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



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




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

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

  • objects are finite sets;

  • morphisms are spans X 1YX 1X_1 \leftarrow Y \to X_1 in FinSet;

  • 2-morphisms are diagrams

    Y X 0 X 1 Y \array{ && Y \\ & \swarrow && \searrow \\ X_0 &&\downarrow^{\mathrlap{\simeq}}&& X_1 \\ & \nwarrow && \nearrow \\ && Y' }

    in FinSet with the vertical morphism an isomorphism.


The homotopy category of 2Comm2Comm is the category CommComm that is the Lawvere theory of commutative monoids.


The Lawvere theory of commutative monoids has as objects the free commutative monoids F[k]F[k] on kk \in \mathbb{N} generators, and as morphisms monoid homomorphisms.

By the free property, morphisms

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

are in natural bijection to kk-tuples of elements of F[l]F[l]. Such elements in turn are sums a 1+a 1++a 1+a 2+a 2++a 2+a 3+a_1 + a_1 + \cdots + a_1 + a_2 + a_2 + \cdots + a_2 + a_3 + \cdots of copies of the ll generators, hence are in bijection to sequences of natural numbers (n 1,,n l)(n_{1}, \cdots, n_l). Hence morphisms f:F[k]F[l]f : F[k] \to F[l] are in bijection to k×lk \times l-matrices with entries in the natural numbers.

One checks that under this identification composition of morphisms corresponds to matrix multiplication.


For instance the spans

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


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

describe the operation

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

and the operation

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

respectively. Clearly, in CommComm both these operations are identified. In 2Comm2Comm however they the are only equivalent

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



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

2CommComm. 2Comm \to Comm \,.

This exhibits 2Comm2Comm as being like CommComm but with some additional auto-2-morphisms of the morphism of CommComm.

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


The (,1)(\infty,1)-category 2Comm2Comm has finite products. The products of objects A,BA, B in 2Comm2Comm is their coproduct ABA \coprod B in FinSet.

This appears as (Cranch, prop. 4.7).


An (∞,1)-category with (∞,1)-product is naturally an algebra over the (2,1)(2,1)-theory 2Comm2Comm.

This appears as (Cranch, theorem 4.26).


An algebra over the (2,1)(2,1)-theory 2Comm2Comm in (∞,1)Cat is a symmetric monoidal (∞,1)-category.

This appears as (Cranch, theorem 5.3).


There is a (2,1)(2,1)-algebraic theory E E_\infty whose algebras are at least in parts like E-∞ algebras.

This is (Cranch), prop. 6.12, theorem 6.13 and section 8.


Free algebras

The free algebra over 2Comm2Comm in ∞Grpd on a single generator is 2Comm(*,):2CommGrpd2Comm(*, -) : 2Comm \to \infty Grpd. Its underlying ∞-groupoid is therefore

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

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

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

where Σ n\Sigma_n is the symmetric group on nn elements and BΣ n\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 PP as our model for the E-∞-operad: that has P n=EΣ nP_n = \mathbf{E} \Sigma_n. The free algebra over an operad is given by nP n/Σ n\coprod_{n \in \mathbb{N}} P_n/\Sigma_n, which here is = nEΣ n/Σ n= nBΣ n\cdots = \coprod_{n \in \mathbb{N}} \mathbf{E}\Sigma_n/\Sigma_n = \coprod_n \mathbf{B} \Sigma_n.


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