


\section*{Acknowledgment}

We thank Clemens Berger, Greg Friedman, Kathryn Hess, Chris Kapulkin, Peter May, Manuel Rivera, Paolo Salvatore, Dev Sinha, Dennis Sullivan, and Bruno Vallette for insightful discussion related to this project.

A.M-M. acknowledges financial support from Innosuisse grant 32875.1~IP-ICT-1 and the hospitality of the \textit{Laboratory for Topology and Neuroscience} at EPFL.










\section{Introduction} \label{s:introduction}

Instead of simplices, in his groundbreaking work on fibered spaces Serre considered cubes as the basic shapes used to define cohomology, stating that:
\begin{displaycquote}[p.431]{serre1951homologie}
	Il est en effet evident que ces derniers se pretent mieux que les simplexes a l'etude des produits directs, et, a fortiori, des espaces fibres qui en sont la generalisation.
\end{displaycquote}
Cubical sets, a model for the homotopy category, were considered by Kan \cite{kan1955abstract, kan1956abstract} before introducing simplicial sets, they are central to nonabelian algebraic topology \cite{brown2011nonabelian}, and have become important in Voevodsky's program for univalent foundations and homotopy type theory \cite{kapulkin2020straightening, mortberg2017cubical}.
Other areas that highlight the relevance of cubical methods are applied topology, where cubical complexes are ubiquitous in the study of images \cite{tomasz2004computational}, condensed matter physics, where models on cubical lattices are central \cite{baxter1985exactlysolved}, and geometric group theory \cite{gromov1987hyperbolic}, where fundamental results have been obtained considering actions on certain cube complexes characterized combinatorially \cite{agol2013haken}.

Cubical cochains are equipped with the \textit{Serre algebra structure}, a lift to the cochain level of the graded ring structure in cohomology.
Using an acyclic carrier argument it can be shown that this product is commutative up to coherent homotopies in a non-canonical way.
The study of such objects, referred to as $E_\infty$-algebras, has a long history, where (co)homology operations \cite{steenrod1962cohomology, may1970general}, the recognition of infinite loop spaces \cite{boardman1973homotopy, may1972geometry} and complete algebraic models of the $p$-adic homotopy category \cite{mandell2001padic} are key milestones.
The goal of this work is to introduce a description of an explicit $E_\infty$-algebra structure naturally extending the Serre algebra structure, and relate it to one on simplicial cochains extending the Alexander--Whitney algebra structure.

We use the combinatorial model of the $E_\infty$-operad ${\forget(\M)}$ obtained from the finitely presented prop $\mathcal{M}$ introduced in \cite{medina2020prop1}.
The resulting ${\forget(\M)}$-algebra structure on cubical cochains is induced from a natural $\mathcal{M}$-bialgebra structure on the chains of standard cubes, which is determined by only three linear maps.
To our knowledge, this is the first effective construction of an $E_\infty$-algebra structure on cubical cochains.
Non-constructively, this result could be obtained using a lifting argument based on the cofibrancy of the reduced version of the operad ${\forget(\M)}$ in the model category of operads \cite{hinich1997homological, berger2003modelcategory}, but this existence statement misses the rich combinatorial structure present in our effective construction.

As described in \cite{medina2020prop1}, the operad ${\forget(\M)}$ also acts on simplicial cochains extending the Alexander--Whitney algebra structure.
We use a construction of Cartan and Serre to relate these cubical and simplicial $E_\infty$-structures.
More specifically, in \cite[p. 442]{serre1951homologie}, Serre describes for any topological space $\mathfrak{Z}$ a natural quasi-isomorphism
\begin{equation} \label{e:cs on singular cochains}
	\cScochains(\mathfrak{Z}) \to \sScochains(\mathfrak{Z})
\end{equation}
between its cubical and simplicial singular cochains.
Furthermore, he states this to be a quasi-isomorphism of algebras with respect to the Serre and Alexander--Whitney structures.
In the present work we deduce from a statement at the level of general simplicial sets that this map is in fact a quasi-isomorphisms of $E_\infty$-algebras.
More specifically, let $\cubify$ be the right adjoint to the triangulation functor from cubical to simplicial sets.
We construct a natural quasi-isomorphism of $E_\infty$-algebras
\[
\ccochains(\cubify Y) \to \scochains(Y)
\]
for any simplicial set $Y$, which factors the Cartan--Serre map \eqref{e:cs on singular cochains} when $Y = \sSing(\mathfrak{Z})$ so that the other map is induced from a cubical map.

We now mention three application of the contributions in this paper.
For every prime $p$, the mod $p$ cohomology of a space is equipped with natural stable endomorphisms known as Steenrod operations \cite{steenrod1962cohomology}.
Following an operadic viewpoint developed by May \cite{may1970general}, in \cite{medina2021may_st} we effectively described a May--Steenrod structure on ${\forget(\M)}$, i.e., a compatible choice of elements in ${\forget(\M)}$ that represent Steenrod operations on the mod~$p$ homology of ${\forget(\M)}$-algebras.
Since, as proven in this article, cubical cochains are equipped with a ${\forget(\M)}$-structure, we use this May--Steenrod structure to extend the cubical cup-$i$ products of \cite{kadeishvili1999coproducts} and \cite{pilarczyk2016cubical} to a family of cochain level multioperations representing Steenrod operations at every prime.
Furthermore, the effective nature of these constructions permitted the implementation of these multioperations and associated Steenrod operations in the computer algebra system \href{https://comch.readthedocs.io/en/latest/}{\texttt{ComCH}} \cite{medina2021comch}.

For a closed smooth manifold $M$, in \cite{medina2021flowing} we compared a cochain complex generated by manifolds with corners over $M$, and the complex of cubical cochains defined by a choice of cubulation of $M$.
We used a canonical vector field associated to the cubulation to compare multiplicatively these two models of ordinary cohomology, whose product structures are respectively given by transverse intersection and the Serre product.
With the explicit description introduced here of an $E_\infty$-structure on cubical cochains, we expect to build on this multiplicative comparison and, using a coherent family of vector fields, describe the corresponding $E_\infty$-structure on geometric cochains extending the transverse intersection product.
For more details regarding this geometric model of cohomology please consult \cite{medina2022foundations}.

Our construction of an $E_\infty$-algebra structure on cubical cochains is obtained by dualizing an $E_\infty$-coalgebra structure on cubical chains.
In the fifties, Adams introduced in \cite{adams1956cobar} a comparison map
\[
\theta_\mathfrak{Z} \colon \cobar \sSchains(\mathfrak{Z},z) \to \cSchains(\Omega_z \mathfrak{Z})
\]
from his cobar construction on the simplicial singular chains of a pointed space $(\mathfrak{Z},z)$ to the cubical singular chains on its based loop space $\Omega_z \mathfrak{Z}$.
This comparison map is a quasi-isomorphism of algebras, which was shown by Baues \cite{baues1998hopf} to be one of bialgebras by considering Serre's cubical coproduct, see also \cite{galvez-carrilo202hopf1} for a generalization of Baues construction.
In \cite{medina2021cobar} we use the contributions of this paper to generalize Baues' result by showing that Adams' comparison map is a quasi-isomorphism of $E_\infty$-bialgebras, i.e., of monoids in the category of ${\forget(\M)}$-coalgebras, and to relate the cobar and Kan loop group constructions as functors to the category of $E_\infty$-coalgebras.

\subsection*{Outline}

We recall some basic notions from homological algebra and category theory in \cref{s:preliminaries}.
The required concepts from the theory of operads and props is reviewed in \cref{s:props}, including the definition of the prop $\mathcal{M}$.
\cref{s:action} contains our main contribution; an explicit natural $\mathcal{M}$-bialgebra structure on the chains of standard cubes and, from it, a natural $E_\infty$-coalgebra structure on the chains of cubical sets.
The comparison between simplicial and cubical cochains is presented in \cref{s:comparison}, where we show that the Cartan--Serre map is a quasi-isomorphism of $E_\infty$-algebras.

\section{The Cartan--Serre map} \label{s:comparison}

Let us consider, with their usual CW structures, the topological simplex $\mathbb{\Delta}^n$ and the topological cube $\mathbb{I}^n$.
In \cite[p. 442]{serre1951homologie}, Serre described a quasi-isomorphism of algebras between the cubical and simplicial singular cochains of a topological space.
It is given by precomposing with a canonical cellular map $\cs \colon \mathbb{I}^n \to \mathbb{\Delta}^n$ also considered in \cite[p.199]{eilenberg1953acyclic} where it is attributed to Cartan.

The goal of this section is to deduce from a more general categorical statement that this comparison map between singular cochains of a space is a quasi-isomorphism of $E_\infty$-algebras.

\subsection{Simplicial sets} \label{ss:simplicial sets}


We denote the \textit{simplex category} by $\triangle$, the category of \textit{simplicial sets} $\mathsf{Fun}(\triangle^\mathrm{op}, \mathsf{Set})$ by $\mathsf{sSet}$, and the standard $n$-simplex $\yoneda \big( [n] \big)$ by $\triangle^n$.
As usual, we denote an element in $\triangle^n_m$ by a non-decreasing tuple $[v_0, \dots, v_m]$ with $v_i \in \{0, \dots, n\}$.
The \textit{Cartesian product} of simplicial sets is defined by the product of functors.
The \textit{simplicial $n$-cube} $\scube{n}$ is the $n^\th$-fold Cartesian product of $\triangle^1$.


We will use the following model of the topological $n$-simplex:
\[
\mathbb{\Delta}^n = \big\{ (y_1, \dots, y_n) \in \mathbb{I}^n \mid i \leq j \Rightarrow y_i \geq y_j \big\},
\]
whose cell structure associates $[v_0, \dots, v_m]$ with the subset
\begin{equation} \label{e:cell structure of gsimplex}
	\Big\{ \big( \underbrace{1, \dots, 1}_{v_0}, \underbrace{y'_1, \dots y'_1}_{v_1-v_0}, \dots, \underbrace{y'_m, \dots y'_m}_{v_m-v_{m-1}}, \underbrace{0, \dots, 0}_{n-v_m} \big) \mid y'_1 \geq \dots \geq y'_m \Big\}.
\end{equation}
The spaces $\mathbb{\Delta}^n$ define a functor $\mathbb{\Delta}^{(-)} \colon \triangle \to \mathsf{CW} $ with
\begin{align*}
	\sigma_i(x_1, \dots, x_n) &= (x_1, \dots, \widehat x_i, \dots, x_n) \\
	\delta_0(x_1, \dots, x_n) &= (1, x_1, \dots, x_n), \\
	\delta_i(x_1, \dots, x_n) &= (x_1, \dots, x_i, x_i, \dots, x_n), \\
	\delta_n(x_1, \dots, x_n) &= (x_1, \dots, x_n, 0).
\end{align*}
Its Yoneda extension is the \textit{geometric realization} functor.
It has a right adjoint $\sSing \colon \mathsf{Top} \to \mathsf{sSet}$ referred to as the \textit{simplicial singular complex} satisfying
\[
\sSing(\mathfrak{Z})_n = \mathsf{Top}(\mathbb{\Delta}^n, \mathfrak{Z})
\]
for any topological space $\mathfrak{Z}$.


The functor of (\textit{normalized}) \textit{chains} $\schains \colon \mathsf{sSet} \to \mathsf{Ch}$ is the composition of the geometric realization functor and that of cellular chains.
We denote the composition $\schains \circ \sSing$ by $\sSchains$ and omit the superscript $\triangle$ if no confusion may result from doing so.
For any $n \in \mathbb{N}$, define $\epsilon \colon \chains(\triangle^n) \to \k$ by
\[
\epsilon \big( [v_0, \dots, v_m] \big) =
\begin{cases}
	1 & \text{ if } m = 0, \\ 0 & \text{ if } m>0,
\end{cases}
\]
$\Delta \colon \chains(\triangle^n) \to \chains(\triangle^n)^{\otimes 2}$ by
\[
\Delta \big( [v_0, \dots, v_m] \big) = \sum_{i=0}^m [v_0, \dots, v_i] \otimes [v_i, \dots, v_m],
\]
and $\ast \colon \chains(\triangle^n)^{\otimes 2} \to \chains(\triangle^n)$ by
\[
\left[v_0, \dots, v_p \right] \ast \left[v_{p+1}, \dots, v_m\right] = \begin{cases} (-1)^{p+|\sigma|} \left[v_{\sigma(0)}, \dots, v_{\sigma(m)}\right] & \text{ if } v_i \neq v_j \text{ for } i \neq j, \\
	0 & \text{ if not}, \end{cases}
\]
where $\sigma$ is the permutation that orders the totally ordered set of vertices and $(-1)^{|\sigma|}$ is its sign.
As shown in \cite[Theorem 4.2]{medina2020prop1} the assignment
\[
\counit \mapsto \epsilon, \quad \coproduct \mapsto \Delta, \quad \product \mapsto \ast,
\]
defines a natural $\mathcal{M}$-bialgebra on the chains of standard simplicial sets.

\subsection{The Eilenberg--Zilber maps}

For any permutation $\sigma \in \mathbb{S}_n$ let
\[
\gi_\sigma \colon \mathbb{\Delta}^n \to \mathbb{I}^n
\]
be the inclusion defined by $(x_1, \dots, x_n) \mapsto (x_{\sigma(1)}, \dots, x_{\sigma(n)})$.
If $e$ is the identity permutation, we denote $\mathfrak{i}_{e}$ simply as $\mathfrak{i}$.
The maps $\set{\mathfrak{i}_\sigma}_{\sigma \in \mathbb{S}_n}$ define a subdivision of $\mathbb{I}^n$ making it isomorphic to $\bars[\big]{\scube{n}}$ in $\mathsf{CW}$.
Using this identification, the identity map induces a cellular map
\[
\ez \colon \mathbb{I}^n \to \bars[\big]{\scube{n}}
\]
whose induced chain map is denoted by
\[
\EZ \colon \chains(\square^n) \to \chains \big( \scube{n} \big).
\]
Using the isomorphism $\chains(\square^n) \cong \chains(\square^1)^{\otimes n} \cong \chains(\triangle^1)^{\otimes n}$, this map agrees with the usual Eilenberg--Zilber maps $\chains(\triangle^1)^{\otimes n} \to \chains \big( \scube{n} \big)$.


\subsection{The Cartan-Serre maps}

Let
\[
\cs \colon \mathbb{I}^n \to \mathbb{\Delta}^n
\]
be the cellular map defined by $\cs(x_1, \dots, x_n) = (x_1,\ x_1 x_2, \, \dots \, , \ x_1 x_2 \dotsm x_n)$.
For any topological space $\mathfrak{Z}$ we a chain map
\[
\begin{tikzcd} [column sep=small, row sep=0]
	\CS_{\Schains(\mathfrak{Z})} \colon &[-20] \sSchains(\mathfrak{Z}) \arrow[r] & \cSchains (\mathfrak{Z}) \\ &
	(\mathbb{\Delta}^n \to \mathfrak{Z}) \arrow[r, mapsto] & (\mathbb{I}^n \xra{\cs} \mathbb{\Delta}^n \to \mathfrak{Z}).
\end{tikzcd}
\]
These maps were considered in \cite[p. 442]{serre1951homologie} where it was stated that $\CS_{\Schains(\mathfrak{Z})}$ is a natural quasi-isomorphisms of coalgebras.
We remark that in this reference the topological $n$-simplex is modeled as a subset of $\mathbb{R}^{n+1}$ instead of $\mathbb{R}^n$.


\begin{definition}
	The chain map
	\[
	\CS \colon \chains(\square^n) \to \chains(\triangle^n)
	\]
	is defined by applying the functor of cellular chains to $\cs \colon \mathbb{I}^n \to \mathbb{\Delta}^n$.
\end{definition}

\begin{lemma}
	The chain map $\CS \colon \chains(\square^n) \to \chains(\triangle^n)$ is a quasi-isomorphism of coalgebras.
\end{lemma}

\begin{proof}
	Since $\CS$ is induced from a cellular map between contractible spaces it is a quasi-isomorphism compatible with the counit.
	To study the compatibility of $\CS$ with coproducts consider $n > 0$, since the case $n = 0$ is immediate, and let
	\[
	\Delta \big( [0,1]^{\otimes n} \big) = \sum_{\lambda \in \Lambda} \pm \ x_1^{(\lambda)} \otimes \cdots \otimes x_n^{(\lambda)} \otimes y_1^{(\lambda)} \otimes \cdots \otimes y_n^{(\lambda)},
	\]
	where $\Lambda$ parameterizes all choices of $x_i^{(\lambda)} \in \{[0], [0,1]\}$ and $y_i^{(\lambda)} \in \{[0,1], [1]\}$ such that
	\begin{align*}
		x_i^{(\lambda)} = [0]   & \iff y_i^{(\lambda)} = [0,1], \\
		x_i^{(\lambda)} = [0,1] & \iff y_i^{(\lambda)} = [1].
	\end{align*}
	By \cref{l:cs explicit}, the summands above not sent to $0$ by $\CS \otimes \CS$ are those basis elements for which $x_i^{(\lambda)} = [0]$ implies $x_j^{(\lambda)} = [0]$ for all $i < j$.
	For any one such summand, its sign is positive and its image by $\CS \otimes \CS$ is $[0, \dots, k] \otimes [k, \dots, n]$ where $k+1 = \min \{i \mid x_i^{(\lambda)} = [0]\}$ or $k = n$ if this set is empty.
	The summands $[0, \dots, k] \otimes [k, \dots, n]$ are precisely those appearing when applying the Alexander--Whitney coproduct to $[0, \dots, n] = \CS \big( [0,1]^{\otimes n} \big)$.
\end{proof}

\subsection{No-go results}

Since $\CS$ was shown to be a coalgebra map and $\EZ$ is well known to be one, one may hope for higher structures to be preserved by these maps.
We now provide some examples constraining the scope of these expectations.

\begin{example*}
	Let us consider, as in \cref{e:prop cup-i}, the following cup-$1$ coproduct
	\[
	\Delta_1 = (\ast \otimes \mathsf{id}) \circ (\mathsf{id} \otimes (12) \Delta) \circ \Delta.
	\]
	As a map $\chains(\square^2; {\mathbb{F}_2}) \to \chains(\square^2; {\mathbb{F}_2})^{\otimes 2}$, it satisfies
	\begin{multline*}
		\Delta_1 \big( [01] [01] \big) = \\
		[01][01] \otimes [1][01] + [01][1] \otimes [01][01] + [0][01] \otimes [01][01] - [01][01] \otimes [01][0].
	\end{multline*}
	Therefore,
	\begin{multline*}
		\EZ^{\otimes 2} \circ \, \Delta_1 \big( [01][01] \big) = \\
		\big( 011\times001 + 001\times011 \big) \otimes 11 \times 01 \ +\
		01\times11 \otimes \big( 011\times001 + 001\times011 \big) \ +\ \\
		00\times01 \otimes \big( 011\times001 + 001\times011 \big) \ +\
		\big( 011\times001 + 001\times011 \big) \otimes 01\times00.
	\end{multline*}
	On the other hand, as a map $\chains(\triangle^n; {\mathbb{F}_2}) \to \chains(\triangle^n; {\mathbb{F}_2})^{\otimes 2}$ we have
	\[
	\Delta_1 [0,1,2] = [0,1,2] \otimes [0,1] + [0,2] \otimes [0,1,2] + [0,1,2] \otimes [1,2].
	\]
	Therefore,
	\begin{multline*}
		\Delta_1 \circ \EZ \big( [01] [01] \big) = \Delta_1 \big( 011\times001 + 001\times011 \big) = \\
		011\times001 \otimes 01\times00 \ + \
		01\times01 \otimes 011\times001 \ + \
		011\times001 \otimes 11\times01 \ + \ \\
		001\times011 \otimes 00\times01 \ + \
		01\times01 \otimes 001\times011 \ + \
		001\times011 \otimes 01\times11.
	\end{multline*}
	We conclude that $\EZ^{\otimes 2} \circ \, \Delta_1 \big( [01][01] \big) \neq \Delta_1 \circ \EZ \big( [01] [01] \big)$ since, for example, the basis element $01\times11 \otimes 011\times001$ appears in the left sum but not in the right one.
\end{example*}

\begin{example*}
	The Cartan--Serre map does not preserve $\mathcal{M}$-structures.
	Consider $\CS \colon \chains(\square^2; {\mathbb{F}_2}) \to \chains(\triangle^2; {\mathbb{F}_2})$.
	Since $\CS \big( [0][01] \big) = 0$ we have
	\[
	\CS \big( [1][1] \big) \ast \CS \big( [0][01] \big) = 0
	\]
	but
	\[
	\CS \Big( ([1][1]) \ast ([0][01]) \Big) =
	\CS \Big( [01][01] \Big) = [012].
	\]
	The reason for this incompatibility is that $\ast$ in the simplicial context is commutative whereas in the cubical context it is not.
	From the same observation we deduce the next construction.
\end{example*}

\begin{example*}
	The Cartan--Serre map does not preserve ${\forget(\M)}$-structures.
	Let
	\[
	\widetilde\Delta_1 = (\ast \otimes \mathsf{id}) \circ (12) (\mathsf{id} \otimes (12) \Delta) \circ \Delta.
	\]
	It can be computed that
	\[
	\CS\Big( \widetilde\Delta_1 \big( [01][01] \big) \Big) =
	T \Delta_1 \big( [012] \big)
	\]
	whereas
	\[
	\widetilde\Delta_1 \circ \CS \big( [01][01] \big) =
	\Delta_1 \big( [012] \big).
	\]
	We will restrict the ${\forget(\M)}$-coalgebra structure on simplicial and cubical chains to a sub-$E_\infty$-operad where this symmetry incompatibility is avoided.
\end{example*}

\subsection{$E_\infty$-coalgebra preservation}

The operad ${\forget(\M_{sl})}$ is generated as a suboperad of ${\forget(\M)}$ by all so-called \textit{surjection-like graphs}, i.e., immerse connected graphs of the form:
\input{aux/surjection_like}
where there are no hidden vertices and the strands are joined so that the associated maps $\{1, \dots, k_j\} \to \{1, \dots, n+k\}$ are order-preserving.
We notice that the subcomplex of surjection-like $(1,r)$-graphs is contractible using the same chain contraction employed in \cite{medina2020prop1}.
This implies that the suboperad ${\forget(\M_{sl})}$ of ${\forget(\M)}$ is also $E_\infty$, and that the restriction to ${\forget(\M_{sl})}$ of the ${\forget(\M)}$-structures on simplicial and cubical chains provides them with another $E_\infty$-structure.

We devote the rest of this subsection to the proof of the following key result.

\begin{theorem} \label{t:main local}
	The chain map $\CS \colon \chains(\square^n) \to \chains(\triangle^n)$ is a quasi-isomorphism of ${\forget(\M_{sl})}$-coalgebras.
\end{theorem}

By naturality, it suffices to show that for any $\Gamma \in {\forget(\M)}(r)$ represented by a surjection-like $(1,r)$-graph we have
\begin{equation*}
	\CS^{\otimes r} \! \circ \, \Gamma \big( [0,1]^{\otimes n} \big) =
	\Gamma \circ \CS \big( [0,1]^{\otimes n} \big).
\end{equation*}

Let us start by providing a more explicit description of the $\CS$ map.

\begin{lemma} \label{l:cs explicit}
	Let $x = x_1 \otimes \cdots \otimes x_n \in \chains(\square^n)_m$ be a basis element with $x_{q_i} = [0,1]$ for all $\{q_1 < \dots < q_m\}$.
	Let $p = \min \set[\big]{i \mid x_i = [0]}$ or $p = n+1$ if empty, then
	\[
	\CS(x) =
	\begin{cases}
		\big[ q_1-1, \ \dots , \ q_m-1, \ p-1 \big] & \text{ if } p > q_m, \\
		\hfil 0 & \text{ otherwise}.
	\end{cases}
	\]
\end{lemma}

\begin{proof}
	It can be verified using the cell structure of $\mathbb{\Delta}^n$ described in \cref{e:cell structure of gsimplex}, the fact that the image of $\cs \circ \, \delta^0_i \colon \mathbb{I}^{n-1} \to \mathbb{\Delta}^n$ for $i \in \set{1,\dots,n-1}$ is in a lower dimensional skeleton of $\mathbb{\Delta}^n$, and the identities
	\[
	\cs \circ \, \delta_i^1 = \delta_{i-1} \circ \cs,
	\qquad
	\cs \circ \, \delta_n^0 = \delta_n \circ \cs,
	\]
	for $i \in \set{1, \dots, n}$.
\end{proof}

We will consider the basis of $\chains(\square^n)$ as a poset with
\[
(x_1 \otimes \cdots \otimes x_n) \leq (y_1 \otimes \cdots \otimes y_n)
\]
if and only if $x_i \leq y_i$ for each $i \in \{1, \dots, n\}$ with respect to $[0] < [0,1] < [1]$.

\begin{lemma}
	Let $\Delta^{r-1}$ be the $(r-1)^\th$-fold iterated Serre coproduct.
	If
	\[
	\Delta^{r-1} \big([0,1]^{\otimes n}\big) =
	\sum \pm \ x{(1)} \otimes \cdots \otimes x{(r)}
	\]
	with each $x(i) \in \chains(\square^n)$ a basis element, then $x{(1)} \leq \cdots \leq x{(r)}$.
\end{lemma}

\begin{proof}
	For $r = 2$ we have for every $i \in \{1, \dots, n\}$ that
	\begin{align*}
		x(1)_i = [0]   & \iff x(2)_i = [0,1], \\
		x(1)_i = [0,1] & \iff x(2)_i = [1],
	\end{align*}
	and that neither $x(1)_i = [1]$ or $x(2)_i = [0]$ can occur, hence $x(1) \leq x(2)$.
	The claim for $r > 2$ follows from a straightforward induction argument.
\end{proof}

\begin{lemma}
	Let $x, y, z \in \chains(\square^n)$ be basis elements.
	If $x, y \leq z$ then either $(x \ast y) = 0$ or $(x \ast y) \leq z$.
\end{lemma}

\begin{proof}
	Recall that
	\begin{align*}
		(x_1 \otimes \cdots \otimes x_n) \ast (y_1 \otimes \cdots \otimes y_n) =
		(-1)^{|x|} \sum_{i=1}^n x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i}) \, y_{>i}.
	\end{align*}
	By assumption, for every $i \in \{1, \dots, n\}$ we have $x_{<i} \leq z_{<i}$ and $y_{>i} \leq z_{>i}$.
	If $x_i \ast y_i \neq 0$ then $x_i \ast y_i = [0,1]$ and either $x_i = [1]$ or $y_i = [1]$ which implies $z_i = [1]$ as well, so $x_i \ast y_i \leq z_i$.
\end{proof}

\begin{lemma}
	Let $x, y \in \chains(\square^n)$ be basis elements.
	If $x \leq y$ then
	\begin{equation} \label{e:cs collapse as algebra map}
		\CS(x \ast y) = \CS(x) \ast \CS(y).
	\end{equation}
\end{lemma}

\begin{proof}
	We present this proof in the form of three claims.
	We use \cref{l:cs explicit}, the assumption $x \leq y$, and the fact that the join of basis elements in $\chains(\triangle^n)$ sharing a vertex is $0$ without explicit mention. \newline

	\noindent \textit{Claim 1}.
	If $\CS(x) = 0$ or $\CS(y) = 0$ then for every $i \in \{1, \dots, n\}$
	\begin{equation} \label{e:zero for join}
		\CS \big( x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i}) \, y_{>i} \big) = 0.
	\end{equation}

	Assume $\CS(x) = 0$, that is, there exists a pair $p < q$ such that $x_p = [0]$ and $x_q = [0,1]$, then \eqref{e:zero for join} holds since:
	\begin{enumerate}
		\item If $i > q$, then $x_p$ and $x_q$ are part of $x_{<i}$.
		\item If $i = q$, then $x_q \ast y_q = 0$ for any $y_q$.
		\item If $i < q$, then $\epsilon(x_{>i}) = 0$.
	\end{enumerate}
	Similarly, if there is a pair $p < q$ such that $y_p = [0]$ and $y_q = [0,1]$,  then \eqref{e:zero for join} holds since:
	\begin{enumerate}
		\item If $i < p$, then $y_p$ and $y_q$ are part of $y_{>i}$.
		\item If $i = p$, then $x_i = [0]$ and $x_i \ast y_i = 0$.
		\item If $i > p$, then either $x_i \ast y_i = 0$ or $x_i \ast y_i = [0,1]$ and $x_p = [0]$.
	\end{enumerate}
	This proves the first claim and identity \eqref{e:cs collapse as algebra map} under its hypothesis.

	\noindent \textit{Claim 2}.
	If $\CS(x) \neq 0$ and $\CS(y) \neq 0$ then
	\[
	\CS(x \ast y) =
	\CS \big( x_{<p_x} \epsilon(y_{<p_x}) \otimes \, x_{p_x} \! \ast y_{p_x} \otimes \epsilon(x_{>p_x}) \, y_{>p_x} \big)
	\]
	if $p_x = \min \big\{ i \mid x_i = [0] \big\}$ is well-defined and $x \ast y = 0$ if not.

	Assume $p_x$ is not well-defined, i.e., $x_i \neq [0]$ for all $i \in \{1, \dots, n\}$.
	Given that $x \leq y$ we have that $[0] < x_i$ implies $x_i \ast y_i = 0$, and the claim follows in this case.

	Assume $p_x$ is well-defined.
	We will show that for all $i \in \{1,\dots,n\}$ with the possible exception of $i = p_x$ we have
	\begin{equation} \label{e:case main lemma third claim}
		\CS \big( x_{<i} \, \epsilon(y_{<i}) \otimes \, x_{i} \! \ast y_{i} \otimes \epsilon(x_{>i}) \, y_{>i} \big) = 0
	\end{equation}
	This follows from:
	\begin{enumerate}
		\item If $i < p_x$ and $x_i = [1]$ then $y_i = [1]$ and $x_i \ast y_i = 0$.
		\item If $i < p_x$ and $x_i = [0,1]$ then $x_i \ast y_i = 0$ for any $y_i$.
		\item If $i > p_x$ then \cref{l:cs explicit} implies the claim since $x_{p_x} = [0]$ and $x_i \ast y_i \neq 0$ iff $x_i \ast y_i = [0,1]$.
	\end{enumerate}

	\noindent \textit{Claim 3}.
	If $\CS(x) \neq 0$ and $\CS(y) \neq 0$ then \eqref{e:cs collapse as algebra map} holds.

	Let us assume that $\big\{ i \mid x_i = [0] \big\}$ is empty, which implies the analogous statement for $y$ since $x \leq y$.
	Since neither of $x$ nor $y$ have a factor $[0]$ in them, \cref{l:cs explicit} implies that the vertex $[n]$ is in both $\CS(x)$ and $\CS(y)$, which implies $\CS(x) \ast \CS(y) = 0$ as claimed.

	Assume now that $p_x = \big\{ i \mid x_i = [0] \big\}$ is well defined, and let $\{q_1 < \dots < q_m\}$ with $x_{q_i} = [0,1]$ for $i \in \{1,\dots,m\}$.
	Since $\CS(x) \neq 0$ \cref{l:cs explicit} implies that $p_x > q_m$, so $\epsilon(x_{>p_x}) = 1$ and Claim 2 implies
	\[
	\CS(x \ast y) =
	\CS \big( x_{<p_x} \epsilon(y_{<p_x}) \otimes \, x_{p_x} \! \ast y_{p_x} \otimes y_{>p_x} \big).
	\]
	We have the following cases:
	\begin{enumerate}
		\item If $\epsilon(y_{<p_x}) = 0$ then there is $q_i$ such that $y_{q_i} = [0,1]$ so $[q_i-1]$ is in both $\CS(x)$ and $\CS(y)$.
		\item If $\epsilon(y_{p_x}) \neq 0$ and $y_{p_x} \in \{[0], [0,1]\}$ then $x_{p_x} \ast y_{p_x} = 0$ and $[p_x-1]$ is in both $\CS(x)$ and $\CS(y)$.
		\item If $\epsilon(y_{p_x}) \neq 0$ and $y_{p_x} = [1]$ let $\{\ell_1 < \dots < \ell_k\}$ be such that $y_{\ell_j} = [0,1]$ and let $p_y > \ell_k$ be either $n+1$ or $\min\{j \mid y_j = \{0\}\}$ then
		\begin{align*}
			\CS(x \ast y) & =
			\CS \big( x_{< p_x} \otimes x_{p_x} \ast y_{p_x} \otimes y_{> p_y} \big) \\ & =
			[q_1-1, \dots, q_m-1, p_x-1, \ell_1-1, \dots, \ell_k-1, p_y-1] \\ & =
			\CS(x) \ast \CS(y)
		\end{align*}
	\end{enumerate}
	which concludes the proof of Claim 3 and this lemma.
\end{proof}

This sequence of lemmas provides a proof of \cref{t:main local} using the decomposition of any surjection-like graph into pieces \coproduct \ and \product and the fact that ${\forget(\M_{sl})}$ is generated by these.

In the next subsection we will use \cref{t:main local} to deduce from a more general categorical statement that, for any topological space $\mathfrak{Z}$, the chain map $\CS_{\Schains(\mathfrak{Z})} \colon \sSchains(\mathfrak{Z}) \to \cSchains(\mathfrak{Z})$ is a quasi-isomorphism of $E_\infty$-coalgebras, specifically, of ${\forget(\M_{sl})}$-coalgebras.

\subsection{Categorical reformulation}

The assignment $2^n \mapsto \scube{n}$ defines a functor $\square \to \mathsf{sSet}$ with $\delta_i^\varepsilon \colon \scube{n} \to \scube{(n+1)}$ inserting $[\varepsilon, \dots, \varepsilon]$ as $i^\th$ factor and $\sigma_i \colon \scube{(n+1)} \to \scube{n}$ removing the $i^\th$ factor.
Its Yoneda extension is denoted by
\[
\triangulate \colon \mathsf{cSet} \to \mathsf{sSet}.
\]
This functor admits a right adjoint
\[
\cubify \colon \mathsf{sSet} \to \mathsf{cSet}
\]
defined, as usual, by
\[
\cubify(Y)(2^n) = \mathsf{sSet} \big( \scube{n}, \, Y \big).
\]
Although we do not use this fact, we mention that, as proven in \cite[\S 8.4.30]{cisinski2006presheaves}, the pair $(\triangulate,\, \cubify)$ defines a Quillen equivalence when $\mathsf{sSet}$ and $\mathsf{cSet}$ are considered as model categories.




\begin{definition}
	The simplicial map $\ccs \colon \scube{n} \to \triangle^n$ is defined by
	\[
	[\varepsilon_0^1, \dots, \varepsilon_m^1]
	\times \dots \times
	[\varepsilon_0^n, \dots, \varepsilon_m^n]
	\mapsto
	[v_0, \dots, v_m]
	\]
	where $v_i = \varepsilon_i^1 + \varepsilon_i^1 \varepsilon_i^2 + \dots + \varepsilon_i^1 \dotsm \varepsilon_i^n$.
\end{definition}

We observe that the maps $\cs$ and $\bars{\ccs} \circ \ez$ agree.

\begin{definition}
	Let $Y$ be a simplicial set.
	The map
	\[
	\CS_Y \colon \schains(Y) \to \cchains(\cubify Y)
	\]
	is the linear map induced by sending a simplex $y \in Y_n$ to the composition
	\[
	\scube{n} \xra{\ccs} \triangle^n \xra{\xi_y} Y
	\]
	where $\xi_y \colon \triangle^n \to Y$ is the characteristic map determined by $\xi_y \big( [n] \big) = y$.
\end{definition}

\begin{theorem} \label{t:main comparison}
	For any simplicial set $Y$ the map $\CS_Y \colon \schains(Y) \to \cchains(\cubify Y)$ is a quasi-isomorphism of $E_\infty$-coalgebras, specifically, of ${\forget(\M_{sl})}$-coalgebras.
\end{theorem}

\begin{proof}
	This is a direct consequence of \cref{t:main local} following from standard category theory argument which we now present.
	Consider the isomorphism
	\[
	\chains(\cubify Y) \cong \displaystyle\bigoplus_{n\in\mathbb{N}} \chains(\square^n) \otimes \k\set[\Big]{\mathsf{sSet} \big( \scube{n}, \triangle^n \big)} \Big/ \sim \,
	\]
	and the canonical inclusions:
	\[
	\begin{tikzcd} [row sep=0, column sep=small]
		\chains(\square^n) \arrow[r] &
		\displaystyle\bigoplus_{m \in \mathbb{N}} \mathrm{Hom} \big( \chains(\square^m),\, \chains(\square^n) \big) \\
		(2^m \xra{\delta} 2^n) \arrow[r, mapsto] &
		\big(\chains(\square^m) \xra{\chains(\delta)} \chains(\square^n)\big)
	\end{tikzcd}
	\]
	and
	\[
	\begin{tikzcd} [row sep=0, column sep=small]
		\displaystyle \bigoplus_{n \in \mathbb{N}}
		\k\set[\Big]{\mathsf{sSet}\big(\scube{n}, \triangle^n\big)} \arrow[r] &
		\displaystyle \bigoplus_{n \in \mathbb{N}}
		\mathrm{Hom} \Big( \chains \big( \scube{n} \big),\, \chains(\triangle^n) \Big) \\
		\big( \scube{n} \xra{f} \triangle^n \big) \arrow[r, mapsto] &
		\Big( \chains \big( \scube{n} \big) \xra{\chains(f)} \chains(\triangle^n) \Big).
	\end{tikzcd}
	\]
	We can use these and the naturality of $\EZ$ to construct the following chain map which is an isomorphism onto its image.
	\[
	\begin{tikzcd}[column sep=small, row sep=0]
		\chains(\cubify Y) \arrow[r] &
		\displaystyle\bigoplus_{n\in\mathbb{N}} \mathrm{Hom} \big( \chains(\square^n),\, \chains(Y)\, \big) \\
		(\delta \otimes f) \arrow[r, mapsto]& \big( \chains(f) \circ \EZ \circ \chains(\delta) \, \big).
	\end{tikzcd}
	\]
	Using the naturality of the map $\Gamma^\square \colon \chains(\cubify Y) \to \big( \chains(\cubify Y) \big)^{\otimes r}$ associated to any $\Gamma \in {\forget(\M_{sl})}$ we have that
	$\Gamma^\square(\delta \otimes f)$ corresponds to $(\chains(f) \circ \EZ)^{\otimes r} \circ \Gamma^\square \circ \chains(\delta)$.
	On the other hand, the map $\CS_Y$ corresponds to
	\[
	\begin{tikzcd}[column sep=small, row sep=0]
		\chains(Y)_n \arrow[r] & \chains(\cubify Y)_n \\
		y \arrow[r,mapsto]& \big( \chains(\xi_y) \circ \CS \big)
	\end{tikzcd}
	\]
	where $\xi_y \colon \triangle^n \to Y$ is determined by $\xi_y \big( [n] \big) = y$, and we used that $\CS = \chains(\ccs) \circ \EZ$ to ensure the above assignment is well defined.
	The image of $\Gamma^\triangle(y)$ corresponds to $\chains(\xi_y)^{\otimes r} \circ \Gamma^\triangle \circ \CS$.
	So the claim follows from the identity
	\begin{align*}
		\Gamma^\square \big( 2^n \otimes (\xi_y \circ \ccs) \big) &=
		(\chains(\xi_y \circ \ccs) \circ \EZ)^{\otimes r} \circ \Gamma^\square  \\ &=
		\chains(\xi_y)^{\otimes r} \circ \CS^{\otimes r} \circ \, \Gamma^\square  \\ &=
		\chains(\xi_y)^{\otimes r} \circ \, \Gamma^\triangle \circ \CS,
	\end{align*}
	where we used that $\CS^{\otimes r} \circ \, \Gamma^\square = \Gamma^\triangle \circ \CS$ as proven in \cref{t:main local}.
\end{proof}


\begin{corollary}
	For any topological space $\mathfrak{Z}$ the chain map $\CS_{\Schains(\mathfrak{Z})} \colon \sSchains(\mathfrak{Z}) \to \cSchains(\mathfrak{Z})$ is a quasi-isomorphism of $E_\infty$-coalgebras, specifically, of ${\forget(\M_{sl})}$-coalgebras.
\end{corollary}

\begin{proof}
	The map $\CS_{\Schains(\mathfrak{Z})}$ factors as $\sSchains(\mathfrak{Z}) \to \cchains \big( \cubify \sSing(\mathfrak{Z}) \big) \to \cSchains(\mathfrak{Z})$ where the first map is $\CS_{\sSing(\mathfrak{Z})}$, proven in \cref{t:main comparison} to be a quasi-isomorphism of ${\forget(\M_{sl})}$-coalgebras, and the second is induced by a cubical map $\cubify \sSing(\mathfrak{Z}) \to \cSing(\mathfrak{Z})$ described next, which is therefore a morphisms of ${\forget(\M_{sl})}$-coalgebras.
	The two-out-of-three property of quasi-isomorphisms concludes the proof.
	Using the adjunction
	\[
	\mathsf{sSet} \big( \scube{n}, \sSing(\mathfrak{Z}) \big) \cong \mathsf{Top} \big( \bars{\scube{n}}, \mathfrak{Z} \big)
	\]
	we define
	\[
	\begin{tikzcd} [row sep=0, column sep=small]
		\cubify \sSing(\mathfrak{Z}) \arrow[r] &
		\cSing(\mathfrak{Z}) \\
		\big( \bars{\scube{n} } \xra{f} \mathfrak{Z} \big) \arrow[r, mapsto] &
		\big( \mathbb{I}^n \xra{\ez} \bars{\scube{n}} \xra{f} \mathfrak{Z} \big)
	\end{tikzcd}
	\]
	which is a cubical map given the naturality of $\ez$.
\end{proof}



\section{Conventions and preliminaries} \label{s:preliminaries}

\subsection{Chain complexes}

Throughout this article $\k$ denotes a commutative and unital ring and we work over its associated closed symmetric monoidal category of differential (homologically) graded $\k$-modules $(\mathsf{Ch}, \otimes, \k)$.
We refer to the objects and morphisms of this category as \textit{chain complexes} and \textit{chain maps} respectively.
We denote by $\mathrm{Hom}(C, C^\prime)$ the chain complex of $\k$-linear maps between chain complexes $C$ and $C^\prime$, and refer to the functor $\mathrm{Hom}(-, \k)$ as \textit{linear duality}.

\subsection{Presheaves}

Recall that a category is said to be \textit{small} if its objects and morphisms form sets.
We denote the category of small categories by $\mathsf{Cat}$.
Given categories $\mathsf{B}$ and $\mathsf{C}$ with $\mathsf{B}$ small we denote their associated \textit{functor category} by $\mathsf{Fun}(\mathsf{B}, \mathsf{C})$.
A category is said to be \textit{cocomplete} if any functor to it from a small category has a colimit.
If $\mathsf{A}$ is small and $\mathsf{C}$ cocomplete, then the (\textit{left}) \textit{Kan extension of $g$ along $f$} exists for any pair of functors $f$ and $g$ in the diagram below, and it is the initial object in $\mathsf{Fun}(\mathsf{B}, \mathsf{C})$ making
\begin{equation*}
\begin{tikzcd}[column sep=normal, row sep=normal]
\mathsf{A} \arrow[d, "f"'] \arrow[r, "g"] & \mathsf{C} \\
\mathsf{B} \arrow[dashed, ur, bend right] & \quad
\end{tikzcd}
\end{equation*}
commute.
A Kan extension along the \textit{Yoneda embedding}, i.e., the functor
\[
\yoneda \colon \mathsf{A} \to \mathsf{Fun}(\mathsf{A}^\mathrm{op}, \mathsf{Set})
\]
induced by the assignment
\[
a \mapsto \big( a^\prime \mapsto \mathsf{A}(a^\prime, a) \big),
\]
is referred to as a \textit{Yoneda extension}.
Abusively we use the same notation for a functor and its Yoneda extension.
We refer to the objects in $\mathsf{Fun}(\mathsf{A}^\mathrm{op}, \mathsf{Set})$ as \textit{presheaves on $\mathsf{A}$} and to those in the image of the Yoneda embedding as \textit{representables}.
We remark that any presheaf $P$ on $\mathsf{A}$ can be written as a colimit of representables as:
\[
P \, \cong \colim_{\yoneda(a) \downarrow P} \yoneda(a).
\]

\section{Operads, props and \texorpdfstring{${E_\infty}$}{E-infty}-structures} \label{s:props}

We now review the definition of the finitely presented prop $\mathcal{M}$ introduced in \cite{medina2020prop1} and whose associated operad is a model of the $E_\infty$-operad.
Given its small number of generators and relations, is well suited to define $E_\infty$-structures.
In the next section we use this model to define natural $E_\infty$-structures on cubical chains and cochains.
We start by reviewing the basic material in the theory of operads and props, referring the reader to, for example, \cite{markl2008props} for a more complete treatment.

\subsection{Symmetric (bi)modules}

Let $\mathbb{S}$ be the category whose objects are the non-negative integers $\mathbb{N}$ and whose set of morphisms between $n$ and $n'$ is empty if $n \neq n'$ and is otherwise the symmetric group $\mathbb{S}_n$.
A \textit{left $\mathbb{S}$-module} (resp. \textit{right} $\mathbb{S}$-\textit{module} or $\mathbb{S}$-\textit{bimodule}) is a functor from $\mathbb{S}$ (resp. $\mathbb{S}^\mathrm{op}$ or $\mathbb{S}^\mathrm{op} \times \mathbb{S}$) to $\mathsf{Ch}$.
In this paper we prioritize left module structures over their right counterparts.
As usual, taking inverses makes both perspectives equivalent.
We respectively denote by $\mathsf{Mod}_{\mathbb{S}}$ and $\mathsf{biMod}_{\mathbb{S}}$ the categories of left $\mathbb{S}$-modules and of $\mathbb{S}$-bimodules with morphisms given by natural transformations.

Given a chain complex $C$, we have the following key examples of a left and a right $\mathbb{S}$-module
\begin{gather*}
	\mathrm{End}^C(n) = \mathrm{Hom}(C, C^{\otimes n}), \qquad
	\mathrm{End}_C(m) = \mathrm{Hom}(C^{\otimes m}, C),
\end{gather*}
and of an $\mathbb{S}$-bimodule
\[
\mathrm{End}^C_C(m,n) = \mathrm{Hom}(C^{\otimes m}, C^{\otimes n}),
\]
where the symmetric actions are given by permutation of tensor factors.

The group homomorphisms $\mathbb{S}_n \to \mathbb{S}_1^\mathrm{op} \times \mathbb{S}_n$ induce a forgetful functor
\begin{equation} \label{e:forgetful}
	\forget \colon \mathsf{biMod}_{\mathbb{S}} \to \mathsf{Mod}_{\mathbb{S}}
\end{equation}
defined explicitly on an object $\mathcal{P}$ by $\forget(\mathcal{P})(n) = \mathcal{P}(1, n)$ for $n \in \mathbb{N}$.
The similarly defined forgetful functor to right $\mathbb{S}$-modules will not be considered.

\subsection{Composition structures}

\textit{Operads} and \textit{props} are obtained by enriching $\mathbb{S}$-modules and $\mathbb{S}$-bimodules with certain composition structures.
Intuitively, these are obtained by abstracting the composition structure naturally present in the left $\mathbb{S}$-module $\mathrm{End}^C$ (or right $\mathbb{S}$-module $\mathrm{End}_C$), naturally an operad, and the $\mathbb{S}$-bimodule $\mathrm{End}^C_C$, naturally a prop.
More explicitly, an operad $\mathcal{O}$ is a left $\mathbb{S}$-module with chain maps
\begin{gather*}
	\k \to \mathcal{O}(1), \\
	\mathcal{O}(n_1) \otimes \dotsb \otimes \mathcal{O}(n_r) \otimes \mathcal{O}(r) \to \mathcal{O}(n_1 + \dots + n_r),
\end{gather*}
satisfying relations of associativity, equivariance and unitality.
Similarly, a prop $\mathcal{P}$ is a $\mathbb{S}$-bimodule together with chain maps
\begin{gather*}
	\k \to \mathcal{P}(n,n), \\
	\mathcal{P}(m,k) \otimes \mathcal{P}(k,n) \to \mathcal{P}(m,n), \\
	\mathcal{P}(m,n) \otimes \mathcal{P}(m',n') \to \mathcal{P}(m+m',n+n'),
\end{gather*}
satisfying certain natural relations.
For a complete presentation of these concepts we refer to Definition~11 and 54 of \cite{markl2008props}.
We respectively denote the category of operads and props with structure preserving morphisms by $\mathsf{Oper}$ and $\mathsf{Prop}$.

Let $C$ be a chain complex, $\mathcal{O}$ an operad, and $\mathcal{P}$ a prop.
An $\mathcal{O}$-\textit{coalgebra} (resp. $\mathcal{O}$-\textit{algebra} or $\mathcal{P}$-\textit{bialgebra}) structure on $C$ is an structure preserving morphism $\mathcal{O} \to \mathrm{End}^C$ (resp. $\mathcal{O} \to \mathrm{End}_C$ or $\mathcal{P} \to \mathrm{End}_C^C$).

We remark that the forgetful functor \eqref{e:forgetful} restricts to
\[
\forget \colon \mathsf{Prop} \to \mathsf{Oper},
\]
so any $\mathcal{P}$-bialgebra structure on $C$
\[
\mathcal{P} \to \mathrm{biEnd}_C^C
\]
induces a $\forget(\mathcal{P})$-coalgebra structure on it
\[
\forget(\mathcal{P}) \to \forget(\mathrm{biEnd}_C^C) \cong \mathrm{coEnd}^C.
\]

\subsection{\texorpdfstring{${E_\infty}$}{E-infty}-operads}

Recall that a \textit{projective $\mathbb{S}_n$-resolution} of a chain complex $C$ is a quasi-isomorphism $R \to C$ from a chain complex $R$ of projective $\k[\mathbb{S}_n]$-modules.

An $\mathbb{S}$-module $M$ is said to be $E_\infty$ if there exists a morphism of $\mathbb{S}$-modules $M \to \underline{\k}$ inducing for each $n \in \mathbb{N}$ a free $\mathbb{S}_n$-resolution $M(n) \to \k$.
For example, we can obtain one such $\mathbb{S}$-module by using the functor of singular chains and the set $\{\mathrm{E} \mathbb{S}_n \to \ast\}_{n \in \mathbb{N}}$ of maps to the terminal space from models of the universal $\mathbb{S}_n$-bundle.

An operad is said to be an \textit{$E_\infty$-operad} if its underlying $\mathbb{S}$-module is $E_\infty$.
A prop $\mathcal{P}$ is said to be an \textit{$E_\infty$-prop} if $\forget(\mathcal{P})$ is an $E_\infty$-operad.

\subsection{Presentations} \label{ss:presentation}

\begin{figure}
	\centering
	\input{aux/immersion}
	\caption{Immersed graphs represent labeled directed graphs with the direction implicitly given from top to bottom and the labeling from left to right.}
	\label{f:immersion}
\end{figure}

The \textit{free prop} $\free(M)$ generated by an $\mathbb{S}$-bimodule $M$ is constructed using isomorphism classes of directed graphs with no directed loops that are enriched with the following labeling structure.
We think of each directed edge as built from two compatibly directed half-edges.
For each vertex $v$ of a directed graph $\Gamma$, we have the sets $in(v)$ and $out(v)$ of half-edges that are respectively incoming to and outgoing from $v$.
Half-edges that do not belong to $in(v)$ or $out(v)$ for any $v$ are divided into the disjoint sets $in(\Gamma)$ and $out(\Gamma)$ of incoming and outgoing external half-edges.
For any positive integer $n$ let $\overline{n} = \{1, \dots, n\}$ and set $\overline{0} = \emptyset$.
For any finite set $S$, denote the cardinality of $S$ by $|S|$.
The labeling is given by bijections
\[
\overline{|in(\Gamma)|} \to in(\Gamma), \qquad
\overline{|out(\Gamma)|} \to out(\Gamma),
\]
and
\[
\overline{|in(v)|} \to in(v), \qquad
\overline{|out(v)|} \to out(v),
\]
for every vertex $v$.
We refer to the isomorphism classes of such labeled directed graphs with no directed loops as $(m,n)$\textit{-graphs} denoting the set of these by $\mathfrak{G}(m,n)$, where $m$ and $n$ denote respectively the number of incoming and outgoing half-edges of any $\Gamma \in \mathfrak{G}(m,n)$.
We use graphs immersed in the plane to represent elements in $\mathfrak{G}(m,n)$, please see \cref{f:immersion}.
We consider the right action of $\mathbb{S}_m$ and the left action of $\mathbb{S}_n$ on a $(m,n)$-graph given respectively by permuting the labels of $in(\Gamma)$ and $out(\Gamma)$.
This action defines the $\mathbb{S}$-bimodule structure on the free prop
\begin{equation} \label{e:free prop}
\free(M)(m,n) \ =
\bigoplus_{\Gamma \in \mathfrak{G}(m,n)} \
\bigotimes_{v \in Vert(\Gamma)} out(v) \otimes_{\mathbb{S}_p} M(p, q) \otimes_{\mathbb{S}_p} in(v),
\end{equation}
where we simplified the notation writing $p$ and $q$ for $\overline{|in(v)|}$ and $\overline{|out(v)|}$ respectively, the differential $\bd_{\free(M)}$ is the extension of that of $M$ to the tensor product \eqref{e:free prop}, and the composition structure is defined by
\[
\begin{tikzcd}[column sep=small, row sep=0]
	\k \arrow[r]& \free(M)(r,r) \\
	1 \arrow[r, mapsto] & \identity \ \identity \dotsb \identity
\end{tikzcd}
\]
and (relabeled) grafting and disjoint union.

Let $G$ be assignment of a set $G(m,n)_d$ to each $m,n \in \mathbb{N}$ and $d \in \mathbb{Z}$.
Denote by $\k[\mathbb{S}^\mathrm{op} \times \mathbb{S}] \set{G}$ the $\mathbb{S}$-bimodule assigning to $(m,n)$
the chain complex with trivial differential and degree $d$ part equal to
\[
\k[\mathbb{S}^\mathrm{op}_m \times \mathbb{S}_n] \set[\big]{G(m,n)_d}.
\]
We will denote by $\free(G)$ the free prop generated by this $\mathbb{S}$-bimodule.
Let $\bd \colon \k[\mathbb{S}^\mathrm{op} \times \mathbb{S}] \set{G} \to \free(G)$ be a morphism of $\mathbb{S}$-bimodule whose monoidal extension $\bd \colon \free(G) \to \free(G)$ defines a differential.
We denote by $\free_{\bd}(G)$ the prop obtained by endowing $\free(G)$ with this differential.
Let $R$ be a collection of elements in $\free(G)$ and denote by $\angles{R}$ the smallest ideal containing $R$.
The prop \textit{generated by $G$ modulo $R$ with boundary $\bd$} is defined to be $\free_{\bd}(G) \big/ \angles{R}$.

\subsection{The prop $\mathcal{M}$}

We now recall the $E_\infty$-prop that is central to our constructions.

\begin{definition}
	Let $\mathcal{M}$ be the prop generated by
	\begin{equation} \label{e:generators of M}
	\counit \,, \qquad
	\coproduct \,, \qquad
	\product,
	\end{equation}
	in $(1,0)_0$\,, $(1,2)_0$ and $(2,1)_1$ respectively,
	modulo the relations
	\begin{equation} \label{e:relations of M}
		\leftcounitality \,, \qquad
		\rightcounitality \,, \qquad
		\productcounit,
	\end{equation}
	with boundary defined by
	\begin{equation} \label{e:boundary of M}
	\bd\ \counit = 0, \qquad
	\bd\, \coproduct = 0, \qquad
	\bd \product = \ \boundary\,.
	\end{equation}
\end{definition}

Explicitly, any element in $\mathcal{M}(m,n)$ can be written as a linear combination of the $(m,n)$-graphs generated by those in \eqref{e:generators of M} via grafting, disjoint union and relabeling, modulo the prop ideal generated by the relations in \eqref{e:relations of M}. Its boundary is determined by \eqref{e:boundary of M} using \eqref{e:free prop}.

\begin{proposition}[{\cite[Theorem 3.3]{medina2020prop1}}]
	$\mathcal{M}$ is an $E_\infty$-prop.
\end{proposition}

\begin{remark*}
	The prop $\mathcal{M}$ is obtained from applying the functor of chains to a prop over the category of cellular spaces \cite{medina2021prop2} which can be thought of as a resolution of the operad of stable arc surfaces \cite{kaufmann2009dimension}.
\end{remark*}

\section{An \texorpdfstring{${E_\infty}$}{E-infty}-structure on cubical chains} \label{s:action}

In this section we construct a natural $\mathcal{M}$-bialgebra structure on the chains of standard cubes.
These are determined by three natural linear maps satisfying the relations defining $\mathcal M$.
A Yoneda extension then provides the chains of any cubical set with a natural ${\forget(\M)}$-coalgebra structure.
We begin by recalling the basics of cubical topology.

\subsection{Cubical sets}

The objects of the \textit{cube category} $\square$ are the sets $2^n = \{0, 1\}^n$ with $2^0 = \{0\}$ for $n \in \mathbb{N}$, and its morphisms are generated by the \textit{coface} and \textit{codegeneracy} maps
\begin{align*}
\delta_i^\varepsilon & = \mathrm{id}_{2^{i-1}} \times \delta^\varepsilon \times \mathrm{id}_{2^{n-1-i}} \colon 2^{n-1} \to 2^n, \\
\sigma_i & = \mathrm{id}_{2^{i-1}} \times \, \sigma \times \mathrm{id}_{2^{n-i}} \quad \colon 2^{n} \to 2^{n-1},
\end{align*}
where $\varepsilon \in \{0,1\}$ and the functors
\[
\begin{tikzcd} [column sep=16pt]
2^0 \arrow[r, bend left, "\delta^0"] \arrow[r, bend right, "\delta^1"'] & 2^1 \arrow[r, "\sigma"] & 2^0
\end{tikzcd}
\]
are defined by
\[
\delta^0(0) = 0, \qquad \delta^1(0) = 1, \qquad \sigma(0) = \sigma(1) = 0.
\]
We refer to \cite{grandis2003cubical} for a more leisure exposition and for variations on this definition.

We denote by $\square_{\deg}(2^m, 2^n)$ the subset of morphism in $\square(2^m, 2^n)$ of the form $\sigma_i \circ \tau$ with $\tau \in \square(2^m, 2^{n+1})$.

The category of \textit{cubical sets} $\mathsf{Fun}(\square^\mathrm{op}, \mathsf{Set})$ is denoted by $\mathsf{cSet}$ and the representable cubical set $\yoneda(2^n)$ by $\square^n$.
For any cubical set $X$ we write, as usual, $X_n$ instead of $X(2^n)$.

\subsection{Cubical topology}

Consider the topological $n$-cube
\begin{equation*}
\mathbb{I}^{n} = \big\{ (x_1, \dots, x_n) \mid x_i \in [0,1] \big\}.
\end{equation*}
The assignment $2^n \to \mathbb{I}^n$ defines a functor $\square \to \mathsf{Top}$ with
\begin{align*}
\delta^\varepsilon_i(x_1, \dots, x_n) &= (x_1, \dots, x_i, \varepsilon, x_{i+1}, \dots x_n), \\
\sigma_i(x_1,\dots,x_n) &= (x_1, \dots, \widehat{x}_i, \dots, x_n).
\end{align*}
Its Yoneda extension is known as \textit{geometric realization}.
It has a right adjoint $\cSing \colon \mathsf{Top} \to \mathsf{cSet}$ referred to as the \textit{cubical singular complex} satisfying
\[
\cSing(\mathfrak{Z})_n = \mathsf{Top}(\mathbb{I}^n, \mathfrak{Z})
\]
for any topological space $\mathfrak{Z}$.

\subsection{Cubical chains}

The functor of (\textit{normalized}) \textit{chains} $\chains \colon \mathsf{cSet} \to \mathsf{Ch}$ is the Yoneda extension of the functor $\square \to \mathsf{Ch}$ defined next.
It assigns to an object $2^n$ the chain complex having in degree $m$ the module
\[
\frac{\k\{\square(2^m, 2^n)\}}{\k\{\square_{\deg}(2^m, 2^n)\}}
\]
and differential induced by
\[
\bd (\mathsf{id}_{2^n}) = \sum_{i=1}^{n} \ (-1)^i \
\big(\delta_i^1 - \delta_i^0 \big).
\]
To a morphism $\tau \colon 2^n \to 2^{n'}$ it assigns the chain map
\[
\begin{tikzcd}[row sep=-3pt, column sep=normal,
/tikz/column 1/.append style={anchor=base east},
/tikz/column 2/.append style={anchor=base west}]
\chains(\square^n)_m \arrow[r] & \chains(\square^{n'})_m \\
\big( 2^m \to 2^n \big) \arrow[r, mapsto] & \big( 2^m \to 2^n \stackrel{\tau}{\to} 2^{n'} \big).
\end{tikzcd}
\]
The chain complex $\chains(\square^n)$ is isomorphic to both: $\chains(\square^1)^{\otimes n}$ and the cellular chains on the topological $n$-cube with its standard CW structure $\gchains(\mathbb{I}^n)$.
We use the isomorphism $\chains(\square^n) \cong \gchains(\mathbb{I}^1)^{\otimes n}$ when denoting the elements in the basis of $\chains(\square^n)$ by $x_1 \otimes \dotsb \otimes x_n$ with $x_i \in \{[0], [0,1], [1]\}$.

For a topological space $\mathfrak{Z}$, the chain complex $\chains(\cSing \mathfrak{Z})$ is referred to as the \textit{cubical singular chains} of $\mathfrak{Z}$.

\subsection{Serre coalgebra} \label{ss:serre coalgebra}

We now recall the natural (counital and coassociative) coalgebra structure on cubical chains studied by Cartan and Serre, inducing the cup product in the cohomology of cubical sets.

Using a Yoneda extension, it suffices to equip the chains on standard cubes with a natural coalgebra structure.
For any $n \in \mathbb{N}$, define $\epsilon \colon \chains(\square^n) \to \k$ by
\[
\epsilon \left( x_1 \otimes \dotsb \otimes x_d \right) = \epsilon(x_1) \dotsm \epsilon(x_n),
\]
where
\[
\epsilon([0]) = \epsilon([1]) = 1, \qquad \epsilon([0, 1]) = 0,
\]
and $\Delta \colon \chains(\square^n) \to \chains(\square^n)^{\otimes 2}$ by
\[
\Delta (x_1 \otimes \dotsb \otimes x_n) =
\sum \pm \left( x_1^{(1)} \otimes \dotsb \otimes x_n^{(1)} \right) \otimes
\left( x_1^{(2)} \otimes \dotsb \otimes x_n^{(2)} \right),
\]
where the sign is determined using the Koszul convention, and we are using Sweedler's notation
\[
\Delta(x_i) = \sum x_i^{(1)} \otimes x_i^{(2)}
\]
for the chain map $\Delta \colon \chains(\square^1) \to \chains(\square^1)^{\otimes 2}$ defined by
\[
\Delta([0]) = [0] \otimes [0], \quad \Delta([1]) = [1] \otimes [1], \quad \Delta([0,1]) = [0] \otimes [0,1] + [0,1] \otimes [1].
\]

We remark that, using the canonical isomorphism $\chains(\square^n) \cong \chains(\square^1)^{\otimes n}$, the coproduct $\Delta$ can be described as the composition
\[
\begin{tikzcd}
\chains(\square^1)^{\otimes n} \arrow[r, "\Delta^{\otimes n}"] & \left( \chains(\square^1)^{\otimes 2} \right)^{\otimes n} \arrow[r, "\pi"] & \left( \chains(\square^1)^{\otimes n} \right)^{\otimes 2}
\end{tikzcd}
\]
where $\pi$ is the shuffle permutation of tensor factors placing those in odd positions first.

\subsection{Degree 1 product}

For $n \in \mathbb{N}$ define the \textit{product} $\ast  \colon \chains(\square^n)^{\otimes 2} \to \chains(\square^n)$ by
\begin{multline*}
(x_1 \otimes \dotsb \otimes x_n) \ast (y_1 \otimes \dotsb \otimes y_n) =
(-1)^{|x|} \sum_{i=1}^n x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i}) \, y_{>i},
\end{multline*}
where
\begin{align*}
x_{<i} & = x_1 \otimes \dotsb \otimes x_{i-1}, &
y_{<i} & = y_1 \otimes \dotsb \otimes y_{i-1}, \\
x_{>i} & = x_{i+1} \otimes \dotsb \otimes x_n, &
y_{>i} & = y_{i+1} \otimes \dotsb \otimes y_n,
\end{align*}
with the convention
\[
x_{<1} = y_{<1} = x_{>n} = y_{>n} = 1 \in \mathbb{Z},
\]
and the only non-zero values of $x_i \ast y_i$ are
\[
\ast([0] \otimes [1]) = [0, 1], \qquad  \ast([1] \otimes [0]) = -[0, 1].
\]

\subsection{$\mathcal{M}$-bialgebra on representable cubical sets}

\begin{lemma} \label{l:cubical chain bialgebra}
	The assignment
	\[
	\counit \mapsto \epsilon, \quad \coproduct \mapsto \Delta, \quad \product \mapsto \ast,
	\]
	induces a natural $\mathcal M$-bialgebra structure on $\chains(\square^n)$ for every $n \in \mathbb{N}$.
\end{lemma}

\begin{proof}
	We need to show that this assignment is compatible with the relations
	\[
	\productcounit = 0, \qquad
	\leftcounitality = 0, \qquad
	\rightcounitality = 0,
	\]
	and
	\[
	\bd\ \counit = 0, \qquad
	\bd\ \coproduct = 0, \qquad
	\bd\ \product = \ \boundary\,.
	\]
	For the rest of this proof let us consider two basis elements of $\chains(\square^n)$
	\begin{align*}
		x = x_1 \otimes \dotsb \otimes x_n
		\qquad \text{ and } \qquad
		y = y_1 \otimes \dotsb \otimes y_n.
	\end{align*}
	Since the degree of $\ast$ is $1$ and $\epsilon([0,1]) = 0$, we can verify the first relation easily:
	\begin{align*}
		\varepsilon(x \ast y) & =
		\sum (-1)^{|x|} \epsilon(y_{<i}) \epsilon(x_{<i}) \otimes \epsilon(x_i \ast y_i) \otimes \epsilon(x_{>i}) \epsilon(y_{>i}) = 0.
	\end{align*}
	For the second relation we want to show that $(\epsilon \otimes \mathsf{id}) \circ \Delta = \mathsf{id}$.
	Since
	\begin{gather*}
		(\epsilon \otimes \mathsf{id}) \circ \Delta([0]) = \epsilon([0]) \otimes [0] = [0], \\
		(\epsilon \otimes \mathsf{id}) \circ \Delta([1]) = \epsilon([1]) \otimes [1] = [1], \\
		(\epsilon \otimes \mathsf{id}) \circ \Delta([0, 1]) = \epsilon([0]) \otimes [0, 1] + \epsilon([0, 1]) \otimes [1] = [0,1],
	\end{gather*}
	we have
	\begin{multline*}
		(\epsilon \otimes \mathsf{id}) \circ \Delta (x_1 \otimes \dotsb \otimes x_n) = \\
		\sum \pm \left( \epsilon \big(x_1^{(1)}\big) \otimes \dotsb \otimes \epsilon\big(x_n^{(1)}\big) \right) \otimes
		\left( x_1^{(2)} \otimes \dotsb \otimes x_n^{(2)} \right) \\ =
		x_1 \otimes \dotsb \otimes x_n,
	\end{multline*}
	where the sign is obtained by noticing that the only non-zero term occurs when each factor $x_i^{(0)}$ is of degree $0$.
	The third relation is verified analogously.
	The fourth and fifth are precisely the well known facts that $\epsilon$ and $\Delta$ are chain maps.
	To verify the sixth and final relation we need to show that
	\[
	\bd (x \ast y)\ +\ \bd x \ast y\ +\ (-1)^{|x|}x \ast \bd y\ =\ \epsilon(x) y \ -\ \epsilon(y) x.
	\]
	We have
	\[
	x \ast y = \sum (-1)^{|x|} x_{<i} \, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i})\, y_{>i}
	\]
	and
	\begin{align*}
		\bd(x \ast y) & =
		\sum (-1)^{|x|} \, \bd x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i})\, y_{>i} \\ & +
		\sum (-1)^{|x|+|x_{<i}|} \, x_{<i}\, \epsilon(y_{<i}) \otimes \bd (x_i \ast y_i) \otimes \epsilon(x_{>i}) \, y_{>i} \\ & -
		\sum (-1)^{|x|+|x_{<i}|} \, x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i})\, \bd y_{>i}.
	\end{align*}
	Since $|x| = |x_{<i}| + |x_i| + |x_{>i}|$ and $\epsilon(x_{>i}) \neq 0 \Leftrightarrow |x_{>i}| = 0$ as well as $\bd(x_i \ast y_i) \neq 0 \Rightarrow |x_i| = 0$
	we have
	\begin{equation} \label{e:boundary of product 1}
		\begin{split}
			\bd(x \ast y) & =
			\sum (-1)^{|x|} \, \bd x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i})\, y_{>i} \\ & +
			\sum x_{<i} \, \epsilon(y_{<i}) \otimes \bd (x_i \ast y_i) \otimes \epsilon(x_{>i})\, y_{>i} \\ & -
			\sum x_{<i} \, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i})\, \bd y_{>i}.
		\end{split}
	\end{equation}
	We also have
	\begin{align*}
		\bd x \ast y & =
		\sum (-1)^{|x|-1} \, \bd x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i}) \, y_{>i} \\ & +
		\sum (-1)^{|x|-1+|x_{<i}|} \, x_{<i}\, \epsilon(y_{<i}) \otimes \bd x_i \ast y_i \otimes \epsilon(x_{>i}) \, y_{>i} \\ & +
		\sum (-1)^{|x|-1+|x_{<i}|} \, x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(\bd x_{>i}) \, y_{>i}.
	\end{align*}
	Since
	\[
	\epsilon(\bd x_{>i}) = 0, \quad \bd x_i \neq 0 \Leftrightarrow |x_i| = 1,
	\]
	we have
	\begin{equation} \label{e:boundary of product 2}
		\begin{split}
			\bd x \ast y & =
			\sum (-1)^{|x|-1} \, \bd x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i})\, y_{>i} \\ & +
			\sum x_{<i}\, \epsilon(y_{<i}) \otimes \bd x_i \ast y_i \otimes \epsilon(x_{>i})\, y_{>i}.
		\end{split}
	\end{equation}
	We also have
	\begin{align*}
		(-1)^{|x|} \, x \ast \bd y & =
		\sum x_{<i} \, \epsilon(\bd y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i})\, y_{>i} \\ & +
		\sum (-1)^{|y_{<i}|} \, x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast \bd y_i \otimes \epsilon(x_{>i}) \, y_{>i} \\ & +
		\sum (-1)^{|y_{<i}| + |y_i|} \, x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i}) \, \bd y_{>i},
	\end{align*}
	which is equivalent to
	\begin{equation} \label{e:boundary of product 3}
		\begin{split}
			(-1)^{|x|} \, x \ast \bd y & =
			\sum x_{<i} \, \epsilon(y_{<i}) \otimes x_i \ast \bd y_i \otimes \epsilon(x_{>i})\, y_{>i} \\ & +
			\sum x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i})\, \bd y_{>i}.
		\end{split}
	\end{equation}
	Putting identities \eqref{e:boundary of product 1}, \eqref{e:boundary of product 2} and \eqref{e:boundary of product 3} together, we get
	\begin{multline*}
		\bd (x \otimes y) \ +\ \bd x \ast y\ +\, (-1)^{|x|}x \ast \bd y \\
		= \sum \epsilon(y_{<i})\, x_{<i} \otimes \big(\bd(x_i \ast y_i) + \bd x_i \ast y_i + x_i \ast \bd y_i\big) \otimes \epsilon(x_{>i})\, y_{>i}.
	\end{multline*}
	Since
	\begin{align*}
		\bd(x_i \ast y_i)\ +\ \bd x_i \ast y_i\ +\ x_i \ast \bd y_i =
		\epsilon(x_i)y_i\ -\ \epsilon(y_i)x_i,
	\end{align*}
	we have
	\begin{multline*}
		\bd (x \ast y) \ +\ \bd x \ast y\ +\ (-1)^{|x|}x \ast \bd y  = \\
		\sum \epsilon(y_{<i}) \, x_{<i} \otimes \epsilon(x_{\geq i}) y_{\geq i}\ -\
		\epsilon(y_{\leq i}) \, x_{\leq i} \otimes \epsilon(x_{>i}) y_{>i} \\ =
		\epsilon(x)y - \epsilon(y)x,
	\end{multline*}
	as desired, where the last equality follows from a telescopic sum argument.
\end{proof}

\subsection{$\texorpdfstring{E_\infty}{E-infty}$-coalgebra on cubical chains}

\cref{l:cubical chain bialgebra} defines a functor from the cube category to that of $\mathcal{M}$-bialgebras.
This category is not cocomplete so we do not expect to have an $\mathcal{M}$-bialgebra structure on arbitrary cubical sets.
For example, consider the chains on the cubical set $X$ whose only non-degenerate simplices are $v, w \in X_0$.
By degree reasons $v \ast w = 0$ for any degree 1 product $\ast$ in $\chains(X)$.
The third relation in $\mathcal{M}$ would then imply the contradiction $0 = w-v$.
Nevertheless, categories of coalgebras over operads are cocomplete and we have the following.

\begin{theorem} \label{t:lift to e infinity coalgebras}
	The Yoneda extension of the composition of the functor $\square \to \mathsf{biAlg}_{\mathcal{M}}$ defined in \cref{l:cubical chain bialgebra} with the forgetful functor $\mathsf{biAlg}_{\mathcal{M}} \to \mathsf{coAlg}_{{\forget(\M)}}$ endows the chains of a cubical set with a natural $E_\infty$-coalgebra extension of the Serre coalgebra structure.
\end{theorem}

Using linear duality, the same argument can be employed to define a natural $E_\infty$-algebra structure on cubical cochains.
Explicitly, let $X$ be a cubical set, $x$ a simplex and $f \colon \chains(\square^n) \to \chains(X)$ a chain map sending $\mathsf{id}_{2^n}$ to $x$.
The image $\gamma$ in $\mathrm{Hom}(\chains(\square^n), \chains(\square^n)^{\otimes r})$ of a basis element $\Gamma$ in ${\forget(\M)}(r)$ is either $\varepsilon$ or a composition of $\Delta$, permutations of factors, and $\ast$.
For any $\alpha_1, \dots, \alpha_r \in \cochains(X)$ the action of $\Gamma$ on these is given by
\[
\Gamma(\alpha_1, \dots, \alpha_r)(x) = (\alpha_1 \otimes \dotsb \otimes \alpha_r) \circ f^{\otimes r} \circ \gamma(\mathsf{id}_{2^n}).
\]

\subsection{Cohomology operations}

Steenrod introduced in \cite{steenrod1947products} operations on the mod~2 cohomology of spaces via an explicit construction in the simplicial context of natural linear maps $\Delta_i$ satisfying up to signs the following homological relations
\begin{equation} \label{e:cupi homological relations}
\bd \circ \Delta_i + \Delta_i \circ \bd = (1 + T) \Delta_{i-1}
\end{equation}
with the convention $\Delta_{-1} = 0$.
These maps, referred to as (simplicial) cup-$i$ coproducts, are fundamental combinatorially, admitting an axiomatic characterization \cite{medina2022axiomatic} and defining, for example, the nerve of $n$-categories as introduced by Street \cite{street1987orientals, medina2020globular}.

In the cubical case, cup-$i$ coproducts were defined in \cite{kadeishvili1999coproducts} and \cite{pilarczyk2016cubical}.
It is unclear if these are equivalent.
The formulas used by these authors are analogous to those introduced in \cite{medina2021fast_sq} for the simplicial case, a dual yet equivalent version of Steenrod's original description.
By the same methods used in \cite{medina2020globular}, these formulas define a cubical nerve for higher categories, but it remains unclear if either agrees with the one defined by the generalized Gray tensor product.

A new description of cubical cup-$i$ coproducts can be deduced from our $E_\infty$-structure on cubical chains by considering the action of the elements depicted in \cref{f:cup-i}.
\begin{figure}
	\centering
	\input{aux/cupi}
	\caption{Elements in ${\forget(\M)}$ defining cubical cup-$i$ coproducts.}
	\label{f:cup-i}
\end{figure}
These recursively define:
\begin{equation} \label{e:prop cup-i}
	\begin{split}
		& \Delta_0 = \Delta, \\
		& \Delta_i =
		(\ast \otimes \mathsf{id}) \circ (\mathsf{id} \otimes (12) \Delta_{i-1}) \circ \Delta.
	\end{split}
\end{equation}
It is not known if these agree with either of the previous constructions, pointing to the value of an axiomatic characterization as it exists in the simplicial case, where all known constructions agree.

Cup-$i$ coproducts represent at the cochain level Steenrod squares
\[
\begin{tikzcd} [column sep=small, row sep=0]
	Sq^k \colon &[-20] H^{-n} \arrow[r] & H^{-n-k} \\ &
	{[\alpha]} \arrow[r, mapsto] & \big[ (\alpha \otimes \alpha) \Delta_i \big]
\end{tikzcd}
\]
which are primary operations in mod~2 cohomology.
To obtain secondary cohomology operations one studies the cohomological relations these operations satisfy, for example the Cartan and Adem relations \cite{steenrod1962cohomology}.
To do this at the cubical chain level, as it was done in \cite{medina2020cartan,medina2021adem} for the simplicial case, the operadic viewpoint is important, so our $E_\infty$-structure on cubical cochains invites the construction of cochain representatives for secondary operations in the cubical case.

For an odd $p$, Steenrod also introduced operations on the mod $p$ cohomology of spaces using the homology of symmetric groups \cite{steenrod1952reduced, steenrod1953cyclic}.
Using the operadic framework of May \cite{may1970general}, we described in \cite{medina2021may_st} elements in ${\forget(\M)}$ representing multicooperations defining Steenrod operations at any prime.
In particular, as proven in this work, these so-called cup-$(p,i)$ products are defined on cubical cochains and are expressible, similarly to \cref{e:prop cup-i}, in terms of $\Delta$, the permutations of factors, and $\ast$.
The aforementioned construction of cubical cup-$(p,i)$ coproducts has been implemented in the computer algebra system \href{https://comch.readthedocs.io/en/latest/}{\texttt{ComCH}} \cite{medina2021comch}.
