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

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

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

• objects are finite sets;

• morphisms are spans ${X}_{1}←Y\to {X}_{1}$ in FinSet;

• 2-morphisms are diagrams

$\begin{array}{ccc}& & Y\\ & ↙& & ↘\\ {X}_{0}& & {↓}^{\simeq }& & {X}_{1}\\ & ↖& & ↗\\ & & Y\prime \end{array}$\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 $F\left[k\right]$ on $k\in ℕ$ generators, and as morphisms monoid homomorphisms.

By the free property, morphisms

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

are in natural bijection to $k$-tuples of elements of $F\left[l\right]$. 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 $\left({n}_{1},\cdots ,{n}_{l}\right)$. Hence morphisms $f:F\left[k\right]\to F\left[l\right]$ are in bijection to $k×l$-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

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

and

$\left\{1,2\right\}\stackrel{\simeq }{←}\left\{2,1\right\}\to \left\{1\right\}$\{1,2\} \stackrel{\simeq}{\leftarrow} \{2,1\} \to \{1\}

describe the operation

$\left(a,b\right)↦a+b$(a,b) \mapsto a + b

and the operation

$\left(a,b\right)↦b+a\phantom{\rule{thinmathspace}{0ex}},$(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

$\begin{array}{ccc}& & \left\{1,2\right\}\\ & {}^{\mathrm{id}}↙& & ↘\\ \left\{1,2\right\}& & {↓}^{\simeq }& & \left\{1\right\}\\ & {}_{\simeq }↖& & ↗\\ & & \left\{2,1\right\}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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}\phantom{\rule{thinmathspace}{0ex}}.$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 $\left(\infty ,1\right)$-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 $\left(2,1\right)$-theory $2\mathrm{Comm}$.

This appears as (Cranch, theorem 4.26).

Theorem

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

This appears as (Cranch, theorem 5.3).

Theorem

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

$2\mathrm{Comm}\left(*,*\right)=\mathrm{Core}\left(\mathrm{FinSet}\right)\phantom{\rule{thinmathspace}{0ex}},$2Comm(*,*) = Core(FinSet) \,,

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

$\cdots \simeq \coprod _{n\in ℕ}B{\Sigma }_{n}\phantom{\rule{thinmathspace}{0ex}},$\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 ${\coprod }_{n\in ℕ}{P}_{n}/{\Sigma }_{n}$, which here is $\cdots ={\coprod }_{n\in ℕ}E{\Sigma }_{n}/{\Sigma }_{n}={\coprod }_{n}B{\Sigma }_{n}$.

References

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

Revised on October 15, 2013 19:31:24 by Urs Schreiber (80.237.234.132)