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Stream: theory: mathematics

Topic: Algebras for P(ℕ × -)

fosco (Feb 19 2025 at 15:03):

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

fosco (Feb 19 2025 at 15:04):

Note that $P_\mathbb N$ 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 $\mathbb N$ via sup-preserving morphisms.

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

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.

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).

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?

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!

Nathan Corbyn (Feb 20 2025 at 17:49):

Ah sorry I put the wrong codomain in :sweat_smile:

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.

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!)

fosco (Feb 20 2025 at 18:57):

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

Nathan Corbyn (Feb 20 2025 at 19:03):

A lot of abstentions!

Nathan Corbyn (Feb 20 2025 at 19:04):

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

Nathan Corbyn (Feb 20 2025 at 19:04):

Maybe that doesn’t make much sense…

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

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

fosco (Feb 21 2025 at 14:23):

very good, I like this answer!