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

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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)

Last revised on October 15, 2013 at 19:31:24. See the history of this page for a list of all contributions to it.