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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...