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.
The functor sends a set to the set of subsets of the product ; is there a way to describe the -algebras? (As opposed to the much more studied coalgebras, which can be identified with transition systems)?
Note that 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 via sup-preserving morphisms.
Instead, I would like to know something about the bare endofunctor algebras
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.
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).
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?
Hmm, the algebra structure on the terminal set isn't really telling much about what the algebra map does... it's constant!
Ah sorry I put the wrong codomain in :sweat_smile:
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.
(Of course, there can be some quite crooked elections as there’s no guarantee the chosen candidate appears in anyone’s preferences!)
A lovely idea, if it works! What's the candidate associated with the empty set of preferences?
A lot of abstentions!
I guess you could think of this as a ‘default’ choice
Maybe that doesn’t make much sense…
I have no idea what happens in any real world FPtP elections if no-one shows up to vote haha
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
very good, I like this answer!