# Multiset or Bag monad on Finite-Dimensional Hilbert Spaces

Edit: I will be happy if someone can get me the Bag monad on a 2-category of groupoids, regardless of any reference to Hilbert Spaces. (It's a fire sale!!)

I am trying to create the quantum commutative monoid monad (aka Bag, multiset monad). On the one hand, quantum containers are interesting for quantum computing. I have my own reasons for looking for this and they are based in theoretical physics. To this end, I would like to define a 2-category which has the following properties:

1. It's objects are groupoids
2. To place it in the realm of the quantum, I am wondering if it's objects can be finite dimensional Hilbert spaces, with unitary transformations as morphisms, making each object a groupoid. If you have an idea similar to this, that actually works, it would be most appreciated.
3. Since I would like the objects to be something like Hilbert spaces, I would then constrain the morphisms to be maps between Hilbert spaces. I am afraid I do not know how to constrain the morphisms. If that leaves this question unanswerable, then so be it. If you can see how to constrain the morphisms, to achieve the goal of supporting the proposed monad, please answer.
4. It admits a polynomial monad that is the Bag or Multiset or "Free Commutative Monoid" monad.

Can someone define the category and also this polynomial monad?

One could see Jeffery Morton's comment here:

https://golem.ph.utexas.edu/category/2012/07/morton_and_vicary_on_the_categ.html

Since C⊗sn≅C, this is just a sum of a bunch of 1-dimensional spaces, each of which describes an n-particle system, which again has only one state. The only thing to say about this state is that it has n particles in it. Jamie’s original paper explains this by means of a monad on Hilb, which is essentially the “free commutative monoid” monad: the Fock space is the free commutative monoid on C. This fact gives a bunch of special maps, including a bialgebra structure on the Fock space, and the raising and lowering operators can be constructed out of this. The commutation relations are a consequence of that.

• It doesn't mean anything to ask for the objects to be Hilbert spaces without using that to constrain the morphisms. Do you want the morphisms to be linear maps? Unitary maps? You're also not going to have much luck writing down monads on a groupoid, since the multiplication map $M^2 \to M$ must necessarily be an isomorphism, so the only available monads are idempotent. – Qiaochu Yuan Jul 10 '18 at 23:01
• @Qiaochu., I have changed it so that I am asking for a 2-category of groupoids. – Ben Sprott Jul 11 '18 at 2:40
• @QiaochuYuan, could you suggest a way to constrain the morphisms to achieve the goal of supporting the desired monad? – Ben Sprott Jul 12 '18 at 16:25