by the compatibility of the coproducts with the degeneracies (the relations of Figure <ref>)
and this requirement implies the vanishing relation $\rho_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}^{\square} = 0$
since $I^{k-1}$ is null in dimension $k$.

The operators $\rho_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}: \AOp(\stree)\rightarrow\AOp(\ttree)\otimes I^k$
are defined by induction on $k$.
If $k = 0$, then we take $\rho_{\ttree\rightarrow\stree} = \rho_{\ttree\rightarrow\stree}^{\square}$.
The compatibility conditions of Figure <ref>-<ref>
are tautological in this case.

If $k > 0$, then defining a morphism of dg modules $\rho_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}: \AOp(\stree)\rightarrow\AOp(\ttree)\otimes I^k$
amounts to defining a morphism of dg modules
$\rho'_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}: \AOp(\stree)\otimes\DGN_*(\Delta^1)^{\otimes k}\rightarrow\AOp(\ttree)$.

Let $d_i^{\epsilon}: \DGN_*(\Delta^1)^{\otimes k-1}\rightarrow\DGN_*(\Delta^1)^{\otimes k}$ denote the coface operators attached to the cubical complex $\DGN_*(\Delta^1)^{\otimes k}$
dual to the face operators $d^i_{\epsilon}: \DGN^*(\Delta^1)^{\otimes k}\rightarrow\DGN^*(\Delta^1)^{\otimes k-1}$
that we considered so far. We let $\rho'_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}(x\otimes\underline{01}^{\otimes r})
= \rho^{\square}_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}(x)$,
and we define inductively:

\begin{align*}
\rho'_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}(x\otimes d_i^0(\underline{\sigma})) &
= \rho'_{\ttree\rightarrow\dots\rightarrow\widehat{\ttree_i}\rightarrow\dots\rightarrow\stree}(x\otimes\underline{\sigma}), \\
\rho'_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}(x\otimes d_i^1(\underline{\sigma})) &
= \rho'_{\ttree\rightarrow\dots\rightarrow\ttree_i}(\rho'_{\ttree_i\rightarrow\dots\rightarrow\stree}\otimes\id(x\otimes\underline{\sigma})),
\end{align*}

where we use the factorization $\underline{\sigma}\in\DGN_*(\Delta^1)^{\otimes k-1}\Rightarrow\underline{\sigma} = \underline{\sigma}'\otimes\underline{\sigma}''\in\DGN_*(\Delta^1)^{\otimes k-i}\otimes\DGN_*(\Delta^1)^{\otimes i-1}$.
We deduce from the compatibility of the homotopy coproducts with the face operators that our coproduct operator is necessarily given by these formulas
and this observation proves the uniqueness of the coproduct operators
extending our maps $\rho^{\square}_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}$.
We may also use the observations of Remark <ref> to check that our maps satisfy the compatibility relations
of Figure <ref>-<ref>
on the whole dg modules $\AOp(\ttree)\otimes I^k$.

We only need to prove that the map $\rho_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}$
or equivalently the map $\rho'_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}$
commutes with the differential.
We proceed by induction on $k$. For $k = 0$ the statement is obviously equivalent to relation (<ref>).
For $k > 0$, the relation $\delta(\rho'_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}(x\otimes\underline{\sigma}))
= \rho'_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}\delta(x\otimes\underline{\sigma})$
follows from the induction assumption
in the case $\underline{\sigma} = (d_i^{\epsilon})(\underline{\tau})$, for some $1\leq i\leq k$, $\epsilon\in\{0,1\}$, $\underline{\tau}\in\DGN_*(\Delta^1)^{\otimes k-1}$,
and reduces to relation (<ref>)
when $\underline{\sigma} = \underline{01}^{\otimes k}$.
The conclusion follows.

§.§ The application of cobar complexes

The aim of this Subsection is to extend the cobar construction of Subsection <ref>
to homotopy Segal dg cooperads. We follow the same plan.

We make explicit the definition of the structure operations of the cobar construction $\DGB^c(\AOp)$
associated to a homotopy Segal shuffle dg cooperad $\AOp$
in the next paragraph.
We check the validity of these definitions afterwards and we record the definition of this dg operad $\DGB^c(\AOp)$
to conclude this subsection.
We assume all along this subsection that $\AOp$ is a homotopy Segal shuffle dg cooperad.

We also assume that $\AOp$ is connected in the sense that we have $\AOp(\ttree) = 0$ when the tree $\ttree$ is not reduced (has at least one vertex with a single ingoing edge).
This condition implies that every object $\AOp(\stree)$
is the source of finitely many nonzero homotopy coproducts $\rho_{\ttree\rightarrow\dots\rightarrow\stree}: \AOp(\stree)\rightarrow\AOp(\ttree)\otimes I^k$,
because the set of tree morphisms $\ttree\rightarrow\stree$ with $\ttree$ reduced and $\stree$ fixed is finite.

We define the graded modules $\DGB^c(\AOp)(r)$ which form the components of the cobar construction $\DGB^c(\AOp)$
by the same formula as in the strict case

\begin{equation*}
\DGB^c(\AOp)(r) = \bigoplus_{\ttree\in\Tree(r)}\DGSigma^{-\sharp V(\ttree)}\AOp(\ttree),
\end{equation*}

but we now take a cobar differential that involves terms given by higher coproduct operators and which we associate to multiple edge contractions.
We precisely consider the set of pairs $(\ttree,\underline{e})$, where $\ttree\in\Tree(r)$ and $\underline{e} = (e_1,\dots,e_m)$
is an ordered collection of pairwise distinct edges $e_i\in\mathring{E}(\ttree)$.
To any such pair, we associate the sequence of tree morphisms such that

\begin{gather*}
\sigma(\ttree,\underline{e}) = \{\ttree\rightarrow\ttree/{e_1}\rightarrow\ttree/{\{e_1,e_2\}}\rightarrow\dots\rightarrow\ttree/\{e_1,e_2,\dots,e_m\}\}.
\intertext{and the map of degree $-1$}
\partial_{(\ttree,\underline{e})}: \DGSigma^{-\sharp V(\ttree)+m}\AOp(\ttree/\{e_1,\dots,e_m\})\rightarrow\DGSigma^{-\sharp V(\ttree)}\AOp(\ttree)
\intertext{given by top component of the homotopy coproduct $\rho_{\sigma(\ttree,\underline{e})}$}
\partial_{(\ttree,\underline{e})} = \rho^{\square}_{\sigma(\ttree,\underline{e})},
\end{gather*}

such as defined in Lemma <ref>.

Recall that the object $\DGSigma^{-\sharp V(\ttree)}\AOp(\ttree)$
is identified with a tensor product

\begin{equation*}
\DGSigma^{-\sharp V(\ttree)}\AOp(\ttree) = \bigl(\bigotimes_{v\in V(\ttree)}\underline{01}^{\sharp}_v\bigr)\otimes\AOp(\ttree),
\end{equation*}

where we associate a factor of cohomological degree one $\underline{01}^{\sharp}_v$ to every vertex $v\in V(\ttree)$.
In the definition of our map $\partial_{(\ttree,e)}$, we also perform blow-up operations

\begin{equation*}
\underline{01}_{u\equiv v}^{\sharp}\mapsto\underline{01}_{u}^{\sharp}\otimes\underline{01}_{v}^{\sharp},
\end{equation*}

for each edge contraction step $\ttree/{\{e_1,\dots,e_{i-1}\}}\rightarrow\ttree/{\{e_1,\dots,e_{i-1},e_i\}}$,
where $u$ and $v$ represent the vertices of the edge $e_i$ in the tree $\ttree/{\{e_1,\dots,e_{i-1}\}}$,
and $u\equiv v$ represents the result of the fusion of these vertices
in $\ttree/{\{e_1,\dots,e_{i-1}\}}$.
The performance of this sequence of blow-up operations, in parallel to the application of the map $\rho^{\square}_{\sigma(\ttree,\underline{e})}$,
enables us to pass
from the tensor product $\DGSigma^{-\sharp V(\ttree)+m}\AOp(\ttree/\{e_1,\dots,e_m\})
= \bigl(\bigotimes_{x\in V(\ttree/\{e_1,\dots,e_m\})}\underline{01}^{\sharp}_x\bigr)\otimes\AOp(\ttree/\{e_1,\dots,e_m\})$
to $\DGSigma^{-\sharp V(\ttree)}\AOp(\ttree) = \bigl(\bigotimes_{x\in V(\ttree)}\underline{01}^{\sharp}_x\bigr)\otimes\AOp(\ttree)$.
This operation may involve a sign, which we determine as in the strict case in Construction <ref>.

Finally, we take:

\begin{equation*}
\partial_m = \sum_{(\ttree,(e_1,\dots,e_m))}\partial_{(\ttree,(e_1,\dots,e_m))},\quad\text{for $m\geq 1$},
\quad\text{and}\quad\partial = \sum_{m\geq 1}\partial_m.
\end{equation*}

(We just use the connectedness condition on $\AOp$ to ensure that this sum reduces to a finite number of terms
on each summand $\DGSigma^{-\sharp V(\ttree)}\AOp(\ttree)$
of our object $\DGB^c(\AOp)(r)$.)

We prove next that this map $\partial$ also defines a twisting differential on $\DGB^c(\AOp)(r)$, so that we can again provide $\DGB^c(\AOp)(r)$
with a dg module structure with the sum $\delta+\partial: \DGB^c(\AOp)(r)\rightarrow\DGB^c(\AOp)(r)$ as total differential,
where $\delta: \DGB^c(\AOp)(r)\rightarrow\DGB^c(\AOp)(r)$ denotes the differential induced by the internal differential of the objects $\AOp(\ttree)$
in $\DGB^c(\AOp)(r)$ (as in the case of the cobar construction of strict Segal dg cooperads).

We equip the object $\DGB^c(\AOp)$ with composition products

\begin{equation*}
\circ_{i_p}: \DGB^c(\AOp)(\{i_1<\dots<i_k\})\otimes\DGB^c(\AOp)(\{j_1<\dots<j_l\})\rightarrow\DGB^c(\AOp)(\{1<\dots<r\}),
\end{equation*}

which we define exactly as in the strict case.
We explicitly define $\circ_{i_p}$ as the sum of the maps

\begin{equation*}
\circ_{i_p}^{\stree,\ttree}: \DGSigma^{-\sharp V(\stree)}\AOp(\stree)\otimes\DGSigma^{-\sharp V(\ttree)}\AOp(\ttree)
\rightarrow\DGSigma^{-\sharp V(\stree\circ_{i_p}\ttree)}\AOp(\stree\circ_{i_p}\ttree)
\end{equation*}

yielded by the Segal map $i_{\stree\circ_{i_p}\ttree}: \AOp(\stree)\otimes\AOp(\ttree)\rightarrow\AOp(\stree\circ_{i_p}\ttree)$
associated to the composition operation $\stree\circ_{i_p}\ttree = \lambda_{\gammatree}(\stree,\ttree)$
in the category of trees, where $\stree\in\Tree(\{i_1<\dots<i_k\})$ and $\ttree\in\Tree(\{j_1<\dots<j_l\})$.
In a forthcoming lemma, we also check that these operations preserve the above differential, and hence, provide our object with well-defined operations in the category of dg modules.

We first check the validity of the definition of the twisting differential announced in our construction. This result is a consequence of the following more precise lemma.

We have the relation $\delta\partial_m + \partial_m\delta + \sum_{i=1}^{m-1}\partial_i\partial_{m-i} = 0$, for each $m\geq 1$.

If $m = 1$, then the statement reduces to $\delta\partial_1 + \partial_1\delta = 0$, and we readily check, as in the case of the cobar construction of Segal dg cooperads,
that this relation is equivalent to the fact that the coproducts of degree zero $\rho_{\ttree\rightarrow\stree} = \rho_{\ttree\rightarrow\stree}^{\square}$
are morphisms of dg modules.
We now prove the statement for $m>1$. From Equation (<ref>) of Lemma <ref>,
we see that, for every tree $\ttree$ and every sequence of internal edges $\underline{e} = (e_1,\dots,e_m)$,
we have:

\begin{align*}
\delta\partial_{(\ttree,\underline{e})} + \partial_{(\ttree,\underline{e})}\delta &
= \sum_{i=1}^{m-1}\pm\rho^{\square}_{\ttree\rightarrow\dots\rightarrow\widehat{\ttree/\{e_1,\dots,e_i\}}\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_m\}} \\
& + \sum_{i=1}^{m-1}\pm\rho^{\square}_{\ttree\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_i\}}\circ\rho^{\square}_{\ttree/\{e_1,\dots,e_i\}\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_m\}}
\end{align*}

Then, by taking the sum of these expressions over the set of pairs $(\ttree,\underline{e})$, we obtain the formula:

\begin{align*}
\delta\partial_m + \partial_m\delta &
= \sum_{\substack{(\ttree,\underline{e})\\i=1,\dots,m}}\pm\rho^{\square}_{\ttree\rightarrow\dots\rightarrow\widehat{\ttree/\{e_1,\dots,e_i\}}\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_m\}} \\
& + \sum_{i=1}^{m-1}\bigl(\underbrace{\sum_{(\ttree,\underline{e})}\pm\rho^{\square}_{\ttree\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_i\}}
\circ\rho^{\square}_{\ttree/\{e_1,\dots,e_i\}\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_m\}}}_{= \partial_i\partial_{m-i}}\bigr).
\end{align*}

In the first sum of this formula, the term that corresponds to the removal of the node $\ttree/\{e_1,\dots,e_{i-1},e_i\}$
and the term that corresponds to the removal of the node $\ttree/\{e_1,\dots,e_{i-1},e_{i+1}\}$
for the pair $(\ttree,(e_1,\dots,e_{i-1},e_{i+1},e_i,e_{i+2},\dots,e_m))$
with $e_i$ and $e_{i+1}$ switched
are equal up to a sign. We readily check that these signs are opposite, so that these terms cancel out in our sum. The conclusion of the lemma follows.

We still check the validity of our definition of the composition products.

The twisting map $\partial$ and the differential $\delta$ induced by the internal differential of the object $\AOp$ on the cobar construction $\DGB^c(\AOp)$
form derivations with respect the composition products of Construction <ref>,
so that these operations $\circ_{i_p}$ define morphisms of dg modules.

We generalize the arguments used in Lemma <ref>.
We again prove that $\circ_{i_p}$ commutes with the differential on each summand $\DGSigma^{-\sharp V(\stree)}\AOp(\stree)\otimes\DGSigma^{-\sharp V(\ttree)}\AOp(\ttree)$
of the tensor product $\DGB^c(\AOp)(\{i_1<\dots<i_k\})\otimes\DGB^c(\AOp)(\{j_1<\dots<j_l\})$,
where $\stree\in\Tree(\{i_1<\dots<i_k\})$, $\ttree\in\Tree(\{j_1<\dots<j_l\})$.
We can still use that the Segal maps, which induce our composition product componentwise, are morphisms of dg modules to conclude that $\circ_{i_p}$
preserves the internal differentials on our objects.
We therefore focus on the verification that $\circ_{i_p}$ preserves the twisting differential $\partial$.

We set $\thetatree = \stree\circ_{i_p}\ttree$ and we consider a tree $\thetatree'$ such that $\thetatree = \thetatree'/\{e_1,\dots,e_m\}$
for an sequence of edges $e_1,\dots,e_m\in\mathring{E}(\thetatree')$.
We then have $\thetatree' = \stree'\circ_{i_p}\ttree'$, where $\stree'$ and $\ttree'$ represent the pre-image of the subtrees $\stree\subset\thetatree$ and $\ttree\subset\thetatree$
under the map $\thetatree'\rightarrow\thetatree'/\{e_1,\dots,e_m\}$, and $\stree = \stree'/\{e_{\alpha_1},\dots,e_{\alpha_r}\}$, $\ttree = \ttree'/\{e_{\beta_1},\dots,e_{\beta_s}\}$,
for the partition $\{e_{\alpha_1},\dots,e_{\alpha_r}\}\amalg\{e_{\beta_1},\dots,e_{\beta_s}\} = \{e_1,\dots,e_m\}$
such that $e_{\alpha_1},\dots,e_{\alpha_r}\in\mathring{E}(\stree')$ and $e_{\beta_1},\dots,e_{\beta_s}\in\mathring{E}(\ttree')$.
We may note that one the component of this partition can be empty (we take by convention $r = 0$ or $s = 0$ in this case).

The compatibility between the Segal maps and the coproduct operators in Figure <ref>
together with the degeneracy relations of Figure <ref>
imply the commutativity of the following diagram:

\begin{equation*}
\xymatrixcolsep{5pc}\xymatrix{ \AOp(\stree)\otimes\AOp(\ttree)
\ar[r]^-{\rho_{\sigma(\stree',\underline{e}|_{\stree'})}\otimes\rho_{\sigma(\ttree',\underline{e}|_{\ttree'})}}
\ar[ddd]|{i_{\stree\circ_{i_p}\ttree}} &
\AOp(\stree')\otimes I^{r-1}\otimes\AOp(\ttree')\otimes I^{s-1}
\ar[d]|{(\id\otimes s^{m-\beta_*})\otimes(\id\otimes s^{m-\alpha_*})} \\
& \AOp(\stree')\otimes I^{m-1}\otimes\AOp(\ttree)\otimes I^{m-1}\ar[d]^{\simeq} \\
& \AOp(\stree')\otimes\AOp(\ttree')\otimes I^{m-1}\otimes I^{m-1}\ar[d]^{i_{\stree',\ttree',\thetatree}\otimes\mu} \\
\AOp(\thetatree)\ar[r]_{\rho_{\sigma(\thetatree',\underline{e})}} &
\AOp(\thetatree')\otimes I^{m-1} },
\end{equation*}

where we set $\underline{e}|_{\stree'} = (e_{\alpha_1},\dots,e_{\alpha_r})$ and $\underline{e}|_{\ttree'} = (e_{\beta_1},\dots,e_{\beta_s})$ for short,
and $s^{m-\alpha_*} = s^{m-\alpha_1} s^{m-\alpha_2}\cdots s^{m-\alpha_r}$, $s^{m-\beta_*} = s^{m-\beta_1} s^{m-\beta_2}\cdots s^{m-\beta_s}$.
(These composites correspond to the positions of the degeneracies
when we take the pre-image of the subtrees $\stree,\ttree\subset\thetatree$
under the sequence of tree morphisms $\thetatree'\rightarrow\thetatree'/\{e_1\}\rightarrow\dots\rightarrow\thetatree'/\{e_1,\dots,e_m\} = \thetatree$.)

Note that we may have $r = 0$ or $s = 0$ and our diagram is still valid in these cases. (We then take $I^{-1} = \kk$ by convention and $s^0: I^{-1}\rightarrow I^0$
denotes the identity map.)
We actually have three possible cases:

* $r = 0$:
In this case all the edges of our collection $\underline{e}$ belong to $\ttree'$, we have $\stree = \stree'$,
and the commutativity of the diagram implies that $\partial_{\theta',\underline{e}}\circ i_{\stree\circ_{i_p}\ttree}
= i_{\stree\circ_{i_p}\ttree'}\circ(\id\otimes\partial_{\ttree',\underline{e}})$.

* $s = 0$:
In this mirror case, all the edges of our collection $\underline{e}$ belong to $\stree'$, we have $\ttree = \ttree'$,
and we get $\partial_{\theta',\underline{e}}\circ i_{\stree\circ_{i_p}\ttree}
= i_{\stree'\circ_{i_p}\ttree}\circ(\partial_{\stree',\underline{e}}\otimes\id)$.

* $r,s\geq 1$: in this case, the composite of the vertical morphisms on the right hand side of the diagram
does not meet $\AOp(\stree'\circ_{i_p}\ttree')\otimes\underline{01}^{\sharp}{}^{\otimes m-1}$,
because the product of degeneracies carries $I^{r-1}\otimes I^{s-1}$ to a submodule of $I^{m-1}\otimes I^{m-1}$ concentrated in dimension $<m-1$,
whose image under the product can not meet $\underline{01}^{\sharp}{}^{\otimes m-1}$,
so that we have $\partial_{\theta',\underline{e}}\circ i_{\stree\circ_{i_p}\ttree} = 0$.

From these identities, we obtain the derivation relation $\partial_m\circ\circ_{i_p} = \circ_{i_p}\circ(\partial_m\otimes\id + \id\otimes\partial_m)$,
valid for each $m\geq 1$. The conclusion follows.

We still immediately deduce from the associativity of the Segal maps that the composition products of Construction <ref>
satisfy the associativity relations of the composition products of an operad.
We therefore get the following concluding statement:

The collection $\DGB^c(\AOp) = \{\DGB^c(\AOp)(r),r>0\}$ equipped with the differential and structure operations defined in Construction <ref>
forms a shuffle operad in dg modules.
This operad $\DGB^c(\AOp)$ is the cobar construction of the connected homotopy Segal shuffle dg (pre-)cooperad $\AOp$.

§.§ The definition of homotopy morphisms

We devote this section to the study of homotopy morphisms of Segal cooperads.
We always assume that our target object is equipped with a strict Segal cooperad structure for technical reasons,
but our source object can be equipped with a general homotopy Segal cooperad structure.
We explain the definition of these homotopy morphisms in the context of $E_\infty$-Hopf cooperads first.
We examine the forgetting of $E_\infty$-structures afterwards and then we study the application
of homotopy morphisms to the cobar construction.

We assume that $\BOp$ is a strict Segal $E_\infty$-Hopf shuffle cooperad while $\AOp$ can be any homotopy Segal $E_\infty$-Hopf shuffle cooperad.
We then define a homotopy morphism of homotopy Segal $E_\infty$-Hopf shuffle cooperads $\phi: \AOp\rightarrow\BOp$
as a collection of $\EOp$-algebra morphisms

\begin{equation*}
\phi_{\ttree}: \AOp(\ttree)\rightarrow\BOp(\ttree),\quad\text{$\ttree\in\Tree(r)$, $r>0$},
\end{equation*}

referred to as the underlying maps of our homotopy morphism, together with a collection of higher morphism operators

\begin{equation*}
\phi_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}: \AOp(\stree)\rightarrow\BOp(\ttree)\otimes I^{k+1},
\end{equation*}

defined in the category of $\EOp$-algebras as well and associated to sequences of composable tree morphisms $\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree$,
so that the compatibility relations with the face and degeneracy operators expressed by the diagrams of Figure <ref>-<ref>
hold, as well the compatibility relations with the facet operators
expressed by the diagrams of Figure <ref>-<ref>.
(For the underlying maps of our homotopy morphism, we just retrieve the relation of Definition <ref>.)
When $\AOp$ and $\BOp$ are symmetric cooperads, we say that $\phi: \AOp\rightarrow\BOp$
defines a homotopy morphism of homotopy Segal $E_\infty$-Hopf symmetric cooperads
if we have also the compatibility relations with the action of permutations
expressed by the diagrams of Figure <ref>-<ref>.
(For the underlying maps of our homotopy morphism, we just retrieve the relation of Definition <ref>.)

We also have a version of this definition for homotopy Segal shuffle dg cooperads without $E_\infty$-structure.

We assume that $\BOp$ be a strict Segal shuffle dg cooperad while $\AOp$ can be any homotopy Segal shuffle dg cooperad.
We then define a homotopy morphism of homotopy Segal shuffle dg cooperads $\phi: \AOp\rightarrow\BOp$
as a collection of morphisms dg modules

\begin{equation*}
\phi_{\ttree}: \AOp(\ttree)\rightarrow\BOp(\ttree),\quad\text{$\ttree\in\Tree(r)$, $r>0$},
\end{equation*}

to which we again refer as the underlying maps of our homotopy morphism, together with a collection of higher morphism operators

\begin{equation*}
\phi_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}: \AOp(\stree)\rightarrow\BOp(\ttree)\otimes I^{k+1},
\end{equation*}

defined in the category of dg modules as well and associated to sequences of composable tree morphisms $\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree$,
so that we have the compatibility relations with respect to the face and degeneracy operators expressed by the diagrams
of Figure <ref>-<ref> (as in the case of homotopy morphisms $E_\infty$-Hopf cooperads),
together with the compatibility relations with respect to the Segal maps
expressed by the diagrams of Figure <ref>-<ref>.

The compatibility of homotopy morphisms with $0$-faces.
The diagrams commute for all sequences of composable tree morphisms $\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree$
and for all $1\leq i\leq k$, where $\widehat{\ttree_i}$ means that we delete the node $\ttree_i$.
10pc ()

()⊗I^k+1 [d]^𝕀⊗d_0^i

The compatibility of homotopy morphisms with $1$-faces.
The diagrams commute for all sequences of composable tree morphisms $\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree$
and for all $1\leq i\leq k$.

4pc ()

The compatibility of homotopy morphisms with degeneracies.
The diagrams commute for all sequences of composable tree morphisms $\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree$
and for all $0\leq j\leq k+1$.
10pc ()

[rd]_ϕ_→_k→⋯_j = _j⋯→_1→

The preservation of facet operators by the underlying maps of homotopy morphisms.
The diagram commutes for all subtrees $\sigmatree\subset\stree$.

5pc ()[r]^ϕ_     ()

The compatibility of homotopy morphisms with facet operators.
The diagram commutes for all subtrees $\sigmatree\subset\stree$
and for all sequences of composable tree morphisms $\ttree\xrightarrow{f_k}\ttree_k\xrightarrow{f_{k-1}}\dots\xrightarrow{f_1}\ttree_1\xrightarrow{f_0}\stree$.

10pc ()[r]^-ϕ_→_k→…→_1→

The preservation of the action of permutations by the underlying maps of homotopy morphisms.
The diagram commutes for all $s\in\Sigma_r$ and $\ttree\in\Tree(r)$.

5pc (s)[r]^ϕ_s[d]_s^*     (s)[d]^s^*

()[r]^ϕ_     ()

The compatibility of homotopy morphisms with facet operators.
The diagram commutes for all subtrees $\sigmatree\subset\stree$
and for all sequences of composable tree morphisms $\ttree\xrightarrow{f_k}\ttree_k\xrightarrow{f_{k-1}}\dots\xrightarrow{f_1}\ttree_1\xrightarrow{f_0}\stree$.

10pc ()[r]^-ϕ_→_k→…→_1→

The preservation of Segal maps by the underlying map of homotopy morphisms of homotopy Segal dg cooperads.
The diagram commutes for all tree decompositions $\stree = \lambda_{\utree}(\sigmatree_v,v\in V(\utree))$.

5pc ()[r]^ϕ_     ()

The compatibility of homotopy morphisms of homotopy Segal dg cooperads with the Segal maps.
The diagram commutes for all tree decompositions $\stree = \lambda_{\utree}(\sigmatree_v,v\in V(\utree))$
and for all sequences of composable tree morphisms $\ttree\xrightarrow{f_k}\ttree_k\xrightarrow{f_{k-1}}\dots\xrightarrow{f_1}\ttree_1\xrightarrow{f_0}\stree$.

10pc ()

@.>[r]^-⊗_v ϕ_(f_k…f_0)^-1(_v)→…→f_0^-1 (_v)→_v
⊗_v ∈V()(((f_k…f_0)^-1(_v))⊗I^k+1)

We have the following statement, which is the homotopy version of the result of Lemma <ref>,
and which can be proved by the same arguments.
We still write $\alpha^{\square}: X\rightarrow Y$ for the homomorphism of graded modules of degree $k$
associated to a morphism of dg modules $\alpha: X\rightarrow Y \otimes I^k$
such that $\alpha(x) = (-1)^{k\deg(x)}\alpha^{\square}(x)\otimes\underline{01}^{\sharp}{}^{\otimes k} + \text{tensors with a factor of dimension $<k$ in $I^k$}$.

* Let $\phi: \AOp\rightarrow\BOp$ be a homotopy morphism of homotopy Segal shuffle dg cooperads,
where we still assume that $\BOp$ is a strict Segal shuffle dg cooperad
as in Definition <ref>.
The graded homomorphism of degree $k+1$

\begin{equation*}
\phi^{\square}_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}: \AOp(\stree)\rightarrow\AOp(\ttree)
\end{equation*}

that we associate to the dg module morphism $\phi_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}: \AOp(\stree)\rightarrow\AOp(\ttree)\otimes I^{k+1}$,
for any sequence of composable tree morphisms $\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree$,
satisfies the relation

\begin{equation}\tag{*}\label{equation:homotopy-morphism-top-component}
\delta(\phi^{\square}_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree})
\begin{aligned}[t]
& = (-1)^{k+1}\rho^{\BOp}_{\ttree_k\rightarrow\stree}\phi^{\square}_{\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree} \\
& + \sum_{i=1}^k(-1)^i\phi^{\square}_{\ttree\rightarrow\dots\rightarrow\hat{\ttree}_i\rightarrow\dots\rightarrow\stree} \\
& + (-1)^k\phi_{\ttree}\circ\rho^{\AOp\square}_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree} \\
& - \sum_{i=1}^k(-1)^i\phi^{\square}_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_i}\circ\rho^{\AOp\square}_{\ttree_i\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}.
\end{aligned}
\end{equation}

Moreover, if we have a degeneracy $\ttree_j = \ttree_{j+1}$ in our sequence of tree morphisms, for some $0\leq j\leq k$ (with the convention that $\ttree_{k+1} = \ttree$ and $\ttree_0 = \stree$),
then we have the relation

\begin{equation}\tag{**}\label{equation:homotopy-morphism-degeneracy-top-component}
\phi^{\square}_{\ttree\rightarrow\ttree_{k}\rightarrow\dots\ttree_{j+1}=\ttree_j\cdots\rightarrow\ttree_1\rightarrow\stree} = 0.
\end{equation}

* In the converse direction, if we have a collection of dg module morphisms $\phi_{\ttree}: \AOp(\ttree)\rightarrow\BOp(\ttree)$, $\ttree\in\Tree(r)$, $r>0$,
together with a collection of dg graded homomorphisms $\phi^{\square}_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}: \AOp(\stree)\rightarrow\AOp(\ttree)$,
of degree $k+1$,
which satisfy the relations (<ref>)-(<ref>)
of the previous statement,
then there is a unique collection of morphisms of dg modules $\phi_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}: \AOp(\stree)\rightarrow\AOp(\ttree)\otimes I^k$,
which extend these maps on the summands $\AOp(\ttree)\otimes\underline{01}^{\sharp}{}^{\otimes k+1}$
and satisfy the face and degeneracy relations of homotopy morphism operators
of Figure <ref>-<ref>.

We now prove that any homotopy morphism of homotopy Segal shuffle dg cooperads $\phi: \AOp\rightarrow\BOp$, as in Definition <ref>,
gives rise to an induced morphism on the cobar construction $\phi_*: \DGB^c(\AOp)\rightarrow\DGB^c(\BOp)$.
We address the definition of this morphism in the next paragraph.
We assume all along this study that a homotopy morphism $\phi: \AOp\rightarrow\BOp$ is fixed, with $\AOp$ a homotopy Segal shuffle dg cooperad
and $\BOp$ a strict Segal shuffle dg cooperad.
We need to assume that the object $\AOp$ is connected in order to give a sense to the cobar construction $\DGB^c(\AOp)$ (see <ref>).
We also need to assume that $\BOp$ is connected in the construction of our morphisms.
We therefore assume that these connectedness conditions hold in the rest of this subsection.

The underlying maps of our homotopy morphism $\phi: \AOp\rightarrow\BOp$ induce morphisms of graded modules between the components of the cobar construction:

\begin{equation*}
\phi_{\ttree}: \DGSigma^{-\sharp V(\ttree)}\AOp(\ttree)\rightarrow\DGSigma^{-\sharp V(\ttree)}\AOp(\ttree).
\end{equation*}

In addition to these maps, we consider morphisms

\begin{equation*}
\phi_{(\ttree,\underline{e})}: \DGSigma^{-\sharp V(\ttree)+m}\AOp(\ttree/\{e_1,\dots,e_m\})\rightarrow\DGSigma^{-\sharp V(\ttree)}\BOp(\ttree).
\end{equation*}

associated to the pairs $(\ttree,\underline{e})$, where $\ttree$ is a tree and $\underline{e} = (e_1,\dots,e_m)$
is an ordered collection of pairwise distinct edges $e_i\in\mathring{E}(\ttree)$,
as in Construction <ref>.
To define the latter maps, we again consider the sequence of composable tree morphisms

\begin{gather*}
\sigma(\ttree,\underline{e}) = \{\ttree\rightarrow\ttree/{e_1}\rightarrow\ttree/{\{e_1,e_2\}}\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_m\}\},
\intertext{which we associate to any such pair $(\ttree,\underline{e})$ in Construction~\ref{construction:bar-complex-homotopy},
and we set}
\phi_{(\ttree,\underline{e})} = \phi^{\square}_{\sigma(\ttree,\underline{e})},
\end{gather*}

where we take the top component of the morphism $\phi_{\sigma(\ttree,\underline{e})}$ (such as defined in Lemma <ref>).
In this construction, we also use the same blow-up process as in Construction <ref>
to pass from the tensor product $\DGSigma^{-\sharp V(\ttree)+m}\AOp(\ttree/\{e_1,\dots,e_m\})
= \bigl(\bigotimes_{x\in V(\ttree/\{e_1,\dots,e_m\})}\underline{01}^{\sharp}_x\bigr)\otimes\AOp(\ttree/\{e_1,\dots,e_m\})$
to $\DGSigma^{-\sharp V(\ttree)}\BOp(\ttree) = \bigl(\bigotimes_{x\in V(\ttree)}\underline{01}^{\sharp}_x\bigr)\otimes\AOp(\ttree)$
and to determine a possible sign, which we associate to our map $\phi_{(\ttree,\underline{e})}$.
In what follows, we identify the morphisms $\phi_{\ttree}$, induced by the underlying maps of our homotopy morphism $\phi: \AOp\rightarrow\BOp$,
with the case $m=0$ of these homomorphisms $\phi_{(\ttree,\underline{e})}$.

Finally, we take:

\begin{gather*}
\phi_m = \sum_{(\ttree,(e_1,\dots,e_m)}\phi_{(\ttree,(e_1,\dots,e_m))},\quad\text{for $m\geq 0$},
\quad\text{and}\quad\phi = \sum_{m\geq 0}\phi_m
\intertext{to get a map}
\phi_*: \DGB^c(\AOp)(r)\rightarrow\DGB^c(\BOp)(r),\quad\text{for each arity $r>0$}.
\end{gather*}

(Note that we use the connectedness condition on $\BOp$ to ensure that the above sum reduces to a finite number of terms on each summand $\DGSigma^{-\sharp V(\ttree)}\AOp(\ttree)$.)

We aim to prove that this map is compatible with the structure operations of the cobar construction.

We check the preservation of differentials first. This claim follows from the following more precise observation.

We have the relation

\begin{equation*}
\delta^{\BOp}\phi_m = \phi_m \delta^{\AOp} + \partial^{\BOp}\phi_{m-1} - \sum_{i=0}^{m-1}\phi_i\partial_{m-i}^{\AOp},
\end{equation*}

for all $m\geq 0$,
where $\delta = \delta^{\AOp},\delta^{\BOp}$ denotes the term of the differential of the cobar construction induced by the internal differential of the objects $\COp = \AOp,\BOp$,
we denote by $\partial^{\AOp} = \sum_{m = 1}^{\infty}\partial_m^{\AOp}$ the twisting map of the cobar construction
of the homotopy Segal shuffle dg cooperad $\AOp$,
while $\partial^{\BOp}$ denotes the twisting differential of the cobar construction
of the strict Segal shuffle dg cooperad $\AOp$.

We argue as in the proof of Lemma <ref>.
We use the relation of Equation (<ref>) of Lemma <ref>
to write

\begin{align*}
\delta^{\BOp}\phi_{(\ttree,\underline{e})} - \phi_{(\ttree,\underline{e})}\delta^{\AOp} &
= \sum_{i=1}^{m-1}\pm\phi^{\square}_{\ttree\rightarrow\dots\rightarrow\widehat{\ttree/\{e_1,\dots,e_i\}}\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_m\}} \\
& - \rho^{\BOp}_{\ttree\rightarrow\ttree/e_1}\phi^{\square}_{\ttree/e_1\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_m\}}\\
& + \sum_{i=0}^{m-1}\pm\phi^{\square}_{\ttree\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_i\}}\rho^{\AOp\square}_{\ttree/\{e_1,\dots,e_i\}\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_m\}}.
\end{align*}

Then, by taking the sum of these expressions over the set of pairs $(\ttree,\underline{e})$, we obtain the formula:

\begin{align*}
\delta^{\BOp}\phi_m - \phi_m\delta^{\AOp} &
= \sum_{\substack{(\ttree,\underline{e})\\i=1,\dots,m}}\pm\phi^{\square}_{\ttree\rightarrow\dots\rightarrow\widehat{\ttree/\{e_1,\dots,e_i\}}\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_m\}} \\
& + \underbrace{\sum_{(\ttree,\underline{e})}\pm\rho^{\BOp}_{\ttree\rightarrow\ttree/e_1}\phi^{\square}_{\ttree/e_1\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_m\}}}_{=\partial^{\BOp}\phi_{m-1}} \\
& + \sum_{i=0}^{m-1}\bigl(\underbrace{\sum_{(\ttree,\underline{e})}\pm\phi^{\square}_{\ttree\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_i\}}
\circ\rho^{\AOp\square}_{\ttree/\{e_1,\dots,e_i\}\rightarrow\dots\rightarrow\ttree/\{e_1,\dots,e_m\}}}_{= \phi_i\partial_{m-i}}\bigr).
\end{align*}

In the first sum of this formula, the term that corresponds to the removal of the node $\ttree/\{e_1,\dots,e_{i-1},e_i\}$
and the term that corresponds to the removal of the node $\ttree/\{e_1,\dots,e_{i-1},e_{i+1}\}$
for the pair $(\ttree,(e_1,\dots,e_{i-1},e_{i+1},e_i,e_{i+2},\dots,e_m))$
with $e_i$ and $e_{i+1}$ switched
are again equal up to a sign. We can still check that these signs are opposite, so that these terms cancel out in our sum. The conclusion of the lemma follows.

We now check that our morphisms preserves the composition products. This claim follows from the following more precise observation.

We have the relation $\phi_m\circ\circ_{i_p} = \sum_{r+s=m}\circ_{i_p}\circ(\phi_r\otimes\phi_s)$, for all $m\geq 0$.

The proof is similar to that of Lemma <ref>.

For $m = 0$, the relation follows the commutativity of the diagram of Figure <ref> (since $\circ_{i_p}$ is defined as a sum of Segal maps).
We therefore focus on the case $m\geq 1$.
We consider again a summand $\Sigma^{-\sharp V(\stree)}\AOp(\stree)\otimes\Sigma^{-\sharp V(\ttree)}\AOp(\ttree)$
of the tensor product $\DGB^c(\AOp)(\{i_1<\dots<i_k\})\otimes\DGB^c(\AOp)(\{j_1<\dots<j_l\})$,
where $\stree\in\Tree(\{i_1<\dots<i_k\})$, $\ttree\in\Tree(\{j_1<\dots<j_l\})$.
The composition product maps $\circ_{i_p}$ carry this summand into $\Sigma^{-\sharp V(\thetatree)}\AOp(\thetatree)$,
with $\thetatree = \stree\circ_{i_p}\ttree$.
The components of the map $\phi_m$ carry this summand into terms of the form $\Sigma^{-\sharp V(\thetatree')}\AOp(\thetatree')$,
for trees $\thetatree'$ equipped with a set of internal edges $(e_1,\dots,e_m)$
such that $\thetatree'/\{e_1,\dots,e_m\} = \thetatree$.
We still have $\thetatree' = \stree'\circ_{i_p}\ttree'$ and $\stree = \stree'/\{e_{\alpha_1},\dots,e_{\alpha_r}\}$, $\ttree = \ttree'/\{e_{\beta_1},\dots,e_{\beta_s}\}$,
for a partition $\{e_{\alpha_1},\dots,e_{\alpha_r}\}\amalg\{e_{\beta_1},\dots,e_{\beta_s}\} = \{e_1,\dots,e_m\}$
such that $e_{\alpha_1},\dots,e_{\alpha_r}\in\mathring{E}(\stree')$ and $e_{\beta_1},\dots,e_{\beta_s}\in\mathring{E}(\ttree')$.
We then have the following commutative diagram:

\begin{equation*}
\xymatrixcolsep{8pc}\xymatrix{
\AOp(\stree)\otimes\AOp(\ttree)
\ar[r]^-{\phi_{\sigma(\stree',\underline{e}|_{\stree'})}\otimes\phi_{\sigma(\ttree',\underline{e}|_{\ttree'})}}
\ar[ddd]|{i_{\stree\circ_{i_p}\ttree}} &
(\BOp(\stree')\otimes I^r)\otimes(\BOp(\ttree')\otimes I^s)
\ar[d]|{\id\otimes s^{m-\beta_*}\otimes\id\otimes s^{m-\alpha_*}} \\
& (\BOp(\stree')\otimes I^m)\otimes(\BOp(\ttree')\otimes I^m)
\ar[d]^{\simeq} \\
& \BOp(\stree')\otimes\BOp(\ttree')\otimes I^m\otimes I^m
\ar[d]^{i_{\stree'\circ_{i_p}\ttree'}\otimes\mu} \\
\AOp(\thetatree)\ar[r]_{\phi_{\sigma(\thetatree',\underline{e})}} &
\BOp(\thetatree')\otimes I^m }
\end{equation*}

(by the relations of Figure <ref> and Figure <ref>).
We use the same notation as in the proof of Lemma <ref>
in this diagram
and we still consider the morphisms $\phi_{\sigma(\stree',\underline{e}|_{\stree'})}: \AOp(\stree'/\{e_{\alpha_1},\dots,e_{\alpha_r}\})\rightarrow\AOp(\stree')\otimes I^r$
and $\phi_{\sigma(\ttree',\underline{e}|_{\ttree'})}: \AOp(\ttree'/\{e_{\beta_1},\dots,e_{\beta_s}\})\rightarrow\AOp(\ttree')\otimes I^s$
associated to the sequences of tree morphisms
such that $\sigma(\stree',\underline{e}|_{\stree'}) = \{\stree'\rightarrow\stree'/e_{\alpha_1}\rightarrow\dots\rightarrow\stree'/\{e_{\alpha_1},\dots,e_{\alpha_r}\}\}$
and $\sigma(\ttree',\underline{e}|_{\ttree'}) = \{\ttree'\rightarrow\ttree'/e_{\beta_1}\rightarrow\dots\rightarrow\ttree'/\{e_{\beta_1},\dots,e_{\beta_s}\}\}$.
We also use the notation $s^{m-\alpha_*} = s^{m-\alpha_1} s^{m-\alpha_2}\cdots s^{m-\alpha_r}$, $s^{m-\beta_*} = s^{m-\beta_1} s^{m-\beta_2}\cdots s^{m-\beta_s}$,
and $\mu: I^m\otimes I^m\rightarrow I^m$ denotes the product of the dg algebra $I^m$
as usual.

We see, by elaborating on the arguments of the proof of Lemma <ref>,
that the composite of the right-hand side vertical morphisms of this diagram does not meet $\BOp(\thetatree')\otimes\underline{01}^{\sharp}{}^{\otimes m}$
unless we have $\{e_{\alpha_1},\dots,e_{\alpha_r}\} = \{e_1,\dots,e_r\}$ and $\{e_{\beta_1},\dots,e_{\beta_s}) = (e_{r+1},\dots,e_m\}$. (We have in this case
$s^{m-\beta_*}(\underline{01}^{\sharp}{}^{\otimes r}) = \underline{01}^{\sharp}{}^{\otimes r}\otimes 1^{\otimes s}$,
$s^{m-\alpha_*}(\underline{01}^{\sharp}{}^{\otimes s}) = \underline{1}^{\sharp}{}^{\otimes r}\otimes\underline{01}^{\sharp}{}^{\otimes s} + \text{other terms}$,
and $\mu(s^{m-\beta_*}(\underline{01}^{\sharp}{}^{\otimes r})\otimes s^{m-\alpha_*}(\underline{01}^{\sharp}{}^{\otimes s})) = \underline{01}^{\sharp}{}^{\otimes m}$.)
We conclude from this analysis that the composite $\phi_{(\thetatree',\underline{e})}\circ\circ_{i_p}$ vanishes unless the edge collection $\underline{e} = (e_1,\dots,e_m)$
is equipped with an order such that $\{e_{\alpha_1},\dots,e_{\alpha_r}\} = \{e_1,\dots,e_r\}$
and $\{e_{\beta_1},\dots,e_{\beta_s}\} = \{e_{r+1},\dots,e_m\}$.
We get in this case $\phi_{(\thetatree',\underline{e})}\circ\circ_{i_p} = \circ_{i_p}\circ\phi_{(\stree',\underline{e}|_{\stree'})}\otimes\phi_{(\ttree',\underline{e}|_{\ttree'})}$
and summing over the pairs $(\thetatree',(e_1,\dots,e_m))$ with $\thetatree' = \stree'\circ_{i_p}\ttree'$, $e_1,\dots,e_r\in\mathring{E}(\stree')$, $e_{r+1},\dots,e_m\in\mathring{E}(\stree')$,
amounts to summing over the pairs $(\stree',(e_1,\dots,e_r))$ and $(\ttree',(e_{r+1},\dots,e_m))$
such that $\stree'/\{e_1,\dots,e_r\} = \stree$ and $\ttree'/\{e_1,\dots,e_r\} = \ttree$.
We therefore obtain the relation of the lemma $\phi_m\circ\circ_{i_p} = \sum_{r+s=m}\circ_{i_p}\circ(\phi_r\otimes\phi_s)$
when we perform this sum.

We get the following concluding statement:

The collection of morphisms $\phi_*: \DGB^c(\AOp)(r)\rightarrow\DGB^c(\BOp)(r)$, $r>0$, defined in Construction <ref>,
defines a morphism of shuffle dg operads $\phi_*: \DGB^c(\AOp)\rightarrow\DGB^c(\BOp)$,
the morphism induced by the homotopy morphism of connected homotopy Segal shuffle cooperads $\phi: \AOp\rightarrow\BOp$
on the cobar construction.

§.§ The equivalence with strict $ E_\infty $-cooperads

We devote this final subsection to proving the following result.

Every connected homotopy Segal $E_\infty$-Hopf cooperad (either symmetric or shuffle)
is weakly-equivalent to a connected strict Segal $E_\infty$-Hopf cooperad.

We prove Theorem <ref> by constructing a functor $\AOp\mapsto\DGK^c(\AOp)$, from the category of connected homotopy Segal $E_\infty$-Hopf cooperads (symmetric or shuffle)
to the category of strict Segal $E_\infty$-Hopf cooperads, and a zigzag of weak-equivalences between $\AOp$ and $\DGK^c(\AOp)$.

For this purpose, we consider a category $\Tree^{\square}$, enriched in dg modules, which encodes the homotopy coproduct operators of homotopy Segal cooperads.
We explain the definition of this category $\Tree^{\square}$ in the next paragraph.
We have a morphism of enriched categories $\Tree^{\square}\rightarrow\Tree$, where, by an abuse of notation, we denote by $\Tree$ the enriched category in $\kk$-modules
whose hom-objects are the $\kk$-modules spanned by the set-theoretic tree morphisms.
We define our functor $\DGK^c(-)$ as a homotopy Kan extension, by dualizing a two-sided bar complex over the enriched category $\Tree^{\square}$.

Note that, in our statement, we again assume that our Segal cooperad $\AOp$ is connected in the sense of <ref>.
This assumption enables us to simplify our constructions and to avoid technical difficulties
in the verification of our result. We assume that our cooperads satisfy this connectedness condition all along this section.

Recall that a (homotopy or strict) Segal cooperad $\AOp$ is connected if we have $\AOp(\ttree) = 0$
when the tree $\ttree$ is not reduced (has at least one vertex with a single ingoing edge)
and that this condition implies that we can restrict ourselves to the subcategories of reduced trees, denoted by $\widetilde{\Tree}(r)\subset\Tree(r)$, $r>0$,
in the definition of the structure operations that we associate to our objects.
For simplicity, all along this subsection, we keep the notation $\Tree$ for our constructions on tree categories (for instance, we use the notation $\Tree^{\square}$
for our enriched category of trees).
Nevertheless, we restrict ourselves to reduced trees, as permitted by our connectedness assumption on cooperads,
and for this reason, we only define the enriched hom-objects $\Tree^{\square}(\ttree,\stree)$ associated to the full subcategories of reduced trees $\widetilde{\Tree}^{\square}(r)$, $r>0$.

Recall that for reduced trees $\stree,\ttree\in\widetilde{\Tree}(r)$, the set of tree morphisms $\Tree(\ttree,\stree)$
is either empty or reduced to a point (see <cit.>).
For the enriched version of this category, we therefore have:

\begin{equation*}
\Tree(\ttree,\stree) = \begin{cases} \kk, & \text{if we have a morphism $\ttree\rightarrow\stree$}, \\
0, & \text{otherwise}, \end{cases}
\end{equation*}

for any pair of reduced trees $\stree,\ttree\in\widetilde{\Tree}(r)$, $r>0$.
Note that we may still write $\Tree(\ttree,\stree) = *$ in this setting, because we identify the object $\Tree(\ttree,\stree) = \kk$
with the terminal object of the category of cocommutative coalgebras, and we can actually regard $\Tree$
as a category enriched in cocommutative coalgebras.
This observation, to which we go back later on, motivates our abuse of notation.

We define the enriched category $\Tree^{\square}$ in this paragraph. We take the same set of objects as the category of reduced trees $\widetilde{\Tree}$.

In the definition of the hom-objects, we consider the cubical chain complexes $\DGN_*(\Delta^1)^{\otimes k}$, dual to the cubical cochain algebras $I^k = \DGN^*(\Delta^1)^{\otimes k}$
of the definition of homotopy Segal cooperads.
We use the coface operators $d_i^{\epsilon}: \DGN_*(\Delta^1)^{\otimes k-1}\rightarrow\DGN_*(\Delta^1)^{\otimes k}$
and the codegeneracy operators $s_j: \DGN_*(\Delta^1)^{\otimes k}\rightarrow\DGN_*(\Delta^1)^{\otimes k-1}$
dual to the operators $d^i_{\epsilon}: I^k\rightarrow I^{k-1}$, $i = 1,\dots,k$, $\epsilon = 0,1$,
and $s^j: I^{k-1}\rightarrow I^k$, $j = 0,\dots,k$,
considered in Construction <ref>.
We then have $d_i^{\epsilon} = \id^{\otimes k-i}\otimes d^{\epsilon}\otimes\id^{\otimes i-1}$, $s_0 = \id^{\otimes k-1}\otimes s^0$,
$s_j = \id^{\otimes k-j-1}\otimes\nabla_*\otimes\id^{\otimes j-1}$, for $j = 1,\dots,k-1$,
and $s_k = s^k\otimes\id^{\otimes k-1}$,
where $d^{\epsilon}: \kk = \DGN_*(\Delta^0)\rightarrow\DGN_*(\Delta^1)$, $\epsilon = 0,1$, and $s^0: \DGN_*(\Delta^1)\rightarrow\DGN_*(\Delta^0) = \kk$
are the cofaces and the codegeneracy of the normalized chain complex of the one-simplex,
while $\nabla_*: \DGN_*(\Delta^1)\otimes\DGN_*(\Delta^1)\rightarrow\DGN_*(\Delta^1)$
denotes the connection of Construction <ref>.

We precisely define the dg module $\Tree^{\square}(\ttree,\stree)$, which represents the dg hom-object associated to a pair of reduced trees $\stree,\ttree\in\widetilde{\Tree}$
such that $\ttree\not=\stree$ in our enriched category, by the following quotient

\begin{equation*}
\Tree^{\square}(\ttree,\stree)
= \bigoplus_{k\geq 0}\left(\bigoplus_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}
\DGN_*(\Delta^1)^{\otimes k}_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}
\right)/\equiv,
\end{equation*}

where a copy of the cubical chain complex $\DGN_*(\Delta^1)^{\otimes k}$ is assigned to every sequence
of composable tree morphisms $\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree$,
and we mod out by the relations

\begin{align*}
& \equiv\underline{\sigma}_{\ttree\rightarrow\cdots\widehat{\ttree_i}\cdots\rightarrow\stree}, \\
& \equiv\underline{\sigma}_{\ttree\rightarrow\cdots\ttree_j = \ttree_j\cdots\rightarrow\stree},
\end{align*}

where $\underline{\sigma}$ denotes an element of the cubical chain complex (of appropriate dimension).
The composition operations of this enriched category

\begin{equation*}
\circ: \Tree^{\square}(\utree,\stree)\otimes\Tree^{\square}(\ttree,\utree)\rightarrow\Tree^{\square}(\ttree,\stree)
\end{equation*}

are given by

\begin{multline}\tag{$*$}\label{cubical-enriched-category:composition}
\underline{\sigma}_{\utree\rightarrow\utree_k\rightarrow\dots\rightarrow\utree_1\rightarrow\stree}
\circ\underline{\tau}_{\ttree\rightarrow\ttree_l\rightarrow\dots\rightarrow\ttree_1\rightarrow\utree}\\
= (\underline{\tau}\otimes\underline{0}\otimes\underline{\sigma})_{\ttree\rightarrow\ttree_l\rightarrow\dots\rightarrow\ttree_1\rightarrow\utree
\rightarrow\utree_k\rightarrow\dots\rightarrow\utree_1\rightarrow\stree}
\end{multline}

as long as $\utree\not=\stree$ and $\ttree\not=\utree$. We just take in addition $\Tree^{\square}(\stree,\stree) = \kk$ to provide our category with identity homomorphisms.
We easily check that the above formula preserves the relations of our hom-objects
and hence gives a well-defined morphism
of dg modules. We immediately see that these composition operations
are associative too.

We can also define the objects $\Tree^{\square}(\ttree,\stree)$ in terms of a coend.
We then consider an indexing category $\CubeCat$ generated by the $0$-cofaces $d^0_i$ and the codegeneracies $s^j$
attached to our cubical chain complexes
and which reflect the face and degeneracy operations that we apply to the sequences of composable tree morphisms.
The objects of this category are the ordinals $\underline{k+2} = \{k+1>k>\dots>1>0\}$, with $k\geq 0$.
The morphisms are the non decreasing maps $u: \underline{k+2}\rightarrow\underline{l+2}$
such that $u(k+1) = l+1$ and $u(0) = 0$.
The coface $d^0_i$ corresponds to the map $d^0_i: \underline{k+1}\rightarrow\underline{k+2}$ that jumps over $i+1$ in $\underline{k+2}$,
while the codegeneracy $s_j$ corresponds to the map $s_j: \underline{k+2}\rightarrow\underline{k+1}$
such that $s_j(x) = x$ for $x = 0,\dots,j$ and $s_j(x) = x-1$ for $x = j+1,\dots,k+2$.
We easily check that the collection of cubical chain complexes $\DGN_*(\Delta^1)^{\otimes k}$,
equipped with the previously defined operators $d^0_i: \DGN_*(\Delta^1)^{\otimes k-1}\rightarrow\DGN_*(\Delta^1)^{\otimes k}$
and $s_j: \DGN_*(\Delta^1)^{\otimes k}\rightarrow\DGN_*(\Delta^1)^{\otimes k-1}$,
defines a functor $\underline{k+2}\mapsto\DGN_*(\Delta^1)^{\otimes k}$ on this category $\CubeCat$.

In general, we denote by $X_k$ the image of an object $\underline{k+2}\in\CubeCat$
under a (contravariant or covariant) functor $X$
over the category $\CubeCat$.

For a pair of reduced trees $\ttree,\stree\in\widetilde{\Tree}(r)$ with $\ttree\not=\stree$,
we also consider the functor $\NCat(\ttree,\stree): \CubeCat^{op}\rightarrow\Set$
such that

\begin{equation*}
\NCat(\ttree,\stree)_k = \bigl\{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree|\ttree_i\in\widetilde{\Tree}(r)\,(\forall i)\bigr\},
\end{equation*}

where we consider the set of all sequences of composable tree morphisms of length $k+1$ with $\ttree_{k+1} = \ttree$, $\ttree_0 = \stree$,
and we equip this set with the obvious action of the category $\CubeCat$ (we adapt the usual definition
of the simplicial nerve of a category).
We then have:

\begin{equation*}
\Tree^{\square}(\ttree,\stree) = \int^{\underline{k+2}\in\CubeCat}\kk[\NCat(\ttree,\stree)_k]\otimes\DGN_*(\Delta^1)^{\otimes k},
\end{equation*}

where $\kk[\NCat(\ttree,\stree)_k]$ is the $\kk$-module generated by the set $\NCat(\ttree,\stree)_k$.
We can also express the composition operation of the category $\Tree^{\square}$ in terms of a termwise composition operation on this coend
which we define by the above formula (<ref>).

We have well-defined (contravariant) facet operators

\begin{equation*}
i_{\sigmatree,\stree}: \Tree^{\square}(\ttree,\stree)\rightarrow\Tree^{\square}(\thetatree,\sigmatree),
\end{equation*}

which we associate to all subtree inclusions $\sigmatree\subset\stree$, where $\thetatree = f^{-1}(\sigmatree)$ is the pre-image of the subtree $\sigmatree\subset\stree$
under the (at most unique) morphism $f: \ttree\rightarrow\stree$.
We define these facet operators on our coend termwise,
by the map

\begin{equation*}
i_{\sigmatree,\stree}: \kk[\NCat(\ttree,\stree)_k]\otimes\DGN_*(\Delta^1)^{\otimes k}\rightarrow\kk[\NCat(\thetatree,\sigmatree)_k]\otimes\DGN_*(\Delta^1)^{\otimes k}
\end{equation*}

induced by the set-theoretic facet operator $\NCat(\ttree,\stree)_k\rightarrow\NCat(\thetatree,\sigmatree)_k$
which carries any sequence of composable tree morphisms $\ttree\xrightarrow{f_k}\ttree_k\xrightarrow{f_{k-1}}\cdots\xrightarrow{f_1}\ttree_1\xrightarrow{f_0}\stree$
to the sequence of tree morphisms such that $(f_0\cdots f_k)^{-1}(\sigmatree)\rightarrow(f_0\cdots f_{k-1})^{-1}(\sigmatree)\rightarrow\cdots\rightarrow f_0^{-1}(\sigmatree)\rightarrow\sigmatree$,
where we use $\thetatree = (f_0\cdots f_k)^{-1}(\sigmatree)$.
We can also associate a Segal map

\begin{equation*}
i_{\lambda_{\utree}(\sigmatree_*)}: \Tree^{\square}(\ttree,\stree)\rightarrow\bigotimes_{u\in V(\utree)}\Tree^{\square}(\thetatree_u,\sigmatree_u),
\end{equation*}

to every tree decomposition $\stree = \lambda_{\utree}(\sigmatree_*)$, where we set $\thetatree_u = f^{-1}(\sigmatree_u)$, for all factors $\sigmatree_u\subset\stree$.
We then take the tensor product of the product of the above set-theoretic assignments $\NCat(\ttree,\stree)_k\rightarrow\prod_{u\in V(\utree)}\NCat(\thetatree_u,\sigmatree_u)_k$
with the map $\mu^*: \DGN_*(\Delta^1)^{\otimes k}\rightarrow\bigotimes_{u\in V(\utree)}\DGN_*(\Delta^1)^{\otimes k}$
induced by the coassociative coproduct of the cubical chain complex $\DGN_*(\Delta^1)^{\otimes k}$.
We just need to fix an ordering on the set of vertices of our trees since this coproduct is not associative.
We easily check that the facet operators and the Segal maps satisfy natural associativity relations and are compatible, in some natural sense,
with the composition operation of our enriched category structure.

In the symmetric context, we can also observe that the hom-objects $\Tree^{\square}(\ttree,\stree)$
inherit an action of the symmetric group such that

\begin{equation*}
s_*: \Tree^{\square}(\ttree,\stree)\rightarrow\Tree^{\square}(s\ttree,s\stree),
\end{equation*}

for every pair of reduced trees $\stree,\ttree\in\widetilde{\Tree}(r)$, $r>0$,
and for every permutation $s\in\Sigma_r$,
which are induced by the mappings $\ttree_i\mapsto s\ttree_i$ at the level of the sets $\NCat(\ttree,\stree)_k$.
These operators are compatible with the enriched category structure (so that the mapping $s_*: \ttree\mapsto s\ttree$
defines a functor on the enriched category $\Tree^{\square}$)
and with the facet operators. (But the action of permutation is not compatible with the Segal maps with values in the tensor product, since we need to order the vertices
of our trees when we form these maps.)

We use that this enriched category in dg modules $\Tree^{\square}$ can be upgraded to a category enriched over the category of $\EOp$-coalgebras.

Each object $\Tree^{\square}(\ttree,\stree)$ inherits an $\EOp$-coalgebra structure from the cubical chain complexes $\DGN_*(\Delta^1)^{\otimes k}$.
The composition products define morphisms of $\EOp$-coalgebras $\circ: \Tree^{\square}(\ttree,\utree)\otimes\Tree^{\square}(\utree,\stree)\rightarrow\Tree^{\square}(\ttree,\stree)$
(we switch the conventional order of the factors of the composition to make the diagonal action of the Barratt–Eccles operad compatible with these operations).
The facet operators $i_{\sigmatree,\stree}: \Tree^{\square}(\ttree,\stree)\rightarrow\Tree^{\square}(\thetatree,\sigmatree)$
also define morphisms of $\EOp$-coalgebras,
as well as the operators that give the action of permutations $s_*: \Tree^{\square}(\ttree,\stree)\rightarrow\Tree^{\square}(s\ttree,s\stree)$
in the symmetric setting.

The constructions of the previous paragraph accordingly give a category $\Tree^{\square}$ enriched in $\EOp$-coalgebras
and equipped with facet operators (together with an action of permutations), which are defined within the category of $\EOp$-coalgebras
and are compatible with the composition structure of our objects.

For each $k\geq 2$, we use that $\kk[\NCat(\ttree,\stree)_k]$ inherits a cocommutative coalgebra structure (given by the diagonal of the set $\NCat(\ttree,\stree)_k$)
in order to extend the $\EOp$-coalgebra structure of the cubical chain complex $\DGN_*(\Delta^1)^{\otimes k}$
to the tensor product $\kk[\NCat(\ttree,\stree)_k]\otimes\DGN_*(\Delta^1)^{\otimes k}$.
We readily check that this $\EOp$-coalgebra structure is compatible with the action of the category $\CubeCat$
and therefore passes to our coend. (Recall simply that the forgetful functor from a category of coalgebras to a base category creates coends.)

We easily check that the composition operations of the category $\Tree^{\square}$ are also compatible with the $\EOp$-coalgebra structure termwise,
as well the facet operators. Then we just pass to the coend
to get the conclusions of the proposition.

We now consider the enriched category in cocommutative coalgebras such that

\begin{equation*}
\Tree(\ttree,\stree) = \begin{cases} \kk, & \text{if we have a morphism $\ttree\rightarrow\stree$}, \\
0, & \text{otherwise}, \end{cases}
\end{equation*}

for any pair of reduced trees $\stree,\ttree\in\widetilde{\Tree}(r)$, $r>0$ (with the same abuse of notation as in the introduction of this subsection).
We immediately see that this enriched category inherits the same structures (facet operators, action of permutations) within the category of cocommutative coalgebras
as the enriched category in $\EOp$-coalgebras $\Tree^{\square}$.
We also have the following observation:

* The hom-objects of the enriched category $\Tree^{\square}$ are endowed with weak-equivalences

\begin{equation*}
\epsilon: \Tree^{\square}(\ttree,\stree)\xrightarrow{\sim}\kk,
\end{equation*}

which are yielded by the augmentation maps such that $\epsilon(1_{\ttree\rightarrow\stree}) = 1$
and $\epsilon(\underline{\sigma}_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_0\rightarrow\stree}) = 0$
in cubical dimension $k>0$ when $\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_0\rightarrow\stree$
is non degenerate.

* These weak-equivalences also define a morphism of enriched categories

\begin{equation*}
\epsilon: \Tree^{\square}\xrightarrow{\sim}\Tree,
\end{equation*}

which is compatible with the facet operators and with the action of permutations whenever we consider this extra structure.
(We then regard the enriched category in cocommutative coalgebras $\Tree$ as a category enriched in $\EOp$-coalgebras
by restriction of structure through the augmentation map of the Barratt–Eccles operad.)

The morphism $\epsilon: \Tree^{\square}(\ttree,\stree)\xrightarrow{\sim}\kk$ has an obvious section $\eta: \kk\rightarrow\Tree^{\square}(\ttree,\stree)$
given by $\eta(1) = 1_{\ttree\rightarrow\stree}$.
We construct a chain homotopy $h: \Tree^{\square}(\ttree,\stree)\otimes\DGN_*(\Delta^1)\rightarrow\Tree^{\square}(\ttree,\stree)$
between $\eta\circ\epsilon = h(-\otimes\underline{1})$ and $\id = h(-\otimes\underline{0})$
to prove that $\epsilon$ is a weak-equivalence.

We proceed as follows. For $k\geq 0$, we consider the map

\begin{equation*}
h_k: \DGN_*(\Delta^1)^{\otimes k}\otimes\DGN_*(\Delta^1)\rightarrow\DGN_*(\Delta^1)^{\otimes k}
\end{equation*}

defined by the composite

\begin{multline*}
\DGN_*(\Delta^1)^{\otimes k}\otimes\DGN_*(\Delta^1)\xrightarrow{\id\otimes\mu^*}\DGN_*(\Delta^1)^{\otimes k}\otimes\DGN_*(\Delta^1)^{\otimes k} \\
\xrightarrow{\simeq}(\DGN_*(\Delta^1)\otimes\DGN_*(\Delta^1))^{\otimes k}\xrightarrow{(\nabla_*^{\max})^{\otimes k}}\DGN_*(\Delta^1)^{\otimes k},
\end{multline*}

where $\mu^*$ is the ($k$-fold) Alexander–Whitney diagonal and $\nabla_*^{\max} = \DGN_*(\max)\circ\EM$ is the composite of the Eilenberg–MacLane map
with the morphism induced by the map of simplicial sets $\max: \Delta^1\times\Delta^1\rightarrow\Delta^1$
such that $\max: (s,t)\mapsto\max(s,t)$. (Thus, we consider a mirror of the connection $\nabla_* = \nabla_*^{\min}$,
which we use in the definition of the codegeneracies of our cubical complex $\DGN_*(\Delta^1)^{\otimes k}$.)

We claim that these maps preserve the action of the cofaces $d^0_i$ and of the codegeneracies $s_j$ on the cubical complex $\DGN_*(\Delta^1)^{\otimes k}$.
The preservation of the cofaces $d^0_i$ follows from the formulas $\nabla_*^{\max}(\underline{1},\underline{01}) = 0$
and $\nabla_*^{\max}(\underline{1},\underline{\tau}) = \underline{1}$ for $\deg(\underline{\tau}) = 0$.
The preservation of the codegeneracies $s_j$ is immediate in the cases $j = 0$ and $j = k$.
We use the properties of the Alexander–Whitney diagonal and of the Eilenberg–MacLane map to reduce the verification of the preservation of the codegeneracies $s_j$
such that $j = 1,\dots,k-1$ to the case $k=2$.
We then deduce our claim from the distribution relation

\begin{equation*}
\nabla_*^{\max}(\nabla_*^{\min}(\underline{\sigma}_2\otimes\underline{\sigma}_1)\otimes\underline{\tau})
= \sum_{(\underline{\tau})}\nabla_*^{\min}(\nabla_*^{\max}(\underline{\sigma}_2\otimes\underline{\tau}')\otimes\nabla_*^{\max}(\underline{\sigma}_1\otimes\underline{\tau}'')),
\end{equation*}

valid for $\underline{\sigma}_2,\underline{\sigma}_1,\underline{\tau}\in\DGN_*(\Delta^1)$,
and where we write $\mu^*(\underline{\tau}) = \sum_{(\underline{\tau})}\underline{\tau}'\otimes\underline{\tau}''$
for the Alexander–Whitney diagonal of the element $\underline{\tau}\in\DGN_*(\Delta^1)$.
(This relation, which reflects the classical min-max distribution relation, can easily be checked by hand.)
We deduce from these verifications that these maps $h_k$, tensored with the identity of the factor $\kk[\NCat(\ttree,\stree)_k]$,
induce a well-defined map on our coend $h: \Tree^{\square}(\ttree,\stree)\otimes\DGN_*(\Delta^1)\rightarrow\Tree^{\square}(\ttree,\stree)$.

We also have $h_k(\underline{\sigma}\otimes\underline{1}) = \underline{1}^{\otimes k}$ and $h_k(\underline{\sigma}\otimes\underline{0}) = \underline{\sigma}$, for each $k\geq 0$,
and these identities give the relations $\eta\circ\epsilon = h(-\otimes\underline{1})$ and $\id = h(-\otimes\underline{0})$
at the coend level.

This verification completes the proof of the first assertion of the proposition, while an immediate inspection gives the verification of the second assertion.

This proposition has the following corollary, which we use in our verification that the homotopy Kan construction returns Segal $E_\infty$-Hopf pre-cooperads
that satisfy the Segal condition.

The Segal map of Construction <ref> defines weak-equivalence of dg modules

\begin{equation*}
i_{\lambda_{\utree}(\sigmatree_*)}: \Tree^{\square}(\ttree,\stree)\rightarrow\bigotimes_{u\in V(\utree)}\Tree^{\square}(\thetatree_u,\sigmatree_u),
\end{equation*}

for every tree decomposition $\stree = \lambda_{\utree}(\sigmatree_*)$, where we again set $\thetatree_u = f^{-1}(\sigmatree_u)$, for all factors $\sigmatree_u\subset\stree$.

We can reformulate the definition of the structure operators of homotopy Segal $E_\infty$-Hopf shuffle pre-cooperads in terms of the category $\Tree^{\square}$.
We get the following result, which follows from formal verifications.

Let $\AOp$ be a connected homotopy Segal $E_\infty$-Hopf shuffle pre-cooperad (either symmetric or shuffle).
The objects $\AOp(\ttree)$, $\ttree\in\Tree(r)$, $r>0$, inherit an action of the enriched category $\Tree^{\square}$,
given by operators

\begin{equation*}
\rho: \AOp(\stree)\otimes\Tree^{\square}(\ttree,\stree)\rightarrow\AOp(\ttree),
\end{equation*}

defined in the category of dg modules, and which preserve the action of the facet operators in some natural sense (as well as the action of permutations in the symmetric setting).
The homotopy coproduct operators $\rho_{\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree}$
are identified with the adjoint morphisms
of the maps

\begin{multline*}
\AOp(\stree)\otimes[\ttree\rightarrow\ttree_k\rightarrow\dots\rightarrow\ttree_1\rightarrow\stree]\otimes\DGN_*(\Delta^1)^{\otimes k}\\
\rightarrow\AOp(\stree)\otimes\underbrace{\int^{\underline{k+2}\in\CubeCat}\kk[\NCat(\ttree,\stree)_k]\otimes\DGN_*(\Delta^1)^{\otimes k}}_{\Tree^{\square}(\ttree,\stree)}
\rightarrow\AOp(\ttree),
\end{multline*}

which we determine from this action.

In the case where $\AOp$ is a strict Segal $E_\infty$-Hopf shuffle pre-cooperad, we may see that the action of the enriched category $\Tree^{\square}$
defined in this proposition factors through an action of the category enriched in cocommutative coalgebras $\Tree$ (which satisfies the same properties).
Thus we just retrieve the obvious functor structure of the object $\AOp$
in this case.

We now tackle the definition of our homotopy Kan construction. We have to dualize the structure operations attached to the category $\Tree^{\square}$.
For this purpose, we use the following observation.

The dg hom-object $\Tree^{\square}(\ttree,\stree)$ forms a free module of finite rank over the ground ring $\kk$, for all trees $\ttree,\stree\in\widetilde{\Tree}(r)$, $r>0$.

Each element of our coend has a unique representative as a (linear combination) of tensors
of the form

\begin{equation*}
\in\kk[\NCat(\ttree,\stree)_k]\otimes\mathring{\DGN}_*(\Delta^1)^{\otimes k},
\end{equation*}

where $\mathring{\DGN}_*(\Delta^1) = \kk\underline{0}\oplus\kk\underline{01}$. Then we just use that the set of sequences of composable morphisms
of the form $\ttree\xrightarrow{\not=}\ttree_k\xrightarrow{\not=}\dots\xrightarrow{\not=}\ttree_1\xrightarrow{\not=}\stree$, $k\geq 0$,
is finite,
because each tree $\ttree_i$ is given by the contraction of a subset of inner edges $e\in\mathring{E}(\ttree)$,
which are in finite number.

We can now define the two-sided cobar complex which underlies our homotopy Kan construction $\DGK^c(\AOp)$.
We address this construction in the next paragraph.

We use the statement of the previous lemma to dualize the structure operations associated to the enriched category $\Tree^{\square}$.

For a pair of trees $\stree,\ttree\in\Tree(r)$, $r>0$, we let $\Tree^{\square}(\ttree,\stree)^{\sharp}$
denote the dual $\EOp$-algebra
of the $\EOp$-coalgebra $\Tree^{\square}(\ttree,\stree)$.
The composition operations of the enriched category $\Tree^{\square}$ induce a coproduct map

\begin{align*}
\gamma^*: \Tree^{\square}(\ttree,\stree)^{\sharp}
& \rightarrow\prod_{\ttree\rightarrow\utree\rightarrow\stree}\Tree^{\square}(\ttree,\utree)^{\sharp}\otimes\Tree^{\square}(\utree,\stree)^{\sharp},
\intertext{which also forms a morphism of $\EOp$-algebras. We also have an augmentation map}
\eta^*: \Tree^{\square}(\ttree,\stree)^{\sharp} & \rightarrow\kk,
\end{align*}

which we take as the identity in the case $\ttree = \stree$, as the zero map in the case $\ttree\not=\stree$.
The facet operators induce $\EOp$-algebra morphisms

\begin{equation*}
i_{\sigmatree,\stree}:  \Tree^{\square}(\thetatree,\sigmatree)^{\sharp}\rightarrow\Tree^{\square}(\ttree,\stree)^{\sharp},
\end{equation*}

which preserve the above coproduct and counit operations. Note that the cartesian product in the definition of the operation $\gamma^*$ reduces to a direct sum,
since every morphism $\ttree\rightarrow\stree$ admits finitely many factorizations $\ttree\rightarrow\utree\rightarrow\stree$.
In the symmetric setting, we also consider $\EOp$-algebra morphisms

\begin{equation*}
s^*:  \Tree^{\square}(s\ttree,s\stree)^{\sharp}\rightarrow\Tree^{\square}(\ttree,\stree)^{\sharp}
\end{equation*}

given by the action of permutations $s\in\Sigma_r$.

For a homotopy Segal $E_\infty$-Hopf cooperad, the operations of Proposition <ref>
dualize to $\EOp$-algebra morphisms

\begin{equation*}
\rho^*: \AOp(\stree)\rightarrow\prod_{\ttree\rightarrow\stree}\AOp(\ttree)\otimes\Tree^{\square}(\ttree,\stree)^{\sharp},
\end{equation*}

which we can also identify with an end of the homotopy coproducts attached to our object by the adjoint definition
of these operations in our Proposition <ref> (we again use a variant of the observations
of the previous lemma to obtain that the tensor product with $\AOp(\ttree)$
distributes over this end).
These morphisms are coassociative and counital with respect to the coproduct and counit operations of the objects $\Tree^{\square}(\ttree,\stree)^{\sharp}$,
commute with the action of the facet operators on our homotopy Segal $E_\infty$-Hopf cooperad
and on the objects $\Tree^{\square}(\ttree,\stree)^{\sharp}$ (and commute with the action of permutations in the symmetric setting).

We then let $\FOp_{\stree}(\ttree)\in\EAlg$ be a collection of $\EOp$-algebras, defined for any fixed tree $\stree\in\widetilde{\Tree}(r)$, for all $\ttree\in\widetilde{\Tree}(r)/\stree$,
and equipped with coproduct operations

\begin{equation*}
\gamma^*: \FOp_{\stree}(\ttree)\rightarrow\prod_{\ttree\rightarrow\utree\rightarrow\stree}\Tree^{\square}(\ttree,\utree)^{\sharp}\otimes\FOp_{\stree}(\utree),
\end{equation*}

defined in the category of $\EOp$-algebras, and which are again coassociative and counital with respect to the coproduct and counit operations
of the objects $\Tree^{\square}(\ttree,\stree)^{\sharp}$.
We assume that this collection defines a functor in the tree $\stree$ when $\stree$ varies
and that we have facet operators

\begin{equation*}
i_{\sigmatree,\stree}:  \FOp_{\sigmatree}(\thetatree)\rightarrow\FOp_{\stree}(\ttree),
\end{equation*}

associated to all subtrees $\sigmatree\subset\stree$, with $\thetatree = f^{-1}(\sigmatree)$, the pre-image of $\sigmatree$ under the morphism $f: \ttree\rightarrow\stree$,
which again satisfy natural functoriality relations and are compatible with the coproduct operations.
In the symmetric setting, we also assume that we have an action of the permutations $s^*: \FOp_{s\stree}(s\ttree)\rightarrow\FOp_{\stree}(\ttree)$,
which is compatible with the structure operations
attached to our collection.
In what follows, we consider the cases $\FOp_{\stree}(\ttree) = \Tree^{\square}(\ttree,\stree)^{\sharp}$ and $\FOp_{\stree}(\ttree) = \Tree(\ttree,\stree)^{\sharp}$,
where in the latter case $\Tree(\ttree,\stree)^{\sharp}$ denotes the commutative algebra of functions $u: \Tree(\ttree,\stree)\rightarrow\kk$
on the morphism sets of the tree category $\Tree(\ttree,\stree)$.

For each $n\in\NN$, we set

\begin{multline*}
\DGK^n(\AOp,\Tree^{\square},\FOp_{\stree})\\
= \prod_{\ttree_n\rightarrow\cdots\rightarrow\ttree_0\rightarrow\stree}
\AOp(\ttree_n)\otimes\Tree^{\square}(\ttree_n,\ttree_{n-1})^{\sharp}\otimes\dots\otimes\Tree^{\square}(\ttree_1,\ttree_0)^{\sharp}\otimes\FOp_{\stree}(\ttree_0),
\end{multline*}

and we equip this object with the coface operators $d^i: \DGK^{n-1}(\AOp,\Tree^{\square},\FOp_{\stree})\rightarrow\DGK^n(\AOp,\Tree^{\square},\FOp_{\stree})$
defined termwise by the maps such that

\begin{align*}
d^i & = \begin{cases}
\id\otimes\id^{\otimes n-1}\otimes\gamma^*, & \text{for $i = 0$}, \\
\id\otimes\id^{\otimes n-i-1}\otimes\gamma^*\otimes\id^{\otimes i-1}\otimes\id, & \text{for $i = 1,\dots,n-1$}, \\
\rho^*\otimes\id^{\otimes n-1}\otimes\id, & \text{for $i = n$},
\end{cases}
\intertext{and with the codegeneracies $s^j: \DGK^{n+1}(\AOp,\Tree^{\square},\FOp_{\stree})\rightarrow\DGK^n(\AOp,\Tree^{\square},\FOp_{\stree})$
defined termwise by the maps}
s^j & = \id\otimes\id^{\otimes n-j}\otimes\eta^*\otimes\id^{\otimes j}\otimes\id\quad\text{for $j = 0,\dots,n$}.
\end{align*}

We easily check that this definition returns a cosimplicial object in the category of $\EOp$-algebras.
We also have facet operators

\begin{equation*}
i_{\sigmatree,\stree}: \DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\sigmatree})\rightarrow\DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\stree}),
\end{equation*}

compatible with the cosimplicial structure, and defined by the termwise tensor products of facet operators

\begin{multline*}
i_{\thetatree_n,\ttree_n}\otimes i_{\thetatree_{n-1},\ttree_{n-1}}\otimes\dots\otimes i_{\thetatree_0,\ttree_0}\otimes i_{\sigmatree,\stree}:\\
\AOp(\thetatree_n)\otimes\Tree^{\square}(\thetatree_n,\thetatree_{n-1})^{\sharp}\otimes\dots\otimes\Tree^{\square}(\thetatree_1,\thetatree_0)^{\sharp}\otimes\FOp_{\stree}(\thetatree_0)\\
\rightarrow\AOp(\ttree_n)\otimes\Tree^{\square}(\ttree_n,\ttree_{n-1})^{\sharp}\otimes\dots\otimes\Tree^{\square}(\ttree_1,\ttree_0)^{\sharp}\otimes\FOp_{\stree}(\ttree_0),
\end{multline*}

where $\thetatree_i = (f_0\cdots f_i)^{-1}(\sigmatree)$, $i = 0,\dots,n$, denotes the pre-image of the subtree $\sigmatree\subset\stree$
under the composite of the tree morphisms $\ttree_i\xrightarrow{f_i}\cdots\xrightarrow{f_1}\ttree_0\xrightarrow{f_0}\stree$.

In the symmetric setting, we still consider an action of permutations

\begin{equation*}
s^*: \DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{s\stree})\rightarrow\DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\stree}),
\end{equation*}

defined again by an obvious termwise construction, and compatible with the facet operators.

We record the outcome of the previous construction in the next proposition.

The construction of the previous paragraph returns a collection of cosimplicial $\EOp$-algebras

\begin{equation*}
\DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\stree})\in\EAlg,\quad\stree\in\widetilde{\Tree}(r),\quad r>0,
\end{equation*}

equipped with compatible facet operators $i_{\sigmatree,\stree}: \DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\stree})\rightarrow\DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\stree})$,
which satisfy the usual functoriality relations.
If the homotopy Segal $E_\infty$-Hopf cooperad $\AOp$ is symmetric and $\FOp_{\stree}(-)$ is endowed with a symmetric structure as well,
then we also have an action of permutations on our objects $s^*: \DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{s\stree})\rightarrow\DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\stree})$
compatible with the cosimplicial structure and with the facet operators.

In the case of a decomposition $\stree = \lambda_{\utree}(\sigmatree_u,u\in V(\utree))$, we can assemble the facet operators
$i_{\sigma_u,\stree}: \FOp_{\sigmatree_u}(\thetatree_u)\rightarrow\FOp_{\stree}(\ttree)$
associated to a bicollection of $\EOp$-algebras $\FOp_{\stree}(\ttree)\in\EAlg$
as in Construction <ref>
into a Segal map

\begin{equation*}
i_{\lambda_{\utree}(\sigmatree_*)}: \bigvee_{u\in V(\utree)}\FOp_{\sigmatree_u}(\thetatree_u)\rightarrow\FOp_{\stree}(\ttree),
\end{equation*}

and we can define a Segal map similarly on our cosimplicial object $\DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\stree})$.
We say that our bicollection $\FOp_{\stree}(\ttree)\in\EAlg$ satisfies the Segal condition when the above Segal map is a weak-equivalence.
We have the following statement.

If $\AOp$ is a homotopy Segal $E_\infty$-Hopf cooperad and hence satisfies the Segal condition, and if the bicollection $\FOp_{\stree}(\ttree)\in\EAlg$ satisfies the Segal condition as well,
then the Segal maps that we associate to our cosimplicial object $\DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\stree})$
define levelwise weak-equivalences of cosimplicial $\EOp$-algebras

\begin{equation*}
i_{\lambda_{\utree}(\sigmatree_*)}: \bigvee_{u\in V(\utree)}\DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\sigmatree_u})\xrightarrow{\sim}\DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\stree}),
\end{equation*}

for all tree decomposition $\stree = \lambda_{\utree}(\sigmatree_u,u\in V(\utree))$,
so that $\DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\stree})$ also satisfies a form of our Segal condition levelwise.

The Segal maps of the proposition are given, on each term of the cosimplicial object $\DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\stree})$, by expressions of the form

\begin{multline*}
\bigvee_{u\in V(\utree)}
\biggl(\AOp(\thetatree_n^u)\otimes\cdots\otimes\Tree^{\square}(\thetatree_i^u,\thetatree_{i-1}^u)^{\sharp}\otimes\cdots\otimes\FOp_{\stree}(\thetatree_0^u)\biggr)\\
\rightarrow\AOp(\ttree_n)\otimes\cdots\otimes\Tree^{\square}(\ttree_i,\ttree_{i-1})^{\sharp}\otimes\cdots\otimes\FOp_{\stree}(\ttree_0),
\end{multline*}

where $\thetatree_i^u$ denote the pre-images of the subtree $\sigmatree\subset\stree$ under the tree morphisms $\ttree_i\rightarrow\dots\rightarrow\ttree_0\rightarrow\stree$
and we take a tensor product of facet operators
on each factor. We compose this map with the Eilenberg–MacLane map to pass from the coproduct $\bigvee_{u\in V(\utree)}$
to a tensor product $\bigotimes_{u\in V(\utree)}$ (as in Proposition <ref>).
We have an obvious commutative diagram which enables us to identify the obtained Segal map
with a tensor product of the form:

\begin{multline*}
\biggl(\bigotimes_{u\in V(\utree)}\AOp(\thetatree_n^u)\biggr)
\otimes\cdots\otimes\biggl(\bigotimes_{u\in V(\utree)}\Tree^{\square}(\thetatree_i^u,\thetatree_{i-1}^u)^{\sharp}\biggr)\otimes\cdots
\otimes\biggl(\bigotimes_{u\in V(\utree)}\FOp_{\stree}(\thetatree_0^u)\biggr)\\
\rightarrow\AOp(\ttree_n)\otimes\cdots\otimes\Tree^{\square}(\ttree_i,\ttree_{i-1})^{\sharp}\otimes\cdots\otimes\FOp_{\stree}(\ttree_0),
\end{multline*}

where we take a factorwise tensor product of Segal maps associated to the objects $\AOp(-)$, $\Tree^{\square}(-,-)^{\sharp}$ and $\FOp_{\stree}(-)$.
In the case of the objects $\Tree^{\square}(-,-)^{\sharp}$, we retrieve the dual of the Segal maps
considered in Corollary <ref>.
These Segal maps are weak-equivalences therefore, like the Segal maps associated to the objects $\AOp(-)$ and $\FOp_{\stree}(-)$ by assumption.
The conclusion follows.

We now focus on the cases $\FOp_{\stree}(-) = \Tree^{\square}(-,\stree)^{\sharp},\Tree(-,\stree)^{\sharp}$. The coproduct operation on $\Tree^{\square}(-,-){\sharp}$,
such as defined in Construction <ref>,
gives a natural transformation $\Tree^{\square}(-,\stree)^{\sharp}\rightarrow\Tree^{\square}(-,\ttree)^{\sharp}\otimes\Tree^{\square}(\ttree,\stree)^{\sharp}$,
which passes to our cosimplicial object, by functoriality of our construction, and yields a morphism of cosimplicial $\EOp$-algebras

\begin{equation*}
\rho^*: \DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}(-,\stree)^{\sharp})
\rightarrow\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}(-,\ttree)^{\sharp})\otimes\Tree^{\square}(\ttree,\stree)^{\sharp},
\end{equation*}

for every pair of objects $\stree,\ttree\in\widetilde{\Tree}(r)$, $r>0$.
In the case $\FOp_{\stree}(-) = \Tree(-,\stree)^{\sharp}$, we similarly get morphisms of cosimplicial $\EOp$-algebras
of the form

\begin{equation*}
\rho^*: \DGK^{\bullet}(\AOp,\Tree^{\square},\Tree(-,\stree)^{\sharp})\rightarrow\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree(-,\ttree)^{\sharp})\otimes\Tree(\ttree,\stree)^{\sharp}.
\end{equation*}

Note that the map $\Tree^{\square}(-,-)\rightarrow\Tree(-,-)$ of Proposition <ref>
induces a natural transformation in the converse direction between these cosimplicial objects $\DGK^{\bullet}(\AOp,\Tree^{\square},\FOp_{\stree})$
that we associate to $\FOp_{\stree}(-) = \Tree^{\square}(-,\stree)^{\sharp}$ and to $\FOp_{\stree}(-) = \Tree(-,\stree)^{\sharp}$.
We then have the following result.

* The above coproduct operations provide the collection of cosimplicial $\EOp$-algebras

\begin{equation*}
\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}(-,\stree)^{\sharp})\in\cosimp\EAlg,\quad\stree\in\widetilde{\Tree}(r),\quad r>0,
\end{equation*}

with the coproduct operators of a homotopy Segal $E_\infty$-Hopf pre-cooperad structure.
These coproduct operators are compatible with the facet operators and hence $\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}(-,\stree)^{\sharp})$
forms a cosimplicial object in the category of homotopy Segal $E_\infty$-Hopf cooperads (shuffle or symmetric when $\AOp$ is so).

* In the case of the cosimplicial object $\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree(-,\stree)^{\sharp})$,
we similarly obtain a strict Segal $E_\infty$-Hopf pre-cooperad structure
compatible with the cosimplicial structure
on $\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree(-,\stree)^{\sharp})$.
Furthermore, the natural transformations $\Tree(-,\stree)^{\sharp}\rightarrow\Tree^{\square}(-,\stree)^{\sharp}$
induce levelwise weak-equivalences of cosimplicial $\EOp$-algebras

\begin{equation*}
\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree(-,\stree)^{\sharp})\xrightarrow{\sim}\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}(-,\stree)^{\sharp}),
\end{equation*}

which preserve the homotopy Segal $E_\infty$-Hopf pre-cooperad structures, and hence, define a levelwise weak-equivalence
of cosimplicial objects in the category of homotopy Segal $E_\infty$-Hopf pre-cooperads.

* Both objects $\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}(-,\stree)^{\sharp})$ and $\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree(-,\stree)^{\sharp})$
also satisfy the Segal condition levelwise, and hence define (homotopy) Segal $E_\infty$-Hopf cooperads
when $\AOp$ does so.

The first assertion is an immediate consequence of the functoriality properties of our construction.
In the second assertion, we use that the natural transformation $\Tree^{\square}(-,\stree)^{\sharp}\rightarrow\Tree(-,\stree)^{\sharp}$
is dual to the augmentation map of Proposition <ref>,
which is a weak-equivalence by the result of this proposition.
The third assertion follows from the result of Proposition <ref>
since Corollary <ref> implies that the bicollection $\FOp_{\stree}(\ttree) = \Tree^{\square}(\ttree,\stree)^{\sharp}$
satisfies the Segal condition and this is also obviously the case of the bicollection $\FOp_{\stree}(\ttree) = \Tree(\ttree,\stree)^{\sharp}$.

We use a totalization functor to transform the cosimplicial (homotopy) Segal $E_\infty$-Hopf cooperads
of the previous proposition into ordinary (homotopy) Segal $E_\infty$-Hopf cooperads
in dg modules.

Let $R^{\bullet}$ be a cosimplicial object of the category of $\EOp$-algebras. We set

\begin{equation*}
\DGN^*(R^{\bullet}) = \int_n R^n\otimes\DGN^*(\Delta^n),
\end{equation*}

and we equip this object with the $\EOp$-algebra structure induced by the diagonal $\EOp$-algebra structure on $R^n\otimes\DGN^*(\Delta^n)$ termwise.
If we forget about $\EOp$-algebra structures, then we can identify this object with the conormalized complex of cosimplicial dg modules
(see for instance <cit.>),
and as such, this functor carries the levelwise weak-equivalences of cosimplicial objects
to weak-equivalences in the category of dg modules.

For a cosimplicial connected (homotopy) $E_\infty$-Hopf pre-cooperad $\KOp^{\bullet}$, the collection $\DGN^*(\KOp^{\bullet}(\stree))$,
which we obtain by applying this conormalized complex functor termwise,
also inherits the structure of a (homotopy) $E_\infty$-Hopf pre-cooperad (shuffle or symmetric when $\KOp^{\bullet}$ is so)
by functoriality of our conormalized complex construction.

We use the following lemma.

Let $R^{\bullet}$ and $S^{\bullet}$ be cosimplicial $\EOp$-algebras. The $\EOp$-algebra morphism $\DGN^*(R^{\bullet})\vee\DGN^*(S^{\bullet})\rightarrow\DGN^*(R^{\bullet}\vee S^{\bullet})$
induced by the canonical morphisms $R^{\bullet}\rightarrow R^{\bullet}\vee S^{\bullet}$ and $S^{\bullet}\rightarrow R^{\bullet}\vee S^{\bullet}$
is a weak-equivalence.

We have a commutative diagram

\begin{equation*}
\xymatrixcolsep{5pc}\xymatrix{ \DGN^*(R^{\bullet})\vee\DGN^*(S^{\bullet})\ar[r] &
\DGN^*(R^{\bullet}\vee S^{\bullet}) \\
\DGN^*(R^{\bullet})\otimes\DGN^*(S^{\bullet})\ar[r]^{\AW}\ar[u]^{\EM} &
\DGN^*(R^{\bullet}\otimes S^{\bullet})\ar[u]_{\DGN^*(\EM)}
\end{equation*}

where the vertical maps are given by the natural transformations between the tensor product and the coproduct in the category of $\EOp$-algebras,
such as defined in Construction <ref>,
and the bottom horizontal map $\AW$ is the generalization of the Alexander–Whitney diagonal
for the conormalized cochain complex of cosimplicial dg modules.
In the case of cosimplicial dg algebras, this map $\AW$ is used to represent a product operation.
The vertical maps $\EM$ also identifies tensor products with associative products in the coproduct of $\EOp$-algebras
by the definition of Construction <ref>. The commutativity of the diagram
readily follows from this interpretation of our maps.

The vertical maps are weak-equivalences by Proposition <ref>.
The bottom horizontal map is also a weak-equivalence (by the general theory of the Eilenberg–Zilber equivalence).
Therefore the map of the proposition, which represents the upper horizontal map
of our diagram, is also a weak-equivalence.

This lemma has the following immediate consequence.

If $\ROp^{\bullet}$ is a cosimplicial (homotopy) Segal $E_\infty$-Hopf pre-cooperad that satisfies the Segal condition levelwise,
then $\DGN^*(\ROp^{\bullet})$ satisfies the Segal condition as well, and hence forms a (homotopy) Segal $E_\infty$-Hopf cooperad
in the category of dg modules.

Then we have the following statement.

Let $\AOp$ be connected homotopy Segal $E_\infty$-Hopf cooperad (shuffle or symmetric).

* The objects $\DGN^*(\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}))$ and $\DGN^*(\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree))$,
defined by taking the totalization of the cosimplicial (homotopy) Segal $E_\infty$-Hopf cooperads
of Proposition <ref>,
respectively form a homotopy Segal $E_\infty$-Hopf cooperad and a strict homotopy Segal $E_\infty$-Hopf cooperad (which are shuffle or symmetric when $\AOp$ is so).
The levelwise weak-equivalence of Proposition <ref>
induces a weak-equivalence of homotopy Segal $E_\infty$-Hopf cooperads

\begin{equation*}
\DGN^*(\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree(-,\stree)^{\sharp}))\xrightarrow{\sim}\DGN^*(\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}(-,\stree)^{\sharp})),
\end{equation*}

when we pass to this totalization.

* Furthermore, we have a weak-equivalence of homotopy Segal $E_\infty$-Hopf cooperads (shuffle or symmetric)

\begin{equation*}
\AOp(\stree)\xrightarrow{\sim}\DGN^*(\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}(-,\stree)^{\sharp})).
\end{equation*}

This weak-equivalence is natural in $\AOp$.

The first assertion of the proposition immediately follows from the statements of Proposition <ref>
and from the result of Proposition <ref>.

Thus, we focus on the proof of the second assertion.
We define our natural transformation first.
We use that the coproduct map $\rho^*: \AOp(\stree)\rightarrow\prod_{\ttree\rightarrow\stree}\AOp(\ttree)\otimes\Tree^{\square}(\ttree,\stree)^{\sharp}$
which we associate to our object in Construction <ref>
defines a coaugmentation over the cosimplicial object $\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}(-,\stree)^{\sharp})$,
or equivalently, a morphism

\begin{equation*}
\eta: \AOp\rightarrow\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}(-,\stree)^{\sharp}),
\end{equation*}

where we regard $\AOp$ as a constant cosimplicial object (see <cit.> for an account of these concepts).
We immediately see that this morphism is compatible with the coproduct operators, with the facets (and with the action of permutations whenever defined),
and hence, defines a morphism of cosimplicial homotopy Segal $E_\infty$-Hopf cooperads (shuffle or symmetric)
which yields a morphism of homotopy Segal $E_\infty$-Hopf cooperads in dg modules of the form of the proposition
when we pass to conormalized cochain complexes (we just use that we have $\DGN^*(\AOp) = \AOp$
in the case of the constant cosimplicial object $\AOp$).

The weak-equivalence claim follows from the observation that, in the case $F_{\stree}(-) = \Tree^{\square}(-,\stree)^{\sharp}$,
the cosimplicial object $\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}(-,\stree)^{\sharp})$
is endowed with an extra codegeneracy $s^{-1}$,
which is defined by extending the definition of Construction <ref>
to the case $j=-1$:

\begin{equation*}
s^{-1} = \id\otimes\id^{\otimes n+1}\otimes\eta^*.
\end{equation*}

(We again refer to <cit.> for a proof that the existence of this extra codegeneracy forces the contractibility of the conormalized cochain complex
in the cosimplicial direction, and hence, forces the acyclicity of our map.)
This observation finishes the proof of the proposition.

We can now conclude the proof of Theorem <ref>. The results of the previous proposition
returns a zigzag of weak-equivalences of homotopy Segal $E_\infty$-Hopf cooperads (shuffle or symmetric)

\begin{equation*}
\AOp(\stree)\xrightarrow{\sim}\DGN^*(\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree^{\square}(-,\stree)))
\xleftarrow{\sim}\DGN^*(\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree(-,\stree)^{\sharp})),
\end{equation*}

from which the result follows since the Segal $E_\infty$-Hopf cooperad

\begin{equation*}
\DGK^c(\AOp)(\stree) = \DGN^*(\DGK^{\bullet}(\AOp,\Tree^{\square},\Tree(-,\stree)^{\sharp}))
\end{equation*}

is strict by construction.

§ THE BARRATT–ECCLES OPERAD AND $E_\INFTY$-ALGEBRAS

The purpose of this appendix is to prove the results on the category of algebras over the Barratt–Eccles operad that we use throughout this article.
In preliminary paragraphs, we briefly review the definition of the chain Barratt–Eccles operad
and the definition of the associated category of $E_\infty$-algebras. We mostly follow the conventions of [1] for the definition of this operad
and we refer to this article for more detailed explanations
on this subject. We also briefly explain our conventions and basic definitions on permutations.
We devote the next paragraph to this subject.

In <ref>, we recall the definition of a cooperad without counit and we also forget about counits in the definition
of the notions of Segal cooperad that we consider all along this paper. But the Barratt–Eccles operad is more naturally defined as a unital operad.
Therefore we go back to the usual definition of an operad with unit in this appendix. Similarly, as the Barratt–Eccles operad naturally forms a symmetric operad,
we consider composition products in the standard form $\circ_i: \EOp(k)\otimes\EOp(l)\rightarrow\EOp(k+l-1)$
in this appendix, and not the general operations $\circ_{i_p}: \EOp(\{i_1<\dots<i_k\})\otimes\EOp(\{j_1<\dots<j_l\})\rightarrow\EOp(\{1<\dots<r\})$,
since we can deduce the latter from the former by the action of a shuffle permutation
on the Barratt–Eccles operad (see <ref>).

[Conventions on permutations and the associative operad]
We denote the symmetric group on $r$ letters by $\Sigma_r$.
We represent a permutation $s\in\Sigma_r$ by giving the sequence of its values

\begin{equation*}
s = (s(1),\dots,s(r)).
\end{equation*}

We use that the collection of the symmetric groups $\Sigma_r$, $r\in\NN$, form an operad in sets.
The symmetric structure of this operad is given by the left translation action.
The operadic composition $u\circ_i v\in\Sigma_{k+l-1}$ of permutations $u\in\Sigma_k$ and $v\in\Sigma_l$
is obtained by inserting the sequence of values of the permutation $v = (v(1),\dots,v(l))$
at the position of the value $i\in\{1,\dots,k\}$
in the permutation $u = (u(1),\dots,u(k))$,
by performing the value shift $v(y)\mapsto v(y) + i-1$
on the terms of the permutation $v$
and the shift $u(x)\mapsto u(x) + l-1$ on the terms of the permutation $u$ such that $u(x)>i$.
Thus, we have

\begin{equation*}
(u(1),\dots,u(k))\circ_i(v(1),\dots,v(l)) = (u(1)',\dots,\underbrace{v(1)',\dots,v(l)'}_{u(t)},\dots,u(k)'),
\end{equation*}

where $t$ is the position of the value $i$ in the sequence $(u(1),\dots,u(k))$, while $v(y)'$ and $u(x)'$ denote the result of our shift operations
so that we have $v(y)' = v(y) + i-1$, for all terms $v(y)$, we have $u(x)' = u(x)$ when $u(x)<i$
and $u(x)' = u(x) + l-1$ when $u(x)>i$.

This operad in sets governs the category of associative monoids. In our constructions, we also use a counterpart of this operad in our base category of modules.
This associative operad $\AsOp$ is defined by taking the modules spanned by the sets of permutations $\AsOp(r) = \kk[\Sigma_r]$, for $r\in\NN$,
with the induced structure operations.
In what follows, we generally identify a permutation $s\in\Sigma_r$ with the associated basis element in $\AsOp(r)$.
We also use the notation $\mu\in\AsOp(2)$ for the element of the associative operad given by the identity permutation on $2$ letters $\mu = \id_2\in\Sigma_2$,
which governs the product operation when we pass to associative algebras.
We trivially have $\AsOp(0) = \kk$ and we can identify the element given by the trivial permutation $* = \id_0\in\Sigma_0$ with an arity zero operation
which represents a unit for this product structure.

[The Barratt–Eccles operad and $E_\infty$-algebra structures]
The chain Barratt–Eccles operad $\EOp$ is defined by the normalized chain complexes of the homogeneous bar construction
of the symmetric groups.
Thus, we have:

\begin{equation*}
\EOp(r) = \DGN_*(W(\Sigma_r)),
\end{equation*}

for each arity $r\in\NN$, where $W(\Sigma_r)$ denotes the simplicial such that

\begin{equation*}
W(\Sigma_r)_n = \underbrace{\Sigma_r\times\dots\times\Sigma_r}_{n+1},
\end{equation*}

for each dimension $n$, together with the face and degeneracy operators such that

\begin{align*}
d_i(w_0,\dots,w_n) & = (w_0,\dots,\widehat{w_i},\dots,w_n), \\
s_j(w_0,\dots,w_n) & = (w_0,\dots,w_j,w_j,\dots,w_n),
\end{align*}

for any $(w_0,\dots,w_n)\in W(\Sigma_r)_n$.

For simplicity, we do not make any distinction between a simplex $(w_0,\dots,w_n)\in W(\Sigma_r)$
and the class of this simplex in the normalized chain complex $\EOp(r) = \DGN_*(W(\Sigma_r))$
in our notation. We just get $(w_0,\dots,w_j,w_j,\dots,w_n)\equiv 0$ for the degenerate simplices when we pass to the normalized chain complex.
The differential of simplices in $\EOp(r)$
is given by the usual formula:

\begin{equation*}
\delta(w_0,\dots,w_n) = \sum_{i=0}^n (-1)^i(w_0,\dots,\widehat{w_i},\dots,w_n).
\end{equation*}

The action of the symmetric group $\Sigma_r$ on $\EOp(r)$ is induced by the left translation action of permutations
on these simplices. We explicitly have:

\begin{equation*}
s\cdot(w_0,\dots,w_n) = (s w_0,\dots,s w_n),
\end{equation*}

for each permutation $s\in\Sigma_r$.
The operadic composition operations $\circ_i: \EOp(k)\otimes\EOp(l)\rightarrow\EOp(k+l-1)$ are given by the composite of a termwise application
of operadic composition operations on permutations with the Eilenberg–MacLane map
when we pass to normalized chains. For $(u_0,\dots,u_m)\in\EOp(k)$ and $(v_0,\dots,v_n)\in\EOp(l)$,
we explicitly have:

\begin{equation*}
(u_0,\dots,u_m)\circ_i(v_0,\dots,v_n) = \sum_{(i_*,j_*)}\pm(u_{i_0}\circ_i v_{j_0},\dots,u_{i_{m+n}}\circ_i v_{j_{m+n}}),
\end{equation*}

where the sum runs over the set of paths $\{(i_t,j_t),t=0,\dots,m+n\}$ which start at $(i_0,j_0) = (0,0)$ and end at $(i_{m+n},j_{m+n}) = (m,n)$
in an $m\times n$ cartesian diagram, the expression $\pm$ denotes a sign which we associate to any such path,
and along our paths, we take the operadic composites $u_{i_t}\circ_i v_{j_t}\in\Sigma_{k+l-1}$
of the permutations $u_{i_t}\in\Sigma_k$ and $v_{j_t}\in\Sigma_l$.
(The sign $\pm$ is determined by the shuffle of horizontal and vertical moves which we use when we form our path.)
For convenience, we may also represent such a composite by a picture
of the following form:

\begin{equation*}
(u_0,\dots,u_m)\circ_i(v_0,\dots,v_n) = \sum\pm\left(\vcenter{\xymatrix@R=2mm@C=3mm{ u_0\circ_i v_0\ar@{-}[d]\ar@{-}[r] &
u_1\circ_i v_0\ar@{-}[d]\ar@{-}[r] &
*{\cdots}\ar@{-}[r] &
u_m\circ_i v_0\ar@{-}[d] \\
u_0\circ_i v_1\ar@{-}[d]\ar@{-}[r] &
u_1\circ_i v_1\ar@{-}[d]\ar@{-}[r] &
*{\cdots}\ar@{-}[r] &
u_m\circ_i v_1\ar@{-}[d] \\
*{\vdots}\ar@{-}[d] & *{\vdots}\ar@{-}[d] & & *{\vdots}\ar@{-}[d] \\
u_0\circ_i v_n\ar@{-}[r] &
u_1\circ_i v_n\ar@{-}[r] &
*{\cdots}\ar@{-}[r] &
u_m\circ_i v_n }}\right),
\end{equation*}

where we take the sum of the simplices that we may form by running over all paths contained in the diagram materialized in our figure. (To be fully explicit, we take the paths
which go from the upper-left corner to the lower-right corner of the diagram
by a shuffle of horizontal moves $\xymatrix@R=2mm@C=3mm{u_x\circ_i v_y\ar@{-}[r] & u_{x+1}\circ_i v_y }$
and of vertical moves $\xymatrix@R=2mm@C=3mm{u_x\circ_i v_y\ar@{-}[r] & u_x\circ_i v_{y+1}}$.)

Recall that the operad of permutations in sets is identified with the set-theoretic associative operad (the operad which governs the category of associative monoids).
From the relation $W(\Sigma_r)_0 = \Sigma_r$ for any $r\in\NN$, we get an operad embedding $\AsOp\subset\EOp$
which identifies the module-theoretic version of the associative operad $\AsOp$
with the degree zero component of the Barratt-Eccles operad $\EOp$.
In what follows, we still use the notation $\mu\in\EOp(2)$ for the degree $0$ operation, represented by the identity permutation $\mu = \id_2\in\Sigma_2$,
which governs the product operation of associative algebra structures
in the Barratt–Eccles operad.
Note that we still have $\EOp(0) = \AsOp(0) = \kk$ (we take the convention to consider operads with a term in arity zero throughout this paragraph)
and the generating element of this arity zero term $*\in\EOp(0)$ also represents a unit operation
when we pass to the category of algebras over the Barratt–Eccles operad.

The Barratt–Eccles operad $\EOp$ is weakly-equivalent to the operad of commutative algebras $\ComOp$, and forms, as such, an instance of an $E_\infty$-operad.
Recall that we have $\ComOp(r) = \kk$, for any $r\in\NN$. The weak-equivalence $\EOp\xrightarrow{\sim}\ComOp$
is given by the standard augmentation $\DGN_*(W(\Sigma_r))\rightarrow\DGN_*(\pt) = \kk$
on the normalized chain complexes $\EOp(r) = \DGN_*(W(\Sigma_r))$, $r\in\NN$,
and sits in a factorization $\AsOp\hookrightarrow\EOp\xrightarrow{\sim}\ComOp$ of the usual morphism $\AsOp\rightarrow\ComOp$
between the associative operad $\AsOp$ and the commutative operad $\ComOp$.

We take the category of algebras over the Barratt–Eccles operad to get our model of the category of $E_\infty$-algebras.
Recall that we have $\EOp(0) = \kk$ so that our $\EOp$-algebras are equipped with a unit, which is represented by the generating element of this arity zero term of our operad $*\in\EOp(0)$.

By the main result of the article [1], the normalized cochain complex of a simplicial set $\DGN^*(X)$
is endowed with the structure of an algebra over the Barratt–Eccles operad.
This $\EOp$-algebra structure is functorial in $X\in\simp\Set$,
and extends the classical associative algebra structure of normalized cochains.

[The diagonal and the action of the Barratt–Eccles operad on tensor products]
In our constructions, we use that the Alexander–Whitney diagonal on the normalized chain complexes $\EOp(r) = \DGN_*(W(\Sigma_r))$, $r\in\NN$,
induces a morphism of dg operads $\Delta: \EOp\rightarrow\EOp\otimes\EOp$, where $\EOp\otimes\EOp$
is given by the arity-wise tensor product $(\EOp\otimes\EOp)(r) = \EOp(r)\otimes\EOp(r)$,
for any $r\in\NN$.
This map is given by the usual formula:

\begin{equation*}
\Delta(w_0,\dots,w_n) = \sum_{k=0}^n(w_0,\dots,w_k)\otimes(w_k,\dots,w_n),
\end{equation*}

for any $(w_0,\dots,w_n)\in\EOp(r)$.

The existence of this diagonal implies that a tensor product of $\EOp$-algebras $A\otimes B$
inherits an $\EOp$-algebra structure, since we can make an operation $\pi\in\EOp(r)$
act on $A\otimes B$
through its diagonal $\Delta(\pi)\in\EOp(r)\otimes\EOp(r)$. We explicitly take:

\begin{equation*}
\pi(a_1\otimes b_1,\dots,a_r\otimes b_r) = \sum_{(\pi)}\pi'(a_1,\dots,a_r)\otimes\pi''(b_1,\dots,b_r),
\end{equation*}

for all $a_1\otimes b_1,\dots,a_r\otimes b_r\in A\otimes B$, where we use the expression $\Delta(\pi) = \sum_{(\pi)}\pi'\otimes\pi''$
for the expansion of the coproduct of the operation $\pi\in\EOp(r)$
in the Barratt–Eccles operad.

In the paper, we also use that we have a morphism of $\EOp$-algebras

\begin{equation*}
\AW: A\vee B\rightarrow A\otimes B,
\end{equation*}

for any pair of $\EOp$-algebras $A$ and $B$, where we adopt the notation $\vee$ for the coproduct in the category of $\EOp$-algebras.
This morphism is induced by the inclusions $A\otimes *\rightarrow A\otimes B\leftarrow *\otimes B$
on each factor of the coproduct $A\vee B$, where we still use the notation $*$
for the unit of the $\EOp$-algebras $A$ and $B$.
(We will see in the proof of the next proposition that we can identify this map with an instance of an Alexander–Whitney diagonal. We therefore adopt the notation $\AW$ for this morphism.)
We have a morphism of dg modules which goes in the converse direction

\begin{equation*}
\EM: A\otimes B\rightarrow A\vee B,
\end{equation*}

and which is given by the formula $\EM(a\otimes b) = \mu(a,b)$, for each tensor $a\otimes b\in A\otimes B$, where $\mu(a,b)$ denotes the product of the elements $a\in A$ and $b\in B$
in the $\EOp$-algebra $A\vee B$. (We are going to see that we can identify this map with an instance of an Eilenberg–MacLane map.)
Note that neither $\AW$ nor $\EM$ are symmetric, and actually, the tensor product $A\otimes B$
does not define a symmetric bifunctor on the category of $\EOp$-algebras since the Alexander–Whitney diagonal $\Delta: \EOp(r)\rightarrow\EOp(r)\otimes\EOp(r)$
fails to be cocommutative.

We have the following useful property.

The above morphisms $\AW: A\vee B\rightarrow A\otimes B$ and $\EM: A\otimes B\rightarrow A\vee B$ satisfy $\AW\EM = \id$
and we have a natural chain homotopy $H: A\vee B\rightarrow A\vee B$
such that $\delta H + H\delta = \EM\AW - \id$.
Hence, our morphisms induce homotopy inverse weak-equivalences of dg modules

\begin{equation*}
\AW: A\vee B\xrightarrow{\sim} A\otimes B\quad\text{and}\quad\EM: A\otimes B\xrightarrow{\sim} A\vee B,
\end{equation*}

for all $\EOp$-algebras $A$ and $B$.

We consider the case of free $\EOp$-algebras first $A = \EOp(X)$ and $B = \EOp(Y)$.
We represent the elements of a free algebra such as $A = \EOp(X)$
by formal expressions of the form $a = u(x_1,\dots,x_r)$,
where $u\in\EOp(r)$ and $x_1,\dots,x_r\in X$.
We then have the expressions:

\begin{align*}
A\otimes B = \EOp(X)\otimes\EOp(Y) & = \bigoplus_{p,q}\EOp(p)\otimes_{\Sigma_p} X^{\otimes p}\otimes\EOp(p)\otimes_{\Sigma_q} Y^{\otimes q}, \\
A\vee B = \EOp(X\oplus Y) & = \bigoplus_{p,q}\EOp(p+q)\otimes_{\Sigma_p\times\Sigma_q} X^{\otimes p}\otimes Y^{\otimes q}.
\end{align*}

We use that the operadic composite $\id_2(u,v)$ of permutations $u\in\Sigma_p$ and $v\in\Sigma_q$
is identified with the result of a direct sum operation
such that $u\oplus v = (u(1),\dots,u(p),p+v(1),\dots,p+v(q))$.
Recall that $\mu = \id_2$ represents the associative product when we pass to the Barratt–Eccles operad $\EOp$.
For a tensor $a\otimes b = u(x_1,\dots,x_p)\otimes v(y_1,\dots,y_q)$
such that $u = (u_0,\dots,u_m)\in\EOp(p)$ and $v = (v_0,\dots,v_n)\in\EOp(q)$,
we have $\mu(a,b) = \mu(u,v)(x_1,\dots,x_p,y_1,\dots,y_q)$,
where we take the composite $\mu(u,v)$
in the Barratt–Eccles operad.

By definition of this composite in terms of shuffles of termwise composites $\mu(u_i,v_j) = u_i\oplus v_j$ (we apply the Eilenberg–MacLane map),
we obtain an expression of the following form:

\begin{multline*}
\EM(u(x_1,\dots,x_p)\otimes v(y_1,\dots,y_q))\\
= \sum\pm\underbrace{\left(\vcenter{\xymatrix@R=2mm@C=3mm{ u_0\oplus v_0\ar@{-}[d]\ar@{-}[r] &
u_1\oplus v_0\ar@{-}[d]\ar@{-}[r] &
*{\cdots}\ar@{-}[r] &
u_m\oplus v_0\ar@{-}[d] \\
u_0\oplus v_1\ar@{-}[d]\ar@{-}[r] &
u_1\oplus v_1\ar@{-}[d]\ar@{-}[r] &
*{\cdots}\ar@{-}[r] &
u_m\oplus v_1\ar@{-}[d] \\
*{\vdots}\ar@{-}[d] & *{\vdots}\ar@{-}[d] & & *{\vdots}\ar@{-}[d] \\
u_0\oplus v_n\ar@{-}[r] &
u_1\oplus v_n\ar@{-}[r] &
*{\cdots}\ar@{-}[r] &
u_m\oplus v_n }}\right)}_{\in\EOp(p+q)}
\end{multline*}

where the sum runs over all paths that we may form in the diagram of our figure (as in our representation of the composition of the Barratt–Eccles operad in <ref>).
We use this representation to identify our morphism $\EM: A\otimes B\rightarrow A\vee B$ with an instance of an Eilenberg–MacLane map.

The morphism $\AW: A\vee B\rightarrow A\otimes B$, on the other hand, carries any free algebra element $c = w(x_1,\dots,x_p,y_1,\dots,y_q)$
such that $w = (w_0,\dots,w_n)\in\EOp(p+q)$
to the image of the elements $x_1\otimes *,\dots,x_p\otimes *,*\otimes y_1,\dots,*\otimes y_q\in\EOp(X)\otimes\EOp(Y)$
under the action of the operation $w$ on $\EOp(X)\otimes\EOp(Y)$,
and hence to the tensor such that $\AW(w(x_1,\dots,x_p,y_1,\dots,y_q)) = \sum_{(w)} w'(x_1,\dots,x_p,*,\dots,*)\otimes w''(*,\dots,*,y_1,\dots,y_q)$,
where we use the notation $\Delta(w) = \sum_{(w)}w'\otimes w''$
for the expansion of the coproduct of the simplex $w$
in the Barratt–Eccles operad. Thus, if we assume $w = (w_0,\dots,w_n)$, then we have $\sum_{(w)}w'\otimes w'' = \sum_{k=0}^n (w_0,\dots,w_k)\otimes(w_k,\dots,w_n)$.
From this analysis, we deduce that our morphism $\AW$ is given by the following Alexander–Whitney type formula:

\begin{multline*}
\AW(w(x_1,\dots,x_p,y_1,\dots,y_q))\\
= \sum_{k=0}^n (w_0|_I,\dots,w_k|_I)(x_1,\dots,x_p)\otimes(w_k|_J,\dots,w_n|_J)(y_1,\dots,y_q),
\end{multline*}

where we set $I = \{1,\dots,p\}$ and $J = \{1,\dots,q\}$ for short and $|_{I}$, $|_{J}$
denote the obvious restriction operations on permutations
which we apply to our simplices termwise. (In this construction, we also use the canonical bijection $\{p+1,\dots,p+q\}\simeq\{1,\dots,q\}$
to identify the permutations of the set $\{p+1,\dots,p+q\}$ with permutations of the set $\{1,\dots,q\}$.)

For short, we may adopt the notation $x_*$ and $y_*$ for the words of variables $x_* = x_1,\dots,x_p$ and $y_* = y_1,\dots,y_q$
that occur in our expression of free algebra elements.
The chain homotopy $H$ can be given by a formula of the following form:

\begin{multline*}
H(w(x_*,y_*)) = \sum\pm(w_0,\dots,w_k)\\
\star\left(\vcenter{\xymatrix@R=2mm@C=3mm{ \scriptstyle{w_k|_I\oplus w_l|_J}\ar@{-}[d]\ar@{-}[r] &
\scriptstyle{w_{k+1}|_I\oplus w_l|_J}\ar@{-}[d]\ar@{-}[r] &
*{\cdots}\ar@{-}[r] &
\scriptstyle{w_l|_I\oplus w_l|_J}\ar@{-}[d] \\
\scriptstyle{w_k|_I\oplus w_{l+1}|_J}\ar@{-}[d]\ar@{-}[r] &
\scriptstyle{w_{k+1}|_I\oplus w_{l+1}|_J}\ar@{-}[d]\ar@{-}[r] &
*{\cdots}\ar@{-}[r] &
\scriptstyle{w_l|_I\oplus w_{l+1}|_J}\ar@{-}[d] \\
*{\vdots}\ar@{-}[d] & *{\vdots}\ar@{-}[d] & & *{\vdots}\ar@{-}[d] \\
\scriptstyle{w_k|_I\oplus w_n|_J}\ar@{-}[r] &
\scriptstyle{w_{k+1}|_I\oplus w_n|_J}\ar@{-}[r] &
*{\cdots}\ar@{-}[r] &
\scriptstyle{w_l|_I\oplus w_n|_J} }}\right)(x_*,y_*),
\end{multline*}

where $\star$ denotes a “join” operation (given by the obvious concatenation operation in $W(\Sigma_*)$), and the sum runs over the indices $0\leq k\leq l\leq n$,
together with the set of paths $\{(i_t,j_t),t=k,\dots,n\}$ which start at $(i_k,j_k) = (k,l)$ and end at $(i_n,j_n) = (l,n)$
in the $l-k\times n-l$-diagram represented in our figure.

The relation $\AW\EM = \id$ is an instance of the general inversion relation between the Alexander–Whitney diagonal and the Eilenberg–MacLane map.
In our context, we can also deduce this relation from the observation that the element $a\otimes b\in A\otimes B$
in a tensor product of $\EOp$-algebras $A$ and $B$
represents the product of the tensors $a\otimes *,*\otimes b\in A\otimes B$,
so that we have the identity $a\otimes b = \mu(a\otimes *,*\otimes b)$
in $A\otimes B$.

The proof of the chain homotopy relation $\delta H + H\delta = \AW\EM - \id$
is straightforward: the composite $\AW\EM$ corresponds to the $0$-face of the terms with $k=0$ in the expression of $H$
while the identity map corresponds to the $n$-face of the terms with $k=l=n$, and the other faces cancel out
when we form the anti-commutator $\delta(H) = \delta H + H\delta$.

The morphisms $\AW$ and $\EM$, which are defined for all $\EOp$-algebras $A$ and $B$,
are obviously functorial. We check that our chain homotopy $H$ is also functorial
with respect to the action of the morphisms
of free $\EOp$-algebras $\phi: \EOp(X)\rightarrow\EOp(X)$ and $\psi: \EOp(Y)\rightarrow\EOp(Y)$
on the coproduct $\EOp(X)\vee\EOp(Y) = \EOp(X\oplus Y)$.
Thus, we establish that we have the following relation:

\begin{equation*}
H((\phi\vee\psi)(w(x_*,y_*))) = (\phi\vee\psi)(H(w(x_*,y_*))),
\end{equation*}

for all $c = w(x_*,y_*)\in\EOp(X\oplus Y)$.

We use that $\phi = \phi_f: \EOp(X)\rightarrow\EOp(X)$ and $\psi = \psi_g: \EOp(Y)\rightarrow\EOp(Y)$
are induced by morphisms of dg modules $f: X\rightarrow\EOp(X)$
and $g: Y\rightarrow\EOp(Y)$.
We use the short notation $f(x_i) = \sum s^i(\underline{x}_i')$ and $g(y_j) = \sum t^j(\underline{y}_j')$
for the expansion of these free algebra elements $f(x_i)\in\EOp(X)$ and $g(y_j)\in\EOp(Y)$,
which we associate to the factors of a tensor $c = w(x_1,\dots,x_p,y_1,\dots,y_q)$.
We also write $s^i = (s^i_0,\dots,s^i_{d_i})$ and $t^j = (t^j_0,\dots,t^j_{e_j})$, where $d_i$ and $e_j$ are dimension variables.
We have the formula:

\begin{equation}\tag{$*$}\label{eqn:coproduct_functoriality}
(\phi\vee\psi)(w(x_*,y_*)) = w(f(x_*),g(y_*)) = \sum w(s^*,t^*)(\underline{x}_*',\underline{y}_*'),
\end{equation}

where we form the composite $w(s^*,t^*) = w(s^1,\dots,s^p,t^1,\dots,t^q)$ inside the Barratt–Eccles operad.
(In this formula, we also use the notation $s^*$ and $t^*$ for the words of simplices $s^* = s^1,\dots,s^p$ and $t^* = t^1,\dots,t^q$,
as well as the notation $\underline{x}_*'$ and $\underline{y}_*'$
for the composite words $\underline{x}_*' = \underline{x}_1',\dots,\underline{x}_p'$
and $\underline{y}_*' = \underline{y}_1',\dots,\underline{y}_q'$.)
We then use a multidimensional generalization of the picture of <ref>
for the definition of the operadic composition
in the Barratt–Eccles operad. We are going to use that the shuffles of horizontal and vertical moves, which we carry out in this definition of operadic composites,
satisfy natural associativity and commutativity relations when we perform a multidimensional
application of this operation.

We analyze the expression of the composite $(\phi\vee\psi)(H(w(x_*,y_*)))$ first.
We then apply the composition operation $w'\mapsto w'(s^*,t^*)$
to the simplices that occur in the expression of the chain homotopy $H(w(x_*,y_*))$.
We decompose the result of this operation as a join of simplices, using the join decomposition of the simplices that occur in the definition of our chain homotopy.
We identify the first factor of our join with a chain of composite permutations $w_x(s^*_{\alpha_*},t^*_{\beta_*})$,
where $w_x$ runs over the vertices of the simplex $(w_0,\dots,w_k)$,
which represents the first factor of our join in the expression of $H(w(x_*,y_*))$.
We take all shuffles of $w_x$-directional moves in the chain $(w_0,\dots,w_k)$
with $s^i_{\alpha_i}$-directional and $t^j_{\beta_j}$-directional moves
in subchains of the simplices $s^i$ and $t^j$
of the form $s^i{}' = (s^i_0,\dots,s^i_{d_i'})$ and $t^j{}' = (t^j_0,\dots,t^j_{e_j'})$,
where $d_i'\leq d_i$ and $e_j'\leq e_j$.
We identify the second factor of our join with a chain of composite permutations
of the form $w_x|_I(s^*_{\alpha_*})\oplus w_y|_J(t^*_{\beta_*})$,
where $w_x|_I\oplus w_y|_J$
runs over the vertices of the second join factor in the expression of $H(w(x_*,y_*))$, starting at $w_k|_I\oplus w_l|_J$
and ending at $w_l|_I\oplus w_n|_J$.
When we pass from the previous join factor to this second join factor in our computation of $(\phi\vee\psi)(H(w(x_*,y_*)))$,
we carry out a move of the form $\xymatrix@R=2mm@C=3mm{w_k\ar@{-}[r] & w_k|_I\oplus w_l|_J }$,
and hence, our move has to be constant in the $s^i_{\alpha_i}$ directions
and in the $t^j_{\beta_j}$ directions.
This observation implies that we start this simplex at the end point of the chains $s^i{}' = (s^i_0,\dots,s^i_{d_i'})$ in the $s^i_{\alpha_i}$ directions
and at the end point of the chains $t^j{}' = (t^j_0,\dots,t^j_{e_j'})$
in the $t^j_{\beta_j}$ directions.
Thus, to form the chains of composite permutations of our second join factor, we shuffle $w_x|_I$-directional and $w_y|_J$-directional moves
in the $2$-dimensional diagram $(w_k|_I,\dots,w_l|_I)\times(w_l|_J,\dots,w_n|_J)$
with $s^i_{\alpha_i}$-directional moves in the chains $s^i{}'' = (s^i_{d_i'},\dots,s^i_{d_i})$
and $t^j_{\beta_j}$-directional moves in the chains $t^j{}'' = (t^j_{e_j'},\dots,t^j_{e_j})$.

We now analyze the expression of the composite $H((\phi\vee\psi)(w(x_*,y_*)))$,
which we obtain by applying our chain homotopy $H$
to the element $(\phi\vee\psi)(w(x_*,y_*))
= \sum w(s^*,t^*)(\underline{x}_*',\underline{y}_*')$.
The simplices of the composite $w(s^*,t^*)$
consist of chains of composite permutations $w_x(s^*_{\alpha_*},t^*_{\beta_*})$,
which we obtain after shuffling $w_x$-directional moves in the chain $w = (w_0,\dots,w_n)$
with $s^i_{\alpha_i}$-directional moves in the chains $s^i{}'' = (s^i_0,\dots,s^i_{d_i})$
and $t^j_{\beta_j}$-directional moves in the chains $t^j{}'' = (t^j_0,\dots,t^j_{e_j})$.

To form our chain homotopy, we cut this chain at two positions $w_k(s^*_{d_*'},t^*_{e_*'}) = w_k(s^1_{d_1'},\dots,s^p_{d_p'},t^1_{e_1'},\dots,t^q_{e_q'})$
and $w_l(s^*_{d_*''},t^*_{e_*''}) = w_l(s^1_{d_1''},\dots,s^p_{d_p''},t^1_{e_1''},\dots,t^q_{e_q''})$ with $0\leq k\leq l\leq n$
and $0\leq d_i'\leq d_i''\leq d_i$, $0\leq e_j'\leq e_j''\leq e_j$.
We take the subchain of permutations running from $w_0(s^*_0,t^*_0)$
up to $w_k(s^*_{d_*'},t^*_{e_*'})$
to get the first join factor of our chain homotopy. We exactly retrieve the same chains as in our decomposition of $(\phi\vee\psi)(H(w(\underline{x},\underline{y})))$.

We then form the restrictions $w_x(s^*_{\alpha_*},t^*_{\beta_*})|_{I'} = w_x|_I(s^*_{\alpha_*})$
and $w_y(s^*_{\alpha_*},t^*_{\beta_*})|_{J'} = w_y|_J(t^*_{\beta_*})$
where $I'$ denotes the terms of the composite permutations $w_x(s^*_{\alpha_*},t^*_{\beta_*})$
that correspond to the positions of the variables $\underline{x}_*'$,
whereas $J'$ denotes the terms of the composite permutations $w_y(s^*_{\alpha_*},t^*_{\beta_*})$
that correspond to the positions of the variables $\underline{y}_*'$.
We take a chain of direct sums $w_x(s^*_{\alpha_*},t^*_{\beta_*})|_{I'}\oplus w_y(s^*_{\alpha_*},t^*_{\beta_*})|_{J'}$
such that $w_x(s^*_{\alpha_*},t^*_{\beta_*})$ runs from $w_k(s^*_{d_*'},t^*_{e_*'})$ up to $w_l(s^*_{d_*''},t^*_{e_*''})$,
while $w_y(s^*_{\alpha_*},t^*_{\beta_*})$ runs from $w_l(s^*_{d_*''},t^*_{e_*''})$ up to $w_l(s^*_{d_*},t^*_{e_*})$.
(We then take a shuffle of $w_x(s^*_{\alpha_*},t^*_{\beta_*})$-directional moves and of $w_y(s^*_{\alpha_*},t^*_{\beta_*})$-direction moves.)
We easily see that this operation produces a degeneracy in the case where we have $e_j'<e_j''$ for some $j$,
because in this case the subchain of permutations $w_x(s^*_{\alpha_*},t^*_{\beta_*})$
contains a $t^j_{\beta_j}$-directional move, which produces a degeneracy when we pass
to the restriction $w_x(s^*_{\alpha_*},t^*_{\beta_*})|_{I'} = w_x|_I(s^*_{\alpha_*})$.
We similarly see that our operation produces a degeneracy in the case where we have $d_i''<d_i$ for some $i$.
We therefore have to assume $d_i'' = d_i$ for all $i$ and $e_j' = e_j$ for all $j$ in order to avoid possible degeneracies,
and in these cases, we exactly retrieve the same chains as in our expression of the second join factor in our decomposition of $(\phi\vee\psi)(H(w(x_*,y_*)))$.

We conclude from this analysis that the expansions of $H((\phi\vee\psi)(w(x_*,y_*)))$ and $(\phi\vee\psi)(H(w(x_*,y_*)))$ consist of the same joins of simplices,
and therefore these composites agree, as expected.

To finish the proof of our proposition, we use that every object of the category of $\EOp$-algebras has a presentation in terms of a natural reflexive coequalizer of free $\EOp$-algebras.
If we have a pair of objects $A$ and $B$, then we can form a commutative diagram:

\begin{equation*}
\xymatrix{ \EOp(X_1)\otimes\EOp(Y_1)\ar@<-2pt>[r]_{\EM}\ar@<+2pt>[d]\ar@<-2pt>[d] &
\EOp(X_1)\vee\EOp(Y_1)\ar@<-2pt>[l]_{\AW}\ar@<+2pt>[d]\ar@<-2pt>[d]\ar@(ur,dr)[]!UR;[]!DR^{H} \\
\EOp(X_0)\otimes\EOp(Y_0)\ar@<-2pt>[r]_{\EM}\ar<EMAIL_ADDRESS>&
\EOp(X_0)\vee\EOp(Y_0)\ar@<-2pt>[l]_{\AW}\ar@/_1em/[u]\ar@{.>}[d]\ar@(ur,dr)[]!UR;[]!DR^{H} \\
A\otimes B\ar@<-2pt>[r]_{\EM} & A\vee B\ar@<-2pt>[l]_{\AW}\ar@{.>}@(ur,dr)[]!UR;[]!DR^{H} }
\end{equation*}

where we take the presentations of $A$ and $B$ by reflexive coequalizers in the vertical direction and our deformation retract diagram
of Alexander–Whitney and Eilenberg–MacLane maps
in the horizontal direction. We use this diagram to check that our chain homotopy $H$ passes to the quotient
and induces a chain homotopy such that $\delta H + H\delta = \EM\AW - \id$
on $A\vee B$,
as indicated in our figure.

[The action of the Barratt–Eccles operad on the interval, on cubical cochains and the definition of connections]
We already recalled that the normalized cochain complex of a simplicial set $\DGN^*(X)$ inherits the structure of an algebra over the Barratt–Eccles operad.
We refer to [1] for the precise definition.
We consider the cochain algebra of the $1$-simplex $X = \Delta^1$ in our definition of homotopy Segal $E_\infty$-cooperads
together with the cubical cochain algebras

\begin{equation*}
I^m = \underbrace{\DGN^*(\Delta^1)\otimes\dots\otimes\DGN^*(\Delta^1)}_m
\end{equation*}

which we provide with an $\EOp$-algebra structure, using the action of the Barratt–Eccles operad on each cochain complex factor $\DGN^*(\Delta^1)$,
and the diagonal operation of Construction <ref>.

We study structures attached to the cochain algebra $I = \DGN^*(\Delta^1)$ in this paragraph.
We also consider the normalized chain complex $\DGN_*(\Delta^1)$, dual to $\DGN^*(\Delta^1)$.
We have

\begin{equation*}
\DGN_*(\Delta^1) = \kk\underline{0}\oplus\kk\underline{1}\oplus\kk\underline{01},
\end{equation*}

where $\underline{01}$ denotes the class of the fundamental simplex of $\Delta^1$ in the normalized chain complex,
whereas $\underline{0}$ and $\underline{1}$ denote the class of the vertices of $\underline{01}$.
We can identify $\kk\underline{0}\subset\DGN_*(\Delta^1)$ with the image of the map $\DGN_*(d^1): \DGN_*(\Delta^0)\rightarrow\DGN_*(\Delta^1)$
induced by the $1$-coface $d^1: \Delta^0\rightarrow\Delta^1$,
while $\kk\underline{1}\subset\DGN_*(\Delta^1)$ is identified with the image of the map $d^0: \DGN_*(\Delta^0)\rightarrow\DGN_*(\Delta^1)$
induced by the $0$-coface $d^0: \Delta^0\rightarrow\Delta^1$. We have $\delta(\underline{01}) = \underline{1} - \underline{0}$.
For the cochain algebra, we have

\begin{equation*}
\DGN^*(\Delta^1) = \kk\underline{0}^{\sharp}\oplus\kk\underline{1}^{\sharp}\oplus\kk\underline{01}^{\sharp},
\end{equation*}

where we take the basis $(\underline{0}^{\sharp},\underline{1}^{\sharp},\underline{01}^{\sharp})$
dual to $(\underline{0},\underline{1},\underline{01})$.

We now explain the definition of a connection $\nabla^*: \DGN^*(\Delta^1)\otimes\DGN^*(\Delta^1)\rightarrow\DGN^*(\Delta^1)$,
which we use in the construction of degeneracy operators in our definition of homotopy Segal $E_\infty$-cooperads.
We consider the simplicial map $\min: \Delta^1\times\Delta^1\rightarrow\Delta^1$ defined by the mapping $(s,t)\mapsto\min(s,t)$
on topological realizations, or equivalently, by the following representation:

\begin{equation*}
\vcenter{\hbox{\includegraphics{connection-picture.pdf}}},
\end{equation*}

where we take the projection onto the diagonal simplex along the lines depicted in the figure.

We take the composite $\nabla_* = \DGN_*(\min)\circ\EM$ of the induced map on normalized chain complexes $\DGN_*(\min): \DGN_*(\Delta^1\times\Delta^1)\rightarrow\DGN_*(\Delta^1)$
with the Eilenberg–MacLane map $\EM: \DGN_*(\Delta^1)\otimes\DGN_*(\Delta^1)\rightarrow\DGN_*(\Delta^1\times\Delta^1)$.
We get the following formulas:

\begin{align*}
& \nabla_*(\underline{0}\otimes\underline{0}) = \nabla_*(\underline{1}\otimes\underline{0}) = \nabla_*(\underline{0}\otimes\underline{1}) = \underline{0}, \\
& \nabla_*(\underline{1}\otimes\underline{1}) = \underline{1}, \\
& \nabla_*(\underline{1}\otimes\underline{01}) = \nabla_*(\underline{01}\otimes\underline{1}) = \underline{01}, \\
& \nabla_*(\underline{0}\otimes\underline{01}) = \nabla_*(\underline{01}\otimes\underline{0}) = 0, \\
& \nabla_*(\underline{01}\otimes\underline{01}) = 0.
\end{align*}

We define our connection on normalized cochains $\nabla^*$ as the dual map of this morphism $\nabla_*$.
We accordingly take:

\begin{equation*}
\nabla^* = \EM\circ\DGN^*(\min): \DGN^*(\Delta^1)\rightarrow\DGN^*(\Delta^1)\otimes\DGN^*(\Delta^1),
\end{equation*}

and we can determine this morphism by the following formulas on our basis elements:

\begin{align*}
& \nabla^*(\underline{0}^{\sharp}) = \underline{0}^{\sharp}\otimes\underline{0}^{\sharp} + \underline{1}^{\sharp}\otimes\underline{0}^{\sharp}
+ \underline{0}^{\sharp}\otimes\underline{1}^{\sharp}, \\
& \nabla^*(\underline{1}^{\sharp}) = \underline{1}^{\sharp}\otimes\underline{1}^{\sharp}, \\
& \nabla^*(\underline{01}^{\sharp}) = \underline{01}^{\sharp}\otimes\underline{1}^{\sharp} + \underline{1}^{\sharp}\otimes\underline{01}^{\sharp}.
\end{align*}

We crucially need the observation of the next proposition in our constructions.

The map $\nabla^* = \EM\circ\DGN^*(\min): \DGN^*(\Delta^1)\rightarrow\DGN^*(\Delta^1)\otimes\DGN^*(\Delta^1)$ defined in the above paragraph is a morphism of $\EOp$-algebras,
where we use that the Barratt-Eccles operad $\EOp$ acts on the tensor product $\DGN^*(\Delta^1)\otimes\DGN^*(\Delta^1)$
through the operadic diagonal $\Delta: \EOp\rightarrow\EOp\otimes\EOp$
and its action on each factor $\DGN^*(\Delta^1)$.

We go back to the definition of the $\EOp$-algebra structure of normalized cochain complexes of simplicial sets
in terms of a dual $\EOp$-coalgebra structure on normalized chain complexes.
We prove that the morphism $\nabla_*: \DGN_*(\Delta^1)\otimes\DGN_*(\Delta^1)\rightarrow\DGN_*(\Delta^1)$,
dual to the morphism of our claim, is a morphism of $\EOp$-coalgebras.
We may note that the Eilenberg–MacLane map $\EM: \DGN_*(X)\otimes\DGN_*(Y)\rightarrow\DGN_*(X\times Y)$
does not preserve $\EOp$-coalgebra structures
in general. Nevertheless, such a statement holds when one factor is a one-point set, $X = *$ or $Y = *$,
because in this case, we have $\DGN_*(X)\simeq\DGN_*(X)\otimes\kk\simeq\DGN_*(X\times *)$
or $\DGN_*(Y)\simeq \kk\otimes\DGN_*(Y)\simeq\DGN_*(*\times Y)$, and the Eilenberg–MacLane map
reduces to the identity morphism on the functor of normalized chains.
We readily deduce from this observation that our morphism $\nabla_*: \DGN_*(\Delta^1)\otimes\DGN_*(\Delta^1)\rightarrow\DGN_*(\Delta^1)$
preserves $\EOp$-coalgebra structure on the subcomplex generated by the tensors $\underline{\sigma}\otimes\underline{\tau}\in\DGN_*(\Delta^1)\otimes\DGN_*(\Delta^1)$
such that $\underline{\sigma}\in\{\underline{0},\underline{1}\}$
or $\underline{\tau}\in\{\underline{0},\underline{1}\}$
since such tensors lie in the image of the coface maps $d^i\otimes\id: \DGN^*(\Delta^0)\otimes\DGN_*(\Delta^1)\rightarrow\DGN_*(\Delta^1)\otimes\DGN_*(\Delta^1)$
and $\id\otimes d^i: \DGN^*(\Delta^1)\otimes\DGN_*(\Delta^1)\rightarrow\DGN_*(\Delta^0)\otimes\DGN_*(\Delta^1)$,
with $i = 0,1$.

We use the notation $\pi_*: \DGN_*(\Delta^1)\rightarrow\DGN_*(\Delta^1)^{\otimes r}$ for the operation
that we associate to an element of the Barratt–Eccles operad $\pi\in\EOp(r)$
in the definition of the $\EOp$-coalgebra
structure on $\DGN_*(\Delta^1)$.
In general, for a tensor $\underline{\sigma}\otimes\underline{\tau}\in\DGN_*(\Delta^1)\otimes\DGN_*(\Delta^1)$, we have the formula:

\begin{equation*}
\pi_*(\underline{\sigma}\otimes\underline{\tau}) = \sum_{(\pi)}\sh(\pi'_*(\underline{\sigma})\otimes\pi''_*(\underline{\tau})),
\end{equation*}

where $\Delta(\pi) = \sum_{(\pi)}\pi'\otimes\pi''$ denotes the coproduct of the operation $\pi$,
while $\sh: \DGN_*(\Delta^1)^{\otimes r}\otimes\DGN_*(\Delta^1)^{\otimes r}\rightarrow(\DGN^*(\Delta^1)\otimes\DGN^*(\Delta^1))^{\otimes r}$
is the tensor permutation such that $\sh(a_1\otimes\dots\otimes a_r\otimes b_1\otimes\dots\otimes b_r) = (a_1\otimes b_1)\otimes\dots\otimes(a_r\otimes b_r)$.
Thus the statement of our claim is equivalent to the following relation:

\begin{equation}\tag{$*$}\label{eqn:nabla_product}
\pi_*(\nabla_*(\underline{\sigma}\otimes\underline{\tau})) = \sum_{(\pi)}\nabla_*^{\otimes r}\sh(\pi'_*(\underline{\sigma})\otimes\pi''_*(\underline{\tau})),
\end{equation}

for $\pi\in\EOp(r)$, and for any $\underline{\sigma}\otimes\underline{\tau}\in\DGN_*(\Delta^1)\otimes\DGN_*(\Delta^1)$.

We can use the argument of the previous paragraph to establish the validity of this relation in the case where $\underline{\sigma}$ or $\underline{\tau}$
is a vertex $\underline{0},\underline{1}\in\DGN_*(\Delta^1)$.
We therefore focus on the case $\underline{\sigma}\otimes\underline{\tau} = \underline{01}\otimes\underline{01}$.
We then have $\nabla_*(\underline{01}\otimes\underline{01}) = 0$,
so that the above equation (<ref>)
reduces to the following vanishing relation:

\begin{equation}\tag{$*'$}\label{eqn:vanishing_nabla_product}
\sum_{(\pi)}\nabla_*^{\otimes r}\sh(\pi'_*(\underline{01})\otimes\pi''_*(\underline{01})) = 0.
\end{equation}

We devote the rest of this proof to the verification of this relation.

The definition of the action of the Barratt–Eccles operad on chains.
To carry out our verification, we have to go back to the explicit expression of the operation $\varpi_*: \DGN_*(\Delta^1)\rightarrow\DGN_*(\Delta^1)^{\otimes r}$
associated to an element of the Barratt–Eccles operad $\varpi\in\EOp(r)$
in terms of interval cuts associated to a table reduction
of the simplices of the permutations $(s_0,\dots,s_l)$
that occur in the expansion of $\varpi$.
We briefly recall this construction in the general case of a $q$-dimensional simplex $\Delta^q$.
We refer to [1] for details.

The table reduction is a sum of surjective maps $s: \{1,\dots,r+l\}\rightarrow\{1,\dots,r\}$,
which we determine by sequences of values $s = (s(1),\dots,s(r+l))$,
which we arrange on a table, as in the following picture:

\begin{equation*}
s = \left|\begin{array}{l} s(1),\dots,s(e_0-1),s(e_0) \\
s(e_0+1),\ldots,s(e_1-1),s(e_1) \\
\vdots\\
s(e_{l-2}+1),\dots,s(e_{l-1}-1),s(e_{l-1}) \\
\end{array}\right..
\end{equation*}

The caesuras $s(e_0),\dots,s(e_{l-1})$, which terminate the rows of the table, are the terms $y = s(x)$ of the sequence $s = (s(1),\dots,s(r+l))$
that do not form the last occurrence of a value $y\in\{1,\dots,r\}$.
Thus, the complement of the caesuras, which consists of the inner terms of the rows $s(e_{i-1}+1),\dots,s(e_i-1)$, $i = 0,\dots,l-1$,
and of the terms of the last rows $s(e_{l-1}+1),\dots,s(r+l)$, consists of the terms of the sequence $s = (s(1),\dots,s(r+l))$
which are not repeated after their position.

The table reduction of a simplex of permutations $\varpi = (s_0,\dots,s_l)$
is a sum of table arrangements of surjections
of the following form:

\begin{equation*}
\TR(s_0,\dots,s_l) = \sum\left|\begin{array}{l} s_0(1),\dots,s_0(r_0-1),s_0(r_0) \\
s_1(1)',\dots,s_1(r_1-1)',s_1(r_1)' \\
\vdots \\
s_{l-1}(1)',\dots,s_{l-1}(r_{l-1}-1)',s_{l-1}(r_{l-1})' \\
s_l(1)',\dots,s_l(r_l)' \end{array}\right.,
\end{equation*}

and which we obtain by browsing the terms of our permutations $s_i$, $i = 0,\dots,l$.
For $i = 0$, we retain all terms of our permutation $s_0(1),\dots,s_0(x),\dots$ up to the choice of a caesura $s_0(r_0)$, where we decide to stop this enumeration.
For $i > 0$, in the enumeration of the terms of the permutation $s_i$ we only retain the values $s_i(1)',\dots,s_i(x)',\dots$
that do not occur before the caesura on the previous rows of our table.
For $i < l$, we again stop this enumeration at the choice of a caesura $s_i(r_i)$.
For $i = l$, we run this process up to the last term of the permutation $s_l$. We sum over all possible choices of caesuras.

For $0\leq\upsilon_0\leq\dots\leq\upsilon_p\leq q$, we generally denote by $\underline{\upsilon_0\dots\upsilon_p}\in\Delta^q$
the $p$-simplex defined by taking the image of the fundamental simplex of the $q$-simplex $\Delta^q$
under the simplicial operator $u^*: \Delta^q_q\rightarrow\Delta^q_p$
associated to the map $u: \{0<\dots<p\}\rightarrow\{0<\dots<q\}$
such that $u(x) = \upsilon_x$, $x = 0,\dots,p$.
The notation $\underline{0\cdots q}$, for instance, represents the fundamental simplex of $\Delta^q$.

Each surjection $s = (s(1),\dots,s(l+r))$
in the table reduction
of an element of the Barratt–Eccles operad $\varpi = (s_0,\dots,s_l)$
is used to assign a sum of tensors

\begin{equation*}
s_*(\underline{0\cdots q}) = \sum_{\alpha}\pm\underline{\sigma}_{(1)}^{\alpha}\otimes\dots\otimes\underline{\sigma}_{(r)}^{\alpha}\in\DGN_*(\Delta^q)^{\otimes r}
\end{equation*}

to the fundamental simplex $\underline{0\cdots q}\in\Delta^q_q$.
We proceed as follows. We fix a sequence of indices $0 = \rho_0\leq\dots\leq\rho_x\leq\dots\leq\rho_{r+l} = q$,
which we associate to an interval decomposition of the indexing sequence
of the fundamental simplex:

\begin{equation*}
\underline{0\cdots q} = \underline{\rho_0\cdots\rho_1}|\underline{\rho_1\cdots\rho_2}|\cdots\,\cdots|\underline{\rho_{r+l-1}\cdots\rho_{r+l}}.
\end{equation*}

For $x = 1,\dots,r+l$, we label the interval $\underline{\rho_{x-1}\cdots\rho_x}$
with the value of the term $s(x)$ of our surjection $s = (s(1),\dots,s(r+l))$
in $\{1,\dots,r\}$,
as in the following picture:

\begin{equation*}
\underline{\rho_0\overset{s(1)}{\cdots}\rho_1}|\underline{\rho_1\overset{s(2)}{\cdots}\rho_2}|\cdots
\,\cdots|\underline{\rho_{x-1}\overset{s(x)}{\cdots}\rho_x}|\cdots
\,\cdots|\underline{\rho_{r+l-1}\overset{s(r+l)}{\cdots}\rho_{r+l}}.
\end{equation*}

Then, for $i\in\{1,\dots,r\}$, we concatenate the intervals $\underline{\rho_{x-1}\cdots\rho_x}$ labelled by the value $s(x) = i$
in order to form the factor $\sigma_{(i)}^{\alpha}$ of our tensor $s_*(\underline{0\cdots q})\in\DGN_*(\Delta^q)^{\otimes r}$.
We sum over all possible choices of indices $0 = \rho_0\leq\dots\leq\rho_x\leq\dots\leq\rho_{r+l} = q$.

The image of the simplex $\underline{0\cdots q}\in\DGN_*(\Delta^q)$ under the operation $\varpi: \DGN_*(\Delta^q)\rightarrow\DGN_*(\Delta^q)^{\otimes r}$
associated to an element of the Barratt-Eccles operad $\varpi\in\EOp(r)$
is given by the sum of the tensors $s_*(\underline{0\cdots q})\in\DGN_*(\Delta^q)^{\otimes r}$,
where $s$ runs over surjections that occur in the table reduction of $\varpi$.

In general, we can determine the action of the operation $\varpi_*: \DGN_*(X)\rightarrow\DGN_*(X)^{\otimes r}$
on an element $\sigma\in\DGN_*(X)$ in the normalized chains of a simplicial set