nLab (2,1)-algebraic theory of E-infinity algebras

Idea

The (∞,1)-algebraic theory whose algebras are E-∞ algebras is the (2,1)-category of spans of finite sets.

Definition

Definition

Let

$$2Comm := Span(FinSet)$$

be the (2,1)-category of spans of finite sets:

• objects are finite sets;

• morphisms are spans $X_1 \leftarrow Y \to X_1$ in FinSet;

• 2-morphisms are diagrams

  $$\array{ && Y \\ & \swarrow && \searrow \\ X_0 &&\downarrow^{\mathrlap{\simeq}}&& X_1 \\ & \nwarrow && \nearrow \\ && Y' }$$

  in FinSet with the vertical morphism an isomorphism.

Observation

The homotopy category of $2Comm$ is the category $Comm$ that is the Lawvere theory of commutative monoids.

Proof

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

$$f : F[k] \to F[l]$$

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.

Remark

For instance the spans

$$\{1,2\} \stackrel{id}{\leftarrow} \{1,2\} \to \{1\}$$

and

$$\{1,2\} \stackrel{\simeq}{\leftarrow} \{2,1\} \to \{1\}$$

describe the operation

$$(a,b) \mapsto a + b$$

and the operation

$$(a,b) \mapsto b + a \,,$$

respectively. Clearly, in $Comm$ both these operations are identified. In $2Comm$ however they the are only equivalent

$$\array{ && \{1,2\} \\ & {}^{\mathllap{id}}\swarrow && \searrow \\ \{1,2\} &&\downarrow^{\mathrlap{\simeq}}&& \{1\} \\ & {}_{\mathllap{\simeq}}\nwarrow && \nearrow \\ && \{2,1\} } \,.$$

Properties

Observation

Let $Comm$ be the ordinary Lawvere theory of commutative monoids. There is a forgetful 2-functor

$$2Comm \to Comm \,.$$

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).

Proposition

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).

Proposition

An (∞,1)-category with (∞,1)-product is naturally an algebra over the $(2,1)$-theory $2Comm$.

This appears as (Cranch, theorem 4.26).

Theorem

An algebra over the $(2,1)$-theory $2Comm$ in (∞,1)Cat is a symmetric monoidal (∞,1)-category.

This appears as (Cranch, theorem 5.3).

Theorem

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.

Examples

Free algebras

The free algebra over $2Comm$ in ∞Grpd on a single generator is $2Comm(*, -) : 2Comm \to \infty Grpd$. Its underlying ∞-groupoid is therefore

$$2Comm(*,*) = Core(FinSet) \,,$$

the core groupoid of the category FinSet. This is equivalent to

$$\cdots \simeq \coprod_{n \in \mathbb{N}} \mathbf{B} \Sigma_n \,,$$

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 $\cdots = \coprod_{n \in \mathbb{N}} \mathbf{E}\Sigma_n/\Sigma_n = \coprod_n \mathbf{B} \Sigma_n$.

References

• James Cranch, Algebraic Theories and $(\infty,1)$-Categories (arXiv)