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](https://golem.ph.utexas.edu/category/2012/07/morton_and_vicary_on_the_categ.html) 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 (⊗...⊗), 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
>  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.
> 
> 
> 



What is the “free symmetric monoidal category” 2-monad?
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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