TITLE: What is the “free symmetric monoidal category” 2-monad?
QUESTION [5 upvotes]: 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 (⊗...⊗), 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.

REPLY [6 votes]: 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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