symmetric monoidal (∞,1)-category of spectra
The (∞,1)-algebraic theory whose algebras are E-∞ algebras is the (2,1)-category of spans of finite sets.
Let
be the (2,1)-category of spans of finite sets:
objects are finite sets;
2-morphisms are diagrams
in FinSet with the vertical morphism an isomorphism.
The homotopy category of $2Comm$ is the category $Comm$ that is the Lawvere theory of commutative monoids.
The Lawvere theory of commutative monoids has as objects the free commutative monoids $F[k]$ on $k \in \mathbb{N}$ generators, and as morphisms monoid homomorphisms.
By the free property, morphisms
are in natural bijection to $k$-tuples of elements of $F[l]$. Such elements in turn are sums $a_1 + a_1 + \cdots + a_1 + a_2 + a_2 + \cdots + a_2 + a_3 + \cdots$ of copies of the $l$ generators, hence are in bijection to sequences of natural numbers $(n_{1}, \cdots, n_l)$. Hence morphisms $f : F[k] \to F[l]$ are in bijection to $k \times l$-matrices with entries in the natural numbers.
One checks that under this identification composition of morphisms corresponds to matrix multiplication.
For instance the spans
and
describe the operation
and the operation
respectively. Clearly, in $Comm$ both these operations are identified. In $2Comm$ however they the are only equivalent
Let $Comm$ be the ordinary Lawvere theory of commutative monoids. There is a forgetful 2-functor
This exhibits $2Comm$ as being like $Comm$ but with some additional auto-2-morphisms of the morphism of $Comm$.
This is discussed in (Cranch, beginning of section 5.2).
The $(\infty,1)$-category $2Comm$ has finite products. The products of objects $A, B$ in $2Comm$ is their coproduct $A \coprod B$ in FinSet.
This appears as (Cranch, prop. 4.7).
An (∞,1)-category with (∞,1)-product is naturally an algebra over the $(2,1)$-theory $2Comm$.
This appears as (Cranch, theorem 4.26).
An algebra over the $(2,1)$-theory $2Comm$ in (∞,1)Cat is a symmetric monoidal (∞,1)-category.
This appears as (Cranch, theorem 5.3).
There is a $(2,1)$-algebraic theory $E_\infty$ whose algebras are at least in parts like E-∞ algebras.
This is (Cranch), prop. 6.12, theorem 6.13 and section 8.
The free algebra over $2Comm$ in ∞Grpd on a single generator is $2Comm(*, -) : 2Comm \to \infty Grpd$. Its underlying ∞-groupoid is therefore
the core groupoid of the category FinSet. This is equivalent to
where $\Sigma_n$ is the symmetric group on $n$ elements and $\mathbf{B}\Sigma_n$ its one-object delooping groupoid.
Notice that this is indeed the free E-∞-algebra, on the nose so if we use the Barratt-Eccles operad $P$ as our model for the E-∞-operad: that has $P_n = \mathbf{E} \Sigma_n$. The free algebra over an operad is given by $\coprod_{n \in \mathbb{N}} P_n/\Sigma_n$, which here is $\cdots = \coprod_{n \in \mathbb{N}} \mathbf{E}\Sigma_n/\Sigma_n = \coprod_n \mathbf{B} \Sigma_n$.