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: theory: mathematics

Topic: Algebras for P(ℕ × -)


view this post on Zulip fosco (Feb 19 2025 at 15:03):

The functor PN=P(N×)P_\mathbb N = P(\mathbb N \times -) sends a set XX to the set P(N×X)P(\mathbb N \times X) of subsets of the product N×X\mathbb N\times X; is there a way to describe the PNP_\mathbb N -algebras? (As opposed to the much more studied coalgebras, which can be identified with transition systems)?

view this post on Zulip fosco (Feb 19 2025 at 15:04):

Note that PNP_\mathbb N is a monad; if I read correctly from the back of my envelope, the Eilenberg-Moore category is the category of suplattices with an action of the monoid N\mathbb N via sup-preserving morphisms.

Instead, I would like to know something about the bare endofunctor algebras

view this post on Zulip Martin Brandenburg (Feb 19 2025 at 16:41):

Just algebras for a functor have no axioms at all, so not sure what can be said then. PS: I assume you mean the covariant version here.

view this post on Zulip fosco (Feb 19 2025 at 17:05):

Sure, sure, the covariant powerset functor.

As soon as I go back at this problem, I'm going to try restricting to finite subsets (where there is, for example, an initial algebra).

view this post on Zulip Nathan Corbyn (Feb 20 2025 at 12:37):

Just for pure intuition: In case X is terminal I guess you can think of an algebra as an aggregation function over subsets of naturals. So maybe something like aggregation over labelled data?

view this post on Zulip fosco (Feb 20 2025 at 13:08):

Hmm, the algebra structure on the terminal set isn't really telling much about what the algebra map does... it's constant!

view this post on Zulip Nathan Corbyn (Feb 20 2025 at 17:49):

Ah sorry I put the wrong codomain in :sweat_smile:

view this post on Zulip Nathan Corbyn (Feb 20 2025 at 18:52):

Maybe something like an election? Think of X as a set of candidates and N as a countable set of voters. Each subset or N x X then behaves like a set of voter preferences (allowing for multiple votes—n votes for candidate x iff (n,x) in the subset). Then, the algebra selects the candidate based on the voting preferences.

view this post on Zulip Nathan Corbyn (Feb 20 2025 at 18:53):

(Of course, there can be some quite crooked elections as there’s no guarantee the chosen candidate appears in anyone’s preferences!)

view this post on Zulip fosco (Feb 20 2025 at 18:57):

A lovely idea, if it works! What's the candidate associated with the empty set of preferences?

view this post on Zulip Nathan Corbyn (Feb 20 2025 at 19:03):

A lot of abstentions!

view this post on Zulip Nathan Corbyn (Feb 20 2025 at 19:04):

I guess you could think of this as a ‘default’ choice

view this post on Zulip Nathan Corbyn (Feb 20 2025 at 19:04):

Maybe that doesn’t make much sense…

view this post on Zulip Nathan Corbyn (Feb 20 2025 at 19:05):

I have no idea what happens in any real world FPtP elections if no-one shows up to vote haha

view this post on Zulip Nathan Corbyn (Feb 20 2025 at 19:11):

This idea does at least have advantage that the notion of homomorphism is sensible: a function f : X -> Y relabels candidates and a homomorphism is a relabelling that respects election results

view this post on Zulip fosco (Feb 21 2025 at 14:23):

very good, I like this answer!