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Jules Hedges (Sep 10 2020 at 10:30):I was complaining that there's no probability monad on FinSet, and @Jerry Swan suggested using the (finite nonempty) multiset monad on FinSet as a "histogram monad". I think this is a great idea, has anyone ever heard of / considered it?
Jules Hedges (Sep 10 2020 at 10:34):The important formal question is, is there a morphism of monads from the finite nonempty multiset monad on FinSet to the finite support probability monad on Set, taking each multiset to the probability distribution weighted by its frequencies? (Technically these are monads on different categories so I shouldn't talk about monad morphisms, but I think there ought to be an obvious way to make it precise)
Jules Hedges (Sep 10 2020 at 10:37):D'oh, there's no multiset monad on FinSet either, the multi-powerset of a finite set is infinite. I'll leave this here anyway just in case there's anything interesting to say about it
Nathanael Arkor (Sep 10 2020 at 10:40):You can have monad morphisms between monads on different categories: a morphism from to consists of a functor and a natural transformation satisfying a couple of equations (as defined in The formal theory of monads II, for instance.)
Jules Hedges (Sep 10 2020 at 10:47):My second question is still a reasonable question (I just don't have any motivation to ask it anymore) - is "weighting by frequency" a monad morphism from finite nonempty multiset to finite support probability, both as monads on Set
Matteo Capucci (he/him) (Sep 12 2020 at 09:36):This really reminds me of this paper by Fritz and Perrone: https://arxiv.org/abs/1712.05363
Matteo Capucci (he/him) (Sep 12 2020 at 09:38):They basically recover a probability monad as a colimit of finite-samples monads
Matteo Capucci (he/him) (Sep 12 2020 at 09:41):Jules Hedges said:
D'oh, there's no multiset monad on FinSet either, the multi-powerset of a finite set is infinite. I'll leave this here anyway just in case there's anything interesting to say about it
Maybe you can truncate it in some way, as to consider only histograms up to a certain height.
Matteo Capucci (he/him) (Sep 12 2020 at 09:42):Another way to go around this is to consider profinite sets instead of finite sets https://ncatlab.org/nlab/show/profinite+space
Intuitively, these are 'finitely presented' sets, or 'locally finite' sets.
Matteo Capucci (he/him) (Sep 12 2020 at 09:46):Intuitively speaking, I find them more sensible since they are 'locally finite', hence you don't lose many of the perks of a finite world, yet you get enough flexibility to speak about infinite things.
Then I guess the histogram monad works on these, since the multipowerset of a finite set looks very much like a limit of its 'truncated' finite parts.
Paolo Perrone (Sep 12 2020 at 17:44):In the paper to which Matteo is referring, the "finite" part is not quite a monad, but a graded monad, and the probability monad can be obtained by taking the colimit ("union without duplicates") of all those histograms.
Paolo Perrone (Sep 12 2020 at 17:46):I believe the same could be done for finite sets: the monad wouldn't exist since the colimit of all those histograms is an infinite set, however you still get a graded monad, and that's good for most purposes (for example, there are algebras which are pretty much the same).
Paolo Perrone (Sep 12 2020 at 17:46):So think Jules' idea is not at all wrong!
Paolo Perrone (Sep 12 2020 at 17:48):(Monads are useful, not sacred. If something doesn't form a monad it doesn't mean it's wrong, it just means that a monad is not the correct mathematical way to model that phenomenon. Thankfully in category theory we have a million different things that are "kind of like monads", and one of those variants usually works.)