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: learning: questions

Topic: Higher arity algebra

John Onstead (Dec 02 2025 at 11:23):

The free monoid monad, or list monad, is a finitary monad. This is because it sends a set $S$ to the set of finite strings of elements of $S$. Recently, I've been thinking more about how to extend this to infinitary cases, such as an operation that sends $S$ to the set of countable or even uncountable strings of elements of $S$. This operation is functorial; if we denote $F$ as the new functor and $L$ as the list functor, then $F(S) = S^{\mathbb{N}} + L(S)$ or $F(S) = S^{\mathbb{N}} + S^{\mathbb{R}} + L(S)$.

John Onstead (Dec 02 2025 at 11:24):

The basic yes/no question here is: are these functors monads?
I'm not sure, since it seems you run into issues even with the countable case. For instance, in trying to define the monad "collapse" operation, we have to be able to concatenate two or more infinite countable sequences. This seemingly leads to a new kind of sequence where there's a "dot dot dot" in the middle, which might require some ordinal analysis, IE, where $\omega + \omega$ is a thing. But if these functors aren't monads, then is it possible to define a notion of "monoid generated under infinite concatenations" to extend the idea that free monoids are "monoids generated under finite concatenations"?

Kevin Carlson (Dec 02 2025 at 18:10):

Yes, I don't have any detailed background knowledge here, but it seems the most natural way to get a monad would be to set $F(S)=\sum_{\alpha<\omega_1}S^\alpha$ to be the set of all lists of elements of $S$ indexed by (in this case) a countable well-order. Then if you have for each $\beta<\alpha$ a string $a_\beta:\alpha_\beta\to S,$ you can concatenate these to a string $\mu((a_\beta)):\sum_{\beta<\alpha} \alpha_\beta \to S.$ Thus for instance if you have an $\omega$-sequence of $\omega$-sequences, you concatenate them to an $\omega^2$-sequence, which seems reasonable. It seems clear to me this is a monad although I wouldn't be amazed if I'm missing something.

Kevin Carlson (Dec 02 2025 at 18:12):

Then algebras for this monad would have a way of multiplying every countable well-order of elements to a single element, consist with infinitary concatentations. I don't know whether there are examples of these things or whether anybody cares about them.

Kevin Carlson (Dec 02 2025 at 18:28):

I should also point out that there is a monad structure on $S^{\mathbb{N}}$ itself, namely the reader monad, which I think is the only monad structure on this functor. But its multiplication evaluates on the diagonal so is really unrelated to list concatenation (and indeed every set determines its own reader monad in the same way.) In fact, I found a proof by Daniel Schepler here that there is no monad structure on your first proposed $F$ extending the list monad: https://math.stackexchange.com/questions/4784631/monad-of-possibly-infinite-lists

John Onstead (Dec 02 2025 at 23:19):

Kevin Carlson said:

Yes, I don't have any detailed background knowledge here, but it seems the most natural way to get a monad would be to set $F(S)=\sum_{\alpha<\omega_1}S^\alpha$ to be the set of all lists of elements of $S$ indexed by (in this case) a countable well-order.

Thanks! It seems reasonable enough.

Kevin Carlson said:

I don't know whether there are examples of these things or whether anybody cares about them.

Maybe one of them would be the ordinals under ordinal addition? I know they form a monoid, but generally you need extra structure on top of the monoid structure to define "transfinite recursion" or "taking a limit". Perhaps with this kind of algebraic structure that's no longer necessary?

John Onstead (Dec 02 2025 at 23:23):

Kevin Carlson said:

In fact, I found a proof by Daniel Schepler here that there is no monad structure on your first proposed $F$ extending the list monad: https://math.stackexchange.com/questions/4784631/monad-of-possibly-infinite-lists

Ah, that's interesting!
It's also interesting to see someone asking a similar question, though it seems they were thinking of the "collapse" operation as collapsing two infinite lists into the first list, essentially "deleting" the second list, rather than using ordinal indexing.

Kevin Carlson (Dec 03 2025 at 04:54):

Yeah, it seems intuitive that that one isn’t a monad, very unnatural.

Mike Shulman (Dec 03 2025 at 06:22):

You might be interested in the notion of "series monoid" from https://arxiv.org/abs/1704.08787.

Nayan Rajesh (Dec 03 2025 at 11:31):

Partially additive monoids are infinitary versions of commutative (partial) monoids. These are shown to be algebras for a monad on Pfn in https://www.numdam.org/item/CTGDC_1985__26_3_221_0.pdf. The endofunctor for this monad is almost like the one you've described - the object component sends a set X to $\text{Pfn}(\mathbb{N}, X)/\sim$ instead of $\text{Set}(\mathbb{N}, X) + L(X)$. I guess the price to pay to get a well-behaved monad is partiality.

John Onstead (Dec 03 2025 at 19:13):

Mike Shulman said:

You might be interested in the notion of "series monoid" from https://arxiv.org/abs/1704.08787.

Nayan Rajesh said:

Partially additive monoids are infinitary versions of commutative (partial) monoids. These are shown to be algebras for a monad on Pfn in https://www.numdam.org/item/CTGDC_1985__26_3_221_0.pdf. The endofunctor for this monad is almost like the one you've described - the object component sends a set X to $\text{Pfn}(\mathbb{N}, X)/\sim$ instead of $\text{Set}(\mathbb{N}, X) + L(X)$. I guess the price to pay to get a well-behaved monad is partiality.

Very interesting, I will have to take a look at both of these, thanks!