An effective proof of the Cartan formula: the even prime

TL;DR

This paper provides a constructive proof of the Cartan formula at the cochain level over _2, applicable to general algebras over the Barratt-Eccles operad, including singular cochains of spaces.

Contribution

It introduces an explicit, natural coboundary construction that verifies the Cartan formula at the cochain level for _2-cohomology.

Findings

Constructs a natural coboundary for the Cartan formula at the cochain level.

The proof applies to algebras over the Barratt-Eccles operad.

Works for singular cochains of topological spaces.

Abstract

The Cartan formula encodes the relationship between the cup product and the action of the Steenrod algebra in $\mathbb F_p$-cohomology. In this work, we present an effective proof of the Cartan formula at the cochain level when the field is $\mathbb F_2$. More explicitly, for an arbitrary pair of cocycles and any non-negative integer, we construct a natural coboundary that descends to the associated instance of the Cartan formula. Our construction works for general algebras over the Barratt-Eccles operad, in particular, for the singular cochains of spaces.

Equations

$$S{q}^k:H^{\ast }(X;{\mathbb{F}}_2)\to H^{\ast }(X;{\mathbb{F}}_2)$$

$$[\alpha ][\beta ]=[\alpha {⌣}_0\beta ]$$

$${⌣}_0:N^{\ast }(X;\mathbb{Z})\otimes N^{\ast }(X;\mathbb{Z})\to N^{\ast }(X;\mathbb{Z})$$

$${⌣}_i:N^{\ast }(X;\mathbb{Z})\otimes N^{\ast }(X;\mathbb{Z})\to N^{\ast }(X;\mathbb{Z})$$

$$Sq^{k}\big{(}[\alpha]\big{)}=[\alpha\smallsmile_{k-n}\alpha].$$

$$0=\Big{[}(\alpha\smallsmile_{0}\beta)\smallsmile_{i}(\alpha\smallsmile_{0}\beta)\ +\sum_{i=j+k}(\alpha\smallsmile_{j}\alpha)\smallsmile_{0}(\beta\smallsmile_{k}\beta)\Big{]}.$$

$$\delta {\zeta }_i(\alpha \otimes \beta )=(\alpha {⌣}_0\beta ){⌣}_i(\alpha {⌣}_0\beta )+\sum _{i=j+k}(\alpha {⌣}_j\alpha ){⌣}_0(\beta {⌣}_k\beta ).$$

$$\mathrm{Hom}(C,C^{\prime }{)}_n=\{f:c\in C_m\text{ implies }f(c)\in C_{m+n}^{\prime }\}\partial f=\partial \circ f+f\circ \partial .$$

$$\partial h=f+g.$$

$$(C\otimes C^{\prime }{)}_n=\underset{i+j=n}{⨁}{C}_i\otimes C_j^{\prime }\partial =\partial \otimes \mathrm{id}+\mathrm{id}\otimes \partial .$$

$${\delta }_i:[n-1]\to [n]{\sigma }_i:[n+1]\to [n]$$

$${\delta }_i(k)=\{\begin{array}{ll}k& k<i\\ k+1& i\le k\end{array}\text{ and }{\sigma }_i(k)=\{\begin{array}{ll}k& k\le i\\ k-1& i<k\end{array}$$

$$X[n]=X_{n}X({\delta }_i)=d_{i}X({\sigma }_i)=s_{i.}$$

$$(X\times Y{)}_n=X_n\times Y_n{d}_i=d_i\times d_i{s}_i=s_i\times s_{i.}$$

$$N_{\ast }:\mathbf{sSet}\to \mathbf{Ch}$$

$$N_n(X)=\frac{{\mathbb{F}}_2\{X_n\}}{{\mathbb{F}}_2\{s(X_{n-1})\}}{\partial }_n=\sum _{i=0}^n{d}_i$$

$$AW:N_{\ast }(X\times Y)\to N_{\ast }(X)\otimes N_{\ast }(Y)$$

$$AW(x\times y)=\sum _{i=0}^n{d}_{i+1}\cdots d_{n}x\otimes d_0\cdots d_{i-1}y$$

$$EZ:N_{\ast }(X)\otimes N_{\ast }(Y)\to N_{\ast }(X\otimes Y)$$

$$EZ(x\otimes y)=\sum s_{v_p}\cdots s_{v_1}x\otimes s_{w_q}\cdots s_{w_1}y$$

$$\big{(}\{v_{1}<\cdots<v_{p}\},\ \{w_{1}<\cdots<w_{q}\}\big{)}$$

$$SHI:N_n(X\times Y)\to N_{\ast }(X\times Y)$$

$$\begin{array}{rl}SHI(x\times y)=\sum & s_{v_p+m}\cdots s_{v_1+m}{s}_{m-1}{d}_{n-p+1}\cdots d_{n}x\otimes \\ & s_{w_{q+1}+m}\cdots s_{w_1+m}{d}_{n-p-q}\cdots d_{n-p-1}y\end{array}$$

$$\big{(}\{v_{1}<\cdots<v_{p}\},\ \{w_{1}<\cdots<w_{q+1}\}\big{)}$$

$$\Gamma \to \mathrm{Aut}(C)$$

$$\mathcal{O}\to \mathrm{End}(C)$$

$${\circ }_{\Sigma }:{\Sigma }_r\times {\Sigma }_{s_1}\times \cdots \times {\Sigma }_{s_r}\to {\Sigma }_{s_1+\cdots +s_r}$$

$$\begin{array}{rl}E(r{)}_n& =\{({\sigma }_0,\dots ,{\sigma }_n)|{\sigma }_i\in {\Sigma }_r\},\\ d_i({\sigma }_0,\dots ,{\sigma }_n)& =({\sigma }_0,\dots ,{\hat{\sigma }}_i,\dots ,{\sigma }_n),\\ s_i({\sigma }_0,\dots ,{\sigma }_n)& =({\sigma }_0,\dots ,{\sigma }_i,{\sigma }_i,\dots ,{\sigma }_n).\end{array}$$

$$\sigma\big{(}(\sigma_{0},\dots,\sigma_{n})\big{)}=(\sigma\sigma_{0},\dots,\sigma\sigma_{n}).$$

Full text

1. Introduction

Let $X$ be a space. In [Ste47], Steenrod introduced formulae to define his famous Steenrod squares

[TABLE]

and in [SE62], he axiomatically characterized them by the following:

$Sq^{k}$ is natural, 2. 2.

$Sq^{0}$ is the identity, 3. 3.

$Sq^{k}(x)=x^{2}$ for $x\in H^{-k}(X;\mathbb{F}_{2})$, 4. 4.

$Sq^{k}(x)=0$ for $x\in H^{-n}(X;\mathbb{F}_{2})$ with $n>k$, 5. 5.

$Sq^{k}(xy)=\sum_{i+j=k}Sq^{i}(x)Sq^{j}(y)$.

Axiom 5., known as the Cartan formula, is the focus of this work.

To describe our viewpoint and present the contributions of this paper in context, let us revisit some of the history of Steenrod’s construction.

In the late thirties, Alexander, Whitney, and Čech defined the ring structure on cohomology

[TABLE]

using a cochain level construction

[TABLE]

dual to a choice of simplicial chain approximation to the diagonal inclusion.

Steenrod then showed that $\smallsmile_{0}$ is commutative up to coherent homotopies by effectively constructing cup-$i$ products

[TABLE]

enforcing its derived commutativity. (Axioms for these and connections with higher category theory can be found in [MM18a] and [MM19]). Then, with coefficients in $\mathbb{F}_{2}$, Steenrod defined

[TABLE]

This definition of the Steenrod squares makes the Cartan formula equivalent to

[TABLE]

The goal of this work is to effectively construct for any $i\geq 0$ and arbitrary pair of cocycles $\alpha,\beta\in N^{*}(X;\mathbb{F}_{2})$ a natural cochain $\zeta_{i}(\alpha\otimes\beta)$ such that

[TABLE]

Following [May70], we take a more general approach and in doing so we describe a non-necessarily effective construction for any $E_{\infty}$-algebra. We work over a fixed algebraic model $\mathcal{E}$ of the $E_{\infty}$-operad known as the Barratt-Eccles operad. This model, introduced by Berger-Fresse in [BF04], is equipped with a diagonal map and the natural $\mathcal{E}$-algebra structure defined by these authors on the normalized cochains of simplicial sets is suitable for effective constructions.

This paper is part of an ongoing effort spearheaded by Greg Brumfiel and John Morgan to build effective models of classical homotopy-theoretic concepts, see for example [BM16] and [BM18]. Motivation for this especific project came from a question of Anton Kapustin. See [KT17] for an instance where one of our formulae is used in the context of topological phases of matter.

Implementations of the constructions of this paper and of state-of-the-art algorithms for the computation of Steenrod squares and cup-$i$ products, as introduced in [MM18d], can be found in the author’s website.

Acknowledgement

We would like to thank John Morgan, Greg Brumfiel, Dennis Sullivan, Anton Kapustin, Mark Behrens, Marc Stephan, and Kathryn Hess for their insights, questions, and comments about this project.

2. Conventions and preliminaries

For the remainder of this paper all algebraic constructions are considered over $\mathbb{F}_{2}$, the field with two elements.

2.1. Chain complexes and simplicial sets

We denote the category of chain complexes of $\mathbb{F}_{2}$-modules by $\mathbf{Ch}$. Boundary maps decrease degree and for chain complexes $C$ and $C^{\prime}$ the set of linear maps $\mathrm{Hom}(C,C^{\prime})$ is a chain complex with

[TABLE]

Let $f,g,h\in\mathrm{Hom}(C,C^{\prime})$. We remark that $f$ is a chain map if and only if it is a degree [math] cycle. Assume $f$ and $g$ are chain maps, then $h$ is a chain homotopy between them if and only if

[TABLE]

We remark that $f\in\mathrm{Hom}(C,C^{\prime})$ is a chain map if and only if it is a degree [math] cycle, and that $h\in\mathrm{Hom}(C,C^{\prime})$ is a chain homotopy between two chain maps $f$ and $g$ if and only if

[TABLE]

The product of two chain complexes $C$ and $C^{\prime}$ is defined by

[TABLE]

The category $\bm{\Delta}$ is defined to have an object $[n]=\{0,\dots,n\}$ for every non-negative integer $n$ and a morphism $[m]\to[n]$ for each order-preserving function from $\{0,\dots,m\}$ to $\{0,\dots,n\}$.

For integers $0\leq i\leq n$, the morphisms

[TABLE]

defined by

[TABLE]

generate all morphisms in $\bm{\Delta}$.

A simplicial set is a contravariant functor from $\bm{\Delta}$ to the category of sets. We denote the category of simplicial sets by $\mathbf{sSet}$ and use the standard notation

[TABLE]

The product of two simplicial sets $X$ and $Y$ is defined by

[TABLE]

The functor of normalized chains

[TABLE]

is defined by

[TABLE]

where $s(X_{n-1})=\bigcup_{i=0}^{n-1}s_{i}(X_{n-1})$.

The functor of normalized cochains $N^{*}$ is defined by composing $N_{*}$ with the linear duality functor $\mathrm{Hom}(-,\mathbb{F}_{2}):\mathbf{Ch}\to\mathbf{Ch}$. Notice that in this definition cochains are concentrated in non-positive degrees.

Remark 1**.**

The singular cochains with $\mathbb{F}_{2}$-coefficients of a topological space $X$ coincide with $N^{*}(Sing(X))$, where $Sing(X)$ is the simplicial set of continuous maps from standard topological simplices to $X$.

2.2. The Alexander-Whitney and Eilenberg-Zilber maps

The functor of normalized chains does not preserve products, i.e., for a general pair of simplicial sets $X$ and $Y$ there is no isomorphism beetween $N_{*}(X\times Y)$ and $N_{*}(X)\otimes N_{*}(Y)$. Nevertheless, there is a canonical chain homotopy equivalence between them.

The Alexander-Whitney map

[TABLE]

is defined for $x\times y\in N_{n}(X\times Y)$ by

[TABLE]

and the Eilenberg-Zilber map

[TABLE]

is defined for $x\otimes y\in N_{p}(X)\otimes N_{q}(Y)$ by

[TABLE]

where the sum is over all pairs of disjoint subsets

[TABLE]

of $\{0,\dots,p+q-1\}$.

It is well known that the compositions $AWEZ$ is equal to the identity and that the composition $EZAW$ is effectively chain homotopic to the identity. A recursive description for one such chain homotopy was first given in [EML53]. A close formula for it, which we learned from [Rea00] and is credited therein to [Rub91], is given next.

The Shih homotopy

[TABLE]

is defined for $x\times y\in N_{n}(X\otimes Y)$ to be [math] if $n=0$ and if $n>0$ by

[TABLE]

where $m=n-p-q$ and the sum is over all pairs of disjoint subsets

[TABLE]

of $\{0,\dots,p+q\}$ with $0\leq p\leq n-1$ and $0\leq q\leq n-p-1$.

2.3. Group actions and algebras over operads

Let $\Gamma$ be a group. A $\Gamma$-action on an object $C$ is a group morphism

[TABLE]

where the group structure on $\mathrm{Aut}(C)$ is given by composition.

Let $\mathcal{O}$ be an operad. (See for example [LV12].) An $\mathcal{O}$-algebra structure on an object $C$ is an operad morphism

[TABLE]

where the operad structure on $\mathrm{End}(C)=\{\mathrm{Hom}(C^{\otimes n},C)\}_{n\geq 0}$ is induced from composition of linear maps and transpositions of factors.

Let $C$ be an object with a $\Gamma$-action or an $\mathcal{O}$-algebra structure. We identify the elements of $\Gamma$ or $\mathcal{O}$ with their images via $(\ref{equation: group action})$ or $(\ref{equation: algebra over an operad})$.

2.4. The Barratt-Eccles operad $\mathcal{E}$

We review from [BF04] the Barratt-Eccles operad.

For a positive integer $r$ let $\Sigma_{r}$ be the group of permutations of $r$ elements and

[TABLE]

the usual composition of permutations.

For a positive integer $r$ define the simplicial set $E(r)$ by

[TABLE]

We consider $E(r)$ equipped with the action of $\Sigma_{r}$ given by

[TABLE]

Let

[TABLE]

be defined by applying $\bm{\circ}_{\Sigma}$ coordinatewise.

For $r\geq 0$, let

[TABLE]

be edowed with the induced $\Sigma_{r}$-action. Define an operadic compositions $\bm{\circ}_{\mathcal{E}}$ on $\mathcal{E}=\{\mathcal{E}(r)\}_{r\geq 0}$ by

{\mathcal{E}(r)\otimes\mathcal{E}(s_{1})\otimes\cdots\otimes\mathcal{E}(s_{r})}$${N_{*}\big{(}E(r)\otimes E(s_{1})\otimes\cdots\otimes E(s_{r})\big{)}}$${\mathcal{E}(s_{1}+\cdots+s_{r})}$$\scriptstyle{EZ^{r}}$$\scriptstyle{\bm{\circ}_{\mathcal{E}}}$$\scriptstyle{N_{*}(\bm{\circ}_{E})}

where $EZ^{r}$ stand recursively for $EZ(\mathrm{id}\otimes EZ^{r-1})$ with $EZ^{1}=EZ$.

The resulting operad $\mathcal{E}$ is referred to as the Barratt-Eccles operad. It is a model in the category $\mathbf{Ch}$ for the $E_{\infty}$-operad. That is to say, $E(0)=\mathbb{F}_{2}$ and, for $r>0$, $E(r)$ is a resolution of $\mathbb{F}_{2}$ by free $\mathbb{F}_{2}[\Sigma_{r}]$-modules.

3. Cartan coboundaries for $\mathcal{E}$-algebras

3.1. Steenrod cup-$i$ products and Cartan coboundaries

Since $\mathcal{E}$ is an $E_{\infty}$-operad, the orbit complex $\mathcal{E}(2)_{\Sigma_{2}}$ is an algebraic model for $K(\mathbb{F}_{2},1)$, and $H_{i}(\mathcal{E}(2)_{\Sigma_{2}})$ is generated as an $\mathbb{F}_{2}$-module by the orbit of the element

[TABLE]

Definition 2**.**

Let $A$ be an $\mathcal{E}$-algebra. The cup-$i$ product of $A$ is the image of $\tilde{x}_{i}$ in $\mathrm{Hom}(A\otimes A,A)$. We use the notation

[TABLE]

A Cartan $i$-coboundary is any map $\zeta_{i}$ in $\mathrm{Hom}(A\otimes A,A)$ that satisfies

[TABLE]

Definition 3**.**

The chain maps $F,\,G:\mathcal{E}(2)\rightarrow\mathcal{E}(4)$ are defined by

[TABLE]

for any basis element $(\sigma_{0},\dots,\sigma_{n})$.

Lemma 4**.**

Let $A$ be an $\mathcal{E}$-algebra. For $\alpha,\beta\in A$ and $i\geq 0$ we have

[TABLE]

and

[TABLE]

Proof.

For (7) we compute

[TABLE]

For (8) we notice that

[TABLE]

and compute

[TABLE]

as desired. ∎

Lemma 5**.**

Let $H:\mathcal{E}(2)\rightarrow\mathcal{E}(4)$ be a chain homotopy between $(23)\circ F$ and $G$ satisfying

[TABLE]

Then, for any $\mathcal{E}$-algebra $A$ and $i\geq 0$, the map $\zeta_{i}:A\otimes A\to A$ defined by

[TABLE]

is a Cartan $i$-coboundary.

Proof.

By assumption

[TABLE]

so evaluating on $\tilde{x}_{i}$ we get

[TABLE]

Since \big{(}(12)(34)+\mathrm{id}\big{)}(\alpha\otimes\alpha\otimes\beta\otimes\beta)=0, the equivariance assumption implies

[TABLE]

Evaluating (9) on $(\alpha\otimes\alpha\otimes\beta\otimes\beta)$ and using Lemma 4 we obtain

[TABLE]

as desired. ∎

3.2. Statement of the main theorem

We now describe the construction of a chain homotopy as stated in Lemma 5.

The first step of our construction is motivated by the well known fact that two group homomorphisms that are conjugate of each other induce homotopic maps of classifying spaces.

Let us consider the three group inclusions

[TABLE]

of $\Sigma_{2}$ into $\Sigma_{2}\times\Sigma_{2}$.

Definition 6**.**

The homomorphisms $f,g:\Sigma_{2}\to\Sigma_{4}$ are defined by

[TABLE]

or, more explicitly, by

[TABLE]

We notice that the maps $f$ and $g$ are conjugated by $(23)$ in $\Sigma_{4}$, i.e., for $\sigma\in\Sigma_{2}$

[TABLE]

Definition 7**.**

Let $F,G:E(2)\to E(4)$ be the simplicial maps induced from $f$ and $g$ respectively. Explicitly,

[TABLE]

for any $(\sigma_{0},\dots,\sigma_{n})\in E(2)$.

We introduce a linear map $H_{1}$ which, as we will see later, is an appropriately equivariant chain homotopy between $N_{*}(E(f))$ and $N_{*}(E(g))$.

Definition 8**.**

Let $H_{1}:\mathcal{E}(2)\to\mathcal{E}(4)$ be the degree $1$ linear map defined on basis elements by

[TABLE]

The second step relates the maps $F$ and $N_{*}(E(f))$, as well as $G$ and $N_{*}(E(g))$. It turns out, as we will see later, that the maps in the first pair are equal, whereas the maps in the second are chain homotopic via an appropriately equivariant map that we now introduce.

Definition 9**.**

Let $H_{2}:\mathcal{E}(2)\to\mathcal{E}(4)$ be the degree $1$ linear map defined on basis elements by

[TABLE]

We are ready to state our main result.

Theorem 10**.**

Let $A$ be an $\mathcal{E}$-algebra. For any $i\geq 0$ the map $\zeta_{i}$ in $\mathrm{Hom}(A\otimes A,A)$ defined by

[TABLE]

is a Cartan $i$-coboundary.

3.3. Proof of the main theorem

We will prove Theorem 10 after four lemmas.

Lemma 11**.**

In \mathrm{Hom}\big{(}\mathcal{E}(2),\mathcal{E}(4)\big{)}

[TABLE]

Proof.

The essence of the proof is a telescopic sum argument.

[TABLE]

for any basis element $(\sigma_{0},\dots,\sigma_{n})\in\mathcal{E}(2)$. ∎

Lemma 12**.**

The map $H_{1}$ satisfies the following form of equivariance:

[TABLE]

Proof.

Let $H_{1}^{i}$ be the degree $1$ linear map defined on basis elements by

[TABLE]

Notice that $H_{1}=\sum_{i}H_{1}^{i}$. We will verify (12) by checking it for each $H_{1}^{i}$. Using (10) we notice that for any $\sigma$ in $\Sigma_{2}$ we have

[TABLE]

Therefore,

[TABLE]

for any basis element $(\sigma_{0},\dots,\sigma_{n})\in\mathcal{E}(2)$. ∎

Lemma 13**.**

In \mathrm{Hom}\big{(}\mathcal{E}(2),\mathcal{E}(4)\big{)}

[TABLE]

and

[TABLE]

Proof.

We have

[TABLE]

and

[TABLE]

for any basis element $(\sigma_{0},\dots,\sigma_{n})\in\mathcal{E}(2)$. ∎

Lemma 14**.**

The map $H_{2}$ satisfies the following form of equivariance:

[TABLE]

Proof.

Since for any $\sigma,\sigma^{\prime}\in\Sigma_{2}$

[TABLE]

we have

[TABLE]

for any basis element $(\sigma_{0},\dots,\sigma_{n})\in\mathcal{E}(2)$. ∎

Proof of Theorem 10.

We will show that $H=H_{1}+H_{2}$ satisfies the hypothesis of Lemma 5. Notice that the equivariance condition follows from Lemma 12 and Lemma 14. To verify that $H$ is a chain homotopy between $(23)\circ F$ and $G$ we use Lemma 11 and Lemma 13

[TABLE]

as desired. ∎

4. The $\mathcal{E}$-algebra structure on normalized cochains

In this section we effectively describe an $\mathcal{E}$-algebra structure on the normalized cochains of simplicial sets. This will allow us to apply our construction of Cartan $i$-coboundaries to this central example. Following [BF04], we achive this in two steps corresponding to the two operad morphisms

[TABLE]

Here $Sur$ is the Surjection operad of McClure-Smith [MS03] and Berger-Fresse [BF04], the first arrow is the Table Reduction morphism of [BF04], and the second is the natural $Sur$-algebra structure introduced in [MS03] and [BF04].

4.1. The Surjection operad

We review the definition of the Surjection operad of McClure-Smith [MS03] and Berger-Fresse [BF04].

For a non-negative integer $n$ define

[TABLE]

Fix $r\geq 1$ and consider all $n\geq 0$. We make the free $\mathbb{F}_{2}$-module generated by all functions $s:\overline{n}\to\overline{r}$ into a chain complex by declaring the degree of $s:\overline{n}\to\overline{r}$ to be $r-n$ and its boundary to be

[TABLE]

where $\delta_{k}:\overline{n-1}\to\overline{n}$ is the injective order preserving function missing $k$.

We define $Sur(r)$ to be the quotient of this chain complex by the submodule generated by the functions $s:\overline{n}\to\overline{r}$ which are either non-surjective or for which $s(i)$ equals $s(i+1)$ for some $i$.

The chain complex $Sur(r)$ carries an action of $\Sigma_{r}=\{\overline{r}\to\overline{r}\ |\ \text{bijective}\}$ given by composition. The collection $Sur=\{Sur(r)\}_{r\geq 0}$ is an operad with partial composition $\circ_{p}:Sur(r^{\prime})\otimes Sur(r)\to Sur(r+r^{\prime}-1)$ defined on two generators $s:\overline{n}\to\overline{r}$ and $s^{\prime}:\overline{n^{\prime}}\to\overline{r^{\prime}}$ as follows. Represent the surjections $s$ and $s^{\prime}$ by sequences $(s(1),\dots,s(n))$ and $(s^{\prime}(1),\dots,s^{\prime}(n^{\prime}))$ and suppose that $p$ appears $k$ times in the sequence representing $s^{\prime}$ as $p=s^{\prime}(i_{1})=\dots=s^{\prime}(i_{k})$. Denote the set of all tuples

[TABLE]

by $J(k,n)$ and for each such tuple consider the subsequences

[TABLE]

Then, in $(s^{\prime}(1),\dots,s^{\prime}(n))$, replace the value $s^{\prime}(i_{t})$ by $(s(j_{t-1}),\dots,s(j_{t}))$. In addition, increase the terms $s(j)$ by $r-1$ and the terms $s^{\prime}(i)$ such that $s^{\prime}(i)>r$ by $m-1$. The surjection $s^{\prime}\circ_{p}s$ is represented by the sum, parametrized by $J(k,n)$, of these resulting sequences.

Example 15**.**

Let us consider

[TABLE]

Then, $s^{\prime}\circ_{2}s$ equals

[TABLE]

where the internal parenthesis are included solely for expository purposes.

4.2. The Table Reduction morphism

We review the definition of the Table Reduction operad morphism

[TABLE]

introduced in [BF04].

Given a basis element $(\sigma_{0},\dots,\sigma_{n})\in\mathcal{E}(r)_{n}$ we define

[TABLE]

as a sum of surjections

[TABLE]

parametrized by all $a=(a_{0},\dots,a_{n})$ with each $a_{i}\geq 1$ and $a_{0}+\cdots+a_{n}=n+r$.

For one such tuple $a$ we now describe its associated surjection $s_{a}$. Consider the table

[TABLE]

define recursively

[TABLE]

For $k\in\{1,\dots,n+r\}$ we identify $i\in\{1,\dots,n\}$ such that $A_{i-1}<k\leq A_{i}$ and define $s_{a}(k)$ to be the $(k-A_{i-1})$-th element in $(\sigma_{i}(1),\dots,\sigma_{i}(r))$ not in

[TABLE]

Example 16**.**

Let us compute TR\big{(}(23),e,(12)(34)\big{)}. Its associated table is

[TABLE]

and the tuples parametrizing the sum correspond to all permutations of $\{(1,1,4),(1,2,3),(2,2,2)\}$. Then,

[TABLE]

4.3. The diagonal and join maps

We review the natural $Sur$-algebra structure on $N^{*}(X)$ introduced in [MS03] and [BF04] using the perspective presented in [MM18b]. We notice that this $Sur$-algebra structure provides $N^{*}(X)$ with an $\mathcal{E}$-algebra structure

[TABLE]

Let $\bm{\Delta}^{n}$ be the representable simplicial set $\mathrm{Hom}_{\bm{\Delta}}(-,[n])$. We identify a morphism $[m]\to[n]$ with its image $\{v_{0},\dots,v_{m}\}\subseteq\{0,\dots,n\}$. As usual, it suffices to describe the $Surj$-structure for representable simplicial sets only.

Let $\Delta:N^{*}(\bm{\Delta}^{n})\to N^{*}(\bm{\Delta}^{n})^{\otimes 2}$ be defined on basis elements by

[TABLE]

and, for any $k\geq 2$, let $\ast:N^{*}(\bm{\Delta}^{n})^{\otimes k}\to N^{*}(\bm{\Delta}^{n})$ be defined on basis elements by

[TABLE]

We notice that $\Delta$ is equal to the composition of the Alexander-Whitney map and the doubling map

[TABLE]

and that $\ast$ is commutative.

We now describe the $Sur$-algebra structure on $N^{*}(\bm{\Delta}^{n})$. Let

[TABLE]

be a surjection, then $s(\alpha_{1}\otimes\cdots\otimes\alpha_{r})$ is defined by

[TABLE]

where $\Delta^{k}$ is recursively defined by

[TABLE]

and $\ast_{s^{-1}(i)}$ is given by applying $\ast$ to the factors in positions $s^{-1}(i)$.

Remark 17**.**

This $Surj$-algebra structure on normalized cochains of simplicial sets is induced, as explained in [MM18c], from an explicit cellular $E_{\infty}$-bialgebra structure on the geometric realization of $\bm{\Delta^{1}}$.

Appendix A Explicit examples

Let $\bm{\Delta}_{n}$ be the $n$-th representable simplicial set and denote $\{0,\dots,n\}\in N_{n}(\bm{\Delta}^{n})$ by $\mathrm{id}_{n}$. In this appendix, we will compute for an arbitrary pair of homogeneous cocycles $\alpha,\beta\in N^{*}(\bm{\Delta}_{n})$ the value

[TABLE]

in terms of $\alpha$ and $\beta$. We will restrict to $i=0,1,2$ and $n$ the smallest integer for which this value is not identically [math].

Recall that

[TABLE]

and that our Cartan $i$-coboundary is defined by

[TABLE]

where $TR:\mathcal{E}\to Surj$ is the Table Reduction morphism and

[TABLE]

[TABLE]

We remark that the cochain $\zeta_{i}(\alpha\otimes\beta)$ witnesses the relation

[TABLE]

where $-p$ and $-q$ are the degrees of $\alpha$ and $\beta$.

Example 18**.**

For $i=0$, we have

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

Example 19**.**

For $i=1$, we have

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

Example 20**.**

For $i=2$, we have

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

and

[TABLE]