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(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

$$2\mathrm{Comm}:=\mathrm{Span}(\mathrm{FinSet})$$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

  $$\begin{array}{ccc}& & Y\\ & \swarrow & & \searrow \\ X_0& & {\downarrow }^{\simeq }& & X_1\\ & \nwarrow & & \nearrow \\ & & Y\prime \end{array}$$\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 $2\mathrm{Comm}$ is the category $\mathrm{Comm}$ 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

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

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

and

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

describe the operation

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

and the operation

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

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

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

Properties

Observation

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

$$2\mathrm{Comm}\to \mathrm{Comm}.$$2Comm \to Comm \,.

This exhibits $2\mathrm{Comm}$ as being like $\mathrm{Comm}$ but with some additional auto-2-morphisms of the morphism of $\mathrm{Comm}$.

This is discussed in (Cranch, beginning of section 5.2).

Proposition

The $(\mathrm{\infty },1)$-category $2\mathrm{Comm}$ has finite products. The products of objects $A,B$ in $2\mathrm{Comm}$ 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 $2\mathrm{Comm}$.

This appears as (Cranch, theorem 4.26).

Theorem

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

This appears as (Cranch, theorem 5.3).

Theorem

There is a $(2,1)$-algebraic theory $E_{\mathrm{\infty }}$ 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 $2\mathrm{Comm}$ in ∞Grpd on a single generator is $2\mathrm{Comm}(*,-):2\mathrm{Comm}\to \mathrm{\infty }\mathrm{Grpd}$. Its underlying ∞-groupoid is therefore

$$2\mathrm{Comm}(*,*)=\mathrm{Core}(\mathrm{FinSet}),$$2Comm(*,*) = Core(FinSet) \,,

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

$$\cdots \simeq \coprod _{n\in \mathbb{N}}B{\Sigma }_n,$$\cdots \simeq \coprod_{n \in \mathbb{N}} \mathbf{B} \Sigma_n \,,

where ${\Sigma }_n$ is the symmetric group on $n$ elements and $B{\Sigma }_n$ 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 $P_n=E{\Sigma }_n$. The free algebra over an operad is given by ${\coprod }_{n\in \mathbb{N}}{P}_n/{\Sigma }_n$, which here is $\cdots ={\coprod }_{n\in \mathbb{N}}E{\Sigma }_n/{\Sigma }_n={\coprod }_{n}B{\Sigma }_n$.

References

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