as the Dedekind real numbers do not coincide with the Cauchy real numbers.

The relation between the omniscience principles is as follows: Suppose you have an Archimedean ordered field of real numbers $\mathbb{R}_1$ - if an analytic omniscience principle for $\mathbb{R}_1$ holds than the corresponding analytic omniscience principle holds for any ordered subfield $\mathbb{R}_2 \subseteq \mathbb{R}_1$; however, the converse only holds if and only if $\mathbb{R}_2$ coincides with $\mathbb{R}_1$.

Chris Grossack (they/them) (Jul 06 2024 at 01:15):

@Madeleine Birchfield Thanks! With that footnote I was saying that I remember reading somewhere that LPO/etc are equivalent to cauchy-analytic LPO/etc, and the reason we get dedekind-analytic LPO/etc is because of CC collapsing the hierarchy.

But it sounds like you're saying I'm misremembering, and that the nonanalytic statements of LPO/etc are not equivalent to any of the analytic versions in general. Is that right?

Madeleine Birchfield (Jul 06 2024 at 01:32):

Oh. Well, the princples of omniscience for natural numbers are equivalent to the Cauchy-analytic principles of omniscience (I think), and yes one can get the Dedekind-analytic principles of omniscience from the Cauchy-analytic principles of omniscience if one collapses the hierarchy. However, in a general topos, one can have Dedekind-analytic principles of omniscience hold without collapsing the hierarchy - analytic Markov's principle for Dedekind real numbers still holds for the topos of cohesive sets in Mike Shulman's real-cohesive HoTT even though the Dedekind and Cauchy real numbers do not coincide there.

This is why I don't think that "the analytic omniscience principles are always statements about the cauchy reals." is correct.

Morgan Rogers (he/him) (Jul 06 2024 at 12:33):

Chris Grossack (they/them) said:

Morgan Rogers (he/him) said:

"It’s possible that the presence of nonsequential spaces will render a mono in Seq no longer monic in Top!"
Does that actually happen? ;)

Hm... Are you saying it doesn't? :P. I would believe it... the monos in top are the continuous injections, and I guess monos in seq are ALSO continuous injections, since the forgetful functor to set preserves limits.

If you agree, I can go ahead and update that

I was being lazy. I haven't checked for myself, but while it's true that preservation of monos isn't automatic / a corollary of the properties of functors derived before that point, I wouldn't be surprised if they happen to be preserved here.

Chris Grossack (they/them) (Jul 06 2024 at 19:24):

Madeleine Birchfield said:

Oh. Well, the princples of omniscience for natural numbers are equivalent to the Cauchy-analytic principles of omniscience (I think), and yes one can get the Dedekind-analytic principles of omniscience from the Cauchy-analytic principles of omniscience if one collapses the hierarchy. However, in a general topos, one can have Dedekind-analytic principles of omniscience hold without collapsing the hierarchy - analytic Markov's principle for Dedekind real numbers still holds for the topos of cohesive sets in Mike Shulman's real-cohesive HoTT even though the Dedekind and Cauchy real numbers do not coincide there.

This is why I don't think that "the analytic omniscience principles are always statements about the cauchy reals." is correct.

Great! Thanks for taking the time to say all this. Do you know if there's a reference somewhere for natural-omniscience principles being equivalent to cauchy-omniscience principles? I'll make this correction in the blog post rn, but when time comes to write the paper I'll want to put something in the bibliography for it

Chris Grossack (they/them) (Jul 06 2024 at 19:26):

Morgan Rogers (he/him) said:

Chris Grossack (they/them) said:

Morgan Rogers (he/him) said:

"It’s possible that the presence of nonsequential spaces will render a mono in Seq no longer monic in Top!"
Does that actually happen? ;)

Hm... Are you saying it doesn't? :P. I would believe it... the monos in top are the continuous injections, and I guess monos in seq are ALSO continuous injections, since the forgetful functor to set preserves limits.

If you agree, I can go ahead and update that

I was being lazy. I haven't checked for myself, but while it's true that preservation of monos isn't automatic / a corollary of the properties of functors derived before that point, I wouldn't be surprised if they happen to be preserved here.

Yeah, I'm starting to also think they happen to be preserved here... In both cases they're continuous injections. I'll go ahead and update that section soon too!

Madeleine Birchfield (Jul 06 2024 at 19:41):

Chris Grossack (they/them) said:

Great! Thanks for taking the time to say all this. Do you know if there's a reference somewhere for natural-omniscience principles being equivalent to cauchy-omniscience principles? I'll make this correction in the blog post rn, but when time comes to write the paper I'll want to put something in the bibliography for it

I don't know of any references myself since I got my info from the [[principle of omniscience]] article on the nLab, but perhaps @Toby Bartels might know of references, since he was the one who put the statements on the nLab's back in 2012 saying that the omniscience principles for natural numbers are equivalent to the analytic principles of omniscience for the Cauchy real numbers.

Madeleine Birchfield (Jul 06 2024 at 20:20):

(Of course it can be that the nLab is just plain wrong and the principles are not equivalent to each other).

Madeleine Birchfield (Jul 06 2024 at 20:29):

I asked on Mathoverflow whether the statements are equivalent to each other:

https://mathoverflow.net/questions/474578/equivalence-of-omniscience-principles-for-natural-numbers-and-analytic-omniscien

Chris Grossack (they/them) (Jul 06 2024 at 22:47):

Oh cool, thanks @Madeleine Birchfield!

Madeleine Birchfield (Jul 06 2024 at 23:08):

Along with David Madore's proof of both implications for the $\mathrm{LPO}_\mathbb{N}$ in the MathOverflow question, I also have a different proof from a few days ago that the $\mathrm{LPO}_\mathbb{N}$ implies the analytic LPO for Cauchy real numbers:

https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/LPO.20and.20sigma-frame.20structure.20on.20booleans/near/446708929

Namely that $\mathrm{LPO}_\mathbb{N}$ implies the set of booleans is the initial sigma-frame, and thus a sub-sigma-frame of the frame of truth values. Then one can use decidable Dedekind cuts to construct a field of real numbers which I will denote $\mathbb{R}_\mathrm{bool}$, and since the cuts are decidable, the order relation will be decidable as well (analytic LPO for $\mathbb{R}_\mathrm{bool}$), which collapses the hierarchy between $\mathbb{R}_\mathrm{bool}$ and the Cauchy real numbers implying the analytic LPO for Cauchy real numbers. You might have to use some proofs in section 11.2 and 11.4 of the HoTT book to fill in some gaps. Note that since not all Dedekind cuts are decidable we still have the Dedekind reals as different from the Cauchy reals.

Madeleine Birchfield (Jul 07 2024 at 03:50):

For an actual reference, @Mike Shulman 's Brouwer's fixed point theorem in real-cohesive homotopy type theory asserts that "it is well known that" the principles of omniscience for natural numbers are equivalent to the Cauchy-analytic principles of omniscience (see the paragraph after corollary 8.31 on page 60).

However, fair warning, at least in the latest arXiv preprint, Mike got the Cauchy-analytic LPO confused with the Cauchy-analytic WLPO in that paragraph, and similarly the Dedekind-analytic LPO confused with the Dedekind-analytic WLPO a few paragraphs down, something that went undetected until a few months ago. The analytic LPO for both Dedekind and Cauchy reals should be statements about the apartness being decidable rather than equality being decidable. I don't know if the errors have been corrected in the published paper yet.

Graham Manuell (Jul 07 2024 at 21:31):

I haven't had time to read these properly, but there is one place with confusingly nonstandard notation and terminology. The way below relation $\ll$ usally refers to something else, related to local compactness. The normal name for what you want is the rather below relation $\prec$.

As for the prime ideal theorem, I'm pretty sure it's about the internal locales, as you suspected.

Chris Grossack (they/them) (Jul 07 2024 at 22:44):

@Graham Manuell I was hoping you'd chime in!

I wasn't sure how to typeset the "rather below" relation, since Stone Spaces calls it "well inside" and uses an upside down $\leq$ that I couldn't figure out how to typeset on my blog (I know there are ways to flip a symbol upside down in latex, but I don't know if they work with mathjax). I ended up using $\ll$ but I'll switch it to $\prec$ if that's the standard notation.

Chris Grossack (they/them) (Jul 07 2024 at 22:46):

I'm glad to hear you also think the prime ideal theorem doesn't work because of internal locales. There was a part of me worried it didn't work because of a problem with my proof :P

Graham Manuell (Jul 08 2024 at 16:40):

Incidentally, do you happen to know a nice theory that the topological topos classifies?

Chris Grossack (they/them) (Jul 08 2024 at 18:55):

Good question! I haven't thought about it at all, but I agree that would fit in the paper nicely, and it's obviously interesting to think about... We know $\mathcal{T}$ is the topos of sheaves on $M = \text{End}(\mathbb{N}_\infty)$ with the canonical topology, so there's a kind of flippant answer available.

I'm pretty sure the topos of presheaves on $M$ classifies "flat $M$-sets", where an $M$-set $B$ is flat whenever $- \otimes_M B : M\text{-set} \to \mathsf{Set}$ preserves finite limits. But we have a topology on this site, which amounts to adding axioms to the theory of flat $M$-sets.

Since the topology admits a concrete description, it shouldn't be TOO hard to read off the axioms we need to add (and my guess is that someone like @Morgan Rogers (he/him) could do it fairly quickly?), but I'm not sure if there will be any cute description at the end of the day.

Morgan Rogers (he/him) (Jul 08 2024 at 20:10):

Rather than giving you a logical version, it's easiest to say that the models which respect the Grothendieck topology must send a covering family to a jointly epimorphic family. At first glance I would expect that to mean that the sequences must converge to actual values (identified with constant sequences).

Chris Grossack (they/them) (Jul 08 2024 at 20:46):

I think there's a level slip happening here. For instance, there's only ONE $\mathsf{Set}$-valued point of $\mathcal{T}$. Its inverse image part is the underlying set, which I think is tensoring with $\{\star\}$, the trivial $M$-set? I'm not sure, though.

Section 5 of Johnstone's paper uses this to show there's only one $\mathcal{E}$-point in any topos $\mathcal{E}$ with enough points, but that's the section of the paper I'm least familiar with

Morgan Rogers (he/him) (Jul 09 2024 at 05:34):

If what I said is accurate (which I would need to check more carefully: it's not clear to me yet if the required presence of "final subsequences" in a covering family has precisely the effect I said) then I would expect a model to be forced to look pretty much like $\N$, so it's not too surprising that there aren't many of them, but where the receiving topos having enough points comes in is more mysterious. Maybe I should check the paper :stuck_out_tongue_wink:

Morgan Rogers (he/him) (Jul 09 2024 at 06:51):

(Worth noting that the underlying set functor from End(N_infty) is conservative so can't factor through any proper subtopos of that one, so something must be off)

Chris Grossack (they/them) (Jul 11 2024 at 00:40):

Two updates back-to-back! My first paper, The Right Angled Artin Group Functor as a Categorical Embedding just got accepted in AGT! Here I use the language of comonadic descent in order to prove some cute structural theorems in geometric group theory. You can find a (very slightly out of date) preprint here, which I'll update to the revised and accepted version soon ^_^

David Corfield (Jul 11 2024 at 10:02):

Chris Grossack (they/them) said:

Have a 3 part series on the topological topos

Nice series, Chris! Where you discuss condensed mathematics in the final part, you may be interested in this thread which has Peter Scholze explain:

In some sense condensed sets are the version of Johnstone’s topos that replaces convergence along sequences by convergence along ultrafilters.

And further down a comparison with the bornological topos.

Chris Grossack (they/them) (Jul 11 2024 at 17:46):

@David Corfield Thanks! This was a super interesting read. It also clears up some questions I had about exactly how condensed sets are "better behaved" than objects of $\mathcal{T}$. The presence of projectives is a great explanation for why you might prefer condensed sets, and there's a bunch of other fun stuff here too.

David Corfield (Jul 12 2024 at 06:34):

Something I was hoping to see emerge from these discussions was some kind of universal characterization of pyknotic/condensed sets. Scholze here writes:

All of these questions sidestep the question of why condensed sets are not cohesive over sets, when cohesion is meant to model "toposes of spaces" and condensed sets are meant to be "the topos of spaces",

which would suggest some universal structure.

There are plenty of category-theoretic leads to the world of ultra-things.

Nathanael Arkor (Jul 12 2024 at 11:23):

David Corfield said:

Something I was hoping to see emerge from these discussions was some kind of universal characterization of pyknotic/condensed sets.

Are you looking for something distinct from Campbell's claimed characterisation?

David Corfield (Jul 12 2024 at 12:27):

Thanks @Nathanael Arkor, I had seen that but forgotten it. Yes, I think I was hoping for something more clearly demonstrating its value.

I remember a very one-sided conversation I had with myself at the n-Category Cafe where I hoped to glimpse some kind of connection to codensity.

Chris Grossack (they/them) (Jul 12 2024 at 16:51):

@David Corfield, isn't there a comment on the ncatcafe somewhere where someone (Mike Shulman, maybe?) talks about how cohesive topoi really model "spaces" where path-connectedness is taken as the basic notion? From this perspective it's not surprising that $\mathcal{T}$ and $\mathsf{Cond}$ aren't cohesive, since they're not modeling path connected spaces (spaces built on the interval), but instead are modeling spaces with a notion of convergence (either sequential convergence, for $\mathcal{T}$, or ultrafilter convergence for $\mathsf{Cond}$).

Chris Grossack (they/them) (Jul 12 2024 at 16:53):

Indeed, an object of a topos can be thought of as a (nice) gluing of objects in the site. So for $\mathcal{T}$ you get a bunch of copies of $\mathbb{N}_\infty$ glued together and in $\mathsf{Cond}$ youget a bunch of extremely disconnected spaces glued together. In both cases it's believable that the spaces you get wouldn't always play nicely with path-connectedness.

David Corfield (Jul 13 2024 at 06:51):

@Chris Grossack (they/them) There's the idea that standard cohesion, i.e. over sets or $\infty$-groupoids, captures the notion of a thickened point, (Remark 3.2 of cohesive (infinity,1)-topos):

a cohesive $(\infty,1)$-topos $\mathbf{H}$ is, when itself regarded as a little topos, a generalized space, a thickened point. We may think of it as the standard point equipped with a cohesive neighbourhood.

Then we can be interested in relative cohesion, i.e., thickening relative to a certain base, such as to condensed sets, cf. condensed local contractibility.