symmetric monoidal (∞,1)-category of spectra
The Lawvere theory of commutative monoids has as objects the free commutative monoids on generators, and as morphisms monoid homomorphisms.
By the free property, morphisms
are in natural bijection to -tuples of elements of . Such elements in turn are sums of copies of the generators, hence are in bijection to sequences of natural numbers . Hence morphisms are in bijection to -matrices with entries in the natural numbers.
One checks that under this identification composition of morphisms corresponds to matrix multiplication.
For instance the spans
describe the operation
and the operation
respectively. Clearly, in both these operations are identified. In however they the are only equivalent
Let be the ordinary Lawvere theory of commutative monoids. There is a forgetful 2-functor
This exhibits as being like but with some additional auto-2-morphisms of the morphism of .
This is discussed in (Cranch, beginning of section 5.2).
This appears as (Cranch, prop. 4.7).
This appears as (Cranch, theorem 4.26).
This appears as (Cranch, theorem 5.3).
There is a -algebraic theory whose algebras are at least in parts like E-∞ algebras.
This is (Cranch), prop. 6.12, theorem 6.13 and section 8.
Notice that this is indeed the free E-∞-algebra, on the nose so if we use the Barratt-Eccles operad as our model for the E-∞-operad: that has . The free algebra over an operad is given by , which here is .