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Ben Sprott (Sep 13 2020 at 04:26):I am wondering if the category of finite dimensional Hilbert Spaces and general linear maps factorizes a (probability) measures monad on some particular category of spaces. There may be some discussion as to the right choice of maps in the category defining FdHilb.
Andrew Hirsch (Sep 13 2020 at 09:51):It sounds like you're talking about the Giry monad? https://ncatlab.org/nlab/show/Giry+monad
Or did you have something more specific in mind?
Jules Hedges (Sep 13 2020 at 14:19):I don't understand this question, since the space of probability measures on a vector space doesn't itself carry a vector space structure (eg. you can't meaningfully multiply a probability measure by a scalar)
Jules Hedges (Sep 13 2020 at 14:19):There are quite a few known probability monads on various different kinds of spaces, but not vector spaces
Paolo Perrone (Sep 14 2020 at 01:52):I feel like a version of the Kantorovich monad which uses the second Wasserstein metric could do the trick. (But of course then you have to give up finite dimensions.)
Matteo Capucci (he/him) (Sep 14 2020 at 08:42):You could relax 'vector spaces' to 'cones' and then I guess you get at least and endofunctor (see Panangaden's work)