It was recently brought to my attention that the Bag monad, also known as the MultiSet monad, is not polynomial on Set, but is Polynomial on the category of Groupoids, 3.10 Examples. I then started thinking about the category FinHilb, of finitedimensional Hilbert spaces, and the fact that it has a dagger. This allows for the fact that the subcategory of just unitary isomorphisms as morphisms, (ie the core of Hilb) is a groupoid. I believe this means that the core of Hilb supports the polynomial functor of the Bag monad. Can someone show the form of this polynomial monad, and perhaps talk a bit about this monad?
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4$\begingroup$ The trouble is that monads on groupoids are pretty uninteresting, since they are forced to be isomorphic to the identity monad. $\endgroup$ – Todd Trimble♦ Jul 4 '18 at 17:01

1$\begingroup$ it's polynomial in a 2dimensional sense; you'd need a 2category of Hilbert spaces. $\endgroup$ – Eduardo Pareja Tobes Jul 4 '18 at 17:07

$\begingroup$ Hi Todd, does that mean the trivial bag monad would be, like an empty bag? It could have all the natural Isos $TT \rightarrow T$, $1 \rightarrow T$, but the bags are just empty? $\endgroup$ – Ben Sprott Jul 4 '18 at 17:19

1$\begingroup$ I'm having difficulty making sense of the question. In ordinary language, I equate 'bag' to 'multiset' or to an element of a free commutative monoid; the analogous construct for a general monad would be a morphism $1 \to X$ of the Kleisli category. So a "trivial bag structure" of type $X$ I would have to translate as a morphism $1 \to X$ in the Kleisli category of the identity monad, which is just an element of $X$. But all of this is terribly unenlightening. I'd think you'd be much better off pursuing what Eduardo had in mind. $\endgroup$ – Todd Trimble♦ Jul 5 '18 at 0:25
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