We can see here in Jacob's 2013 paper, that he seems to state that a particular kind of multiset monad is actually a vector space monad.

3.2. Vector spaces. For a semiring S one can define the multiset monad MS on Sets by MS(X) = {ϕ: X → S | supp(ϕ) is finite}. Such an element ϕ can be identified with a formal finite sum P i sixi with multiplicities si ∈ S for elements xi ∈ X. The unit of this monad η : X → MS(X) is given by singleton multisets: η(x) = 1x. The multiplication µ: M2 S (X) → MS(X) involves (matrix) multiplication: µ( P i siϕi)(x) = P i si · ϕi(x), where · is the multiplication of the semiring S.

Here Jacobs is talking about hosting the monad on Set. It has been shown that the basic multiset monad does not have a polynomial form on Set. That link also suggests that the 2-category of groupoids (Grpd) supports the polynomial multiset monad. Based on this, I am guessing Set does not support a polynomial form for the vector space monad. Is there a category that does support the polynomial form for the vector space monad? Could it be the 2-category of groupoids? I think this would mean that there is a non-trivial adjunction between the category of Modules over S and the category of groupoids. Here we see that there are no interesting adjunctions between the category FinHilb and Groupoids. This might rule out groupoids as a host for the monad.

I have neglected to say which semigring, S, I am interested in. To be honest, I am just surprised to find there is a "vector space" monad. I am not interested in any particular semiring. I have done a tiny amount of reading as a means to see if any common semirings are interesting to me, but I don't recognize any of them. At the very least, I can understand the statement that the module over a semiring is a commutative monoid (which is the category of algebras for the multiset monad, so that makes sense). I keep trying to think of modules over a semiring as just vector spaces over the complex numbers. This is because I am trying to tie this very abstract result back to quantum mechanics.

It has been pointed out that for the monad to exist on the 2-category of groupoids, that category must satisfy these conditions: it must be cocomplete and cartesian closed. Since Grpd satisfies these, the monad exists, which is very nice to hear for me. What remains to be seen is whether or not there is a polynomial form.

I am interested, firstly, in the existence of the monad on the 2-category of groupoids. I would be especially interested in any case where the semiring is something like the complex numbers. Secondly, I am interested in whether or not there is a polynomial form for this monad on the 2-category of groupoids. An answer to either question will be useful, but an answer to both would be best.

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    $\begingroup$ What is "the vector space monad" and what would it mean for the 2-category of groupoids to "admit" it? $\endgroup$ – Tim Campion Aug 19 '18 at 4:13
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    $\begingroup$ Ok, I think I've worked out part of what you're trying to ask. If $C$ is, say, a cocomplete cartesian closed category and $S$ is a semiring object in $C$, then there is a monad on $C$ whose algebras are modules over the semiring $S$. Following Jacobs, you're calling this "the" vector space monad. But it's not clear to me whether you're asking for conditions for this monad to exist, or rather for conditions under which this monad is polynomial. I've just given some conditions for existence, and they are satisfied in the case of $Gpd$. I'm not sure about polynomialness. $\endgroup$ – Tim Campion Aug 19 '18 at 4:29
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    $\begingroup$ Also, it's not clear which semirings you're interested in. Do you want to consider all semirings on $Gpd$, or are you particularly interested in certain ones? $\endgroup$ – Tim Campion Aug 19 '18 at 4:41
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    $\begingroup$ Note that your earlier question on adjunctions between $Gpd$ and finite dimensional Hilbert spaces is irrelevant here, because there is no finite-dimensionality constraint at play here. $\endgroup$ – Tim Campion Aug 22 '18 at 21:25

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