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I have come across an n-category cafe post where someone describes a monad that generates symmetric monoidal categories. Can someone give details, like what is the base category, what exactly is the endofunctor and natural isomorphisms for “free symmetric monoidal category” 2-monad? Could this generate the category of finite dimensional hilbert spaces and unitary maps?

I have been reading this post by Jeffery Morton. He describes the following “free symmetric monoidal category” 2-monad:

The bosonic Fock space is then ⊕nC⊗sn, the direct sum of all symmetric tensor products of some number of copies of this space. One way to say this is that the symmetric tensor product of a space V with itself is the equalizer of two maps V⊗V→V⊗V, namely the identity and the swap map. Likewise, V⊗sn is the equalizer of all the permutation automorphisms that appear because V⊗n is automatically a representation of Sn. So the symmetric product is the trivial representation.

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.

Now, groupoidifying this is a categorification, so this description has to be weakened. To start with, we take a groupoid describing a system with only one configuration (the “it’s there” state for our particle). This will be the trivial groupoid 1, with one object and only the identity morphism. Then we want to take the “groupoidified Fock space”.

> Since groupoids live in a 2-category, the equivalent of the “free commutative monoid” monad turns out to be a bit weaker, namely the “free symmetric monoidal category” 2-monad. We get a “direct sum” (i.e. in Span(Gpd), the disjoint union) of a bunch of objects which show up as certain 2-limits. In particular, we freely generate a bunch of objects like (𝟙⊗...⊗𝟙), and we must get not EQUATIONS, but ISOMORPHISMS corresponding to all the switch maps. This is essentially where the groupoid of finite sets and bijections come from: think of 𝟙 as the groupoid which contains exactly the 1-element set - the free symmetric monoidal category this generates is the groupoid which contains all finite sets and their bijections.

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There is a 2-monad $P$ on $\mathrm{Cat}$ whose strict algebras are symmetric strict monoidal categories, and whose pseudo-algebras are "unbiased" symmetric monoidal categories. On objects, $PA$ is the category whose objects are finite lists of objects of $A$, and in which a morphism $(a_1,\dots,a_n)\to (b_1,\dots,b_m)$ consists of a bijection $\sigma : \{1,\dots,n\} \to \{1,\dots,m\}$ (so that in particular $n=m$) and morphisms $f_i : a_i \to b_{\sigma i}$ in $A$. The unit $A \to P A$ sends $a$ to the 1-element list $(a)$, and the multiplication $P P A \to P A$ removes parentheses.

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  • $\begingroup$ it looks very interesting. It makes a monoidal category out of any category. Are the functions $f_i$ free to be any morphisms mappingI$a_i$ to $b_{\sigma_i}$?The bijections $\sigma$, are they like connecting the input wires to a box in a string diagram to the output wires? What is their purpose? $\endgroup$ – Ben Sprott Jul 22 '18 at 22:03
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    $\begingroup$ Yes, the $f_i$ are any morphisms in $A$ from $a_i$ to $b_{\sigma i}$. The bijections can indeed be thought of as connecting input to output wires; without them, the monoidal structure would not be symmetric. For instance, the twist $(a) \otimes (b) = (a,b) \cong (b,a) = (b) \otimes (a)$ has $\sigma$ the transposition bijection with $f_1 = \mathrm{id}_{a}$ and $f_2 = \mathrm{id}_{b}$. $\endgroup$ – Mike Shulman Jul 22 '18 at 23:33

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