2.1. Coalgebras

Throughout this article, $\Bbbk $ denotes a commutative and unital ring and we work over its associated symmetric monoidal category of (homologically) graded chain complexes of $\Bbbk $ -modules $(\mathsf {Ch}, \otimes , \Bbbk )$ .

A coalgebra consists of a chain complex C and chain maps $\Delta \colon C \to C \otimes C$ and $\varepsilon \colon C \to \Bbbk $ satisfying the usual coassociativity and counitality relations. Denote by $\mathsf {coAlg}$ the category of coalgebras with morphisms being structure preserving chain maps. The category $\mathsf {coAlg}$ is symmetric monoidal, with braiding induced from $\mathsf {Ch}$ and structure maps of a product $C \otimes C^\prime $ given by

$$ \begin{gather*} C \otimes C^\prime \xrightarrow{\Delta \otimes \Delta^\prime} (C \otimes C) \otimes (C^\prime \otimes C^\prime) \xrightarrow{(23)} (C \otimes C^\prime) \otimes (C \otimes C^\prime), \\ C \otimes C^\prime \xrightarrow{\varepsilon \otimes \varepsilon^\prime} \Bbbk \otimes \Bbbk \xrightarrow{\cong} \Bbbk. \end{gather*} $$

A coaugmentation on a coalgebra C is a coalgebra map $\nu \colon \Bbbk \to C$ . A coalgebra is said to be coaugmented if it is equipped with a coaugmentation. We denote by $\mathsf {coAlg}^\ast $ the category of coaugmented coalgebras with morphisms being coaugmentation preserving coalgebra maps. A coaugmented coalgebra is a connected coalgebra if it is $0$ in negative degrees and the coaugmentation induces an isomorphism $\Bbbk \cong C_0$ of $\Bbbk $ -modules. We denote by $\mathsf {coAlg}^0$ the full subcategory of $\mathsf {coAlg}^\ast $ defined by these.

2.2. Monoids

A monoidal object in a monoidal category $(\mathsf {C}, \otimes , \mathbb{1})$ is an object M together with morphisms $\mu \colon M \otimes M \to M$ and $\eta \colon \mathbb{1} \to M$ satisfying the usual associativity and unital relations. The category of these together with structure preserving morphisms, referred to as monoidal morphisms, is denoted ${\mathsf {Mon}}_{\mathsf {C}}$ . We remark that a lax monoidal functor $\mathsf {C} \to \mathsf {C}^\prime $ induces a functor between their categories of monoids $\mathsf {Mon}_{\mathsf {C}} \to \mathsf {Mon}_{\mathsf {C}^\prime }$ . For example, the forgetful functor from $\mathsf {coAlg}$ to $\mathsf {Ch}$ is monoidal and so, it induces a forgetful functor from monoidal coalgebras to monoidal chain complexes, which are more commonly known as bialgebras and algebras respectively, but this terminology is not well suited for our purposes.

2.3. Simplicial theory

The simplex category is denoted by $\triangle $ and its objects by $[n]$ . The morphisms in $\triangle $ are generated by coface maps, denoted by $\partial ^{j}_n \colon [n-1] \to [n]$ for $0\leq j \leq n$ , and codegeneracy maps, denoted by $\xi ^{j}_n \colon [n+1] \to [n]$ for $0 \leq j \leq n+1$ . These satisfy the usual cosimplicial identities. For simplicity, we omit the subscript in the notation when there is no risk of confusion and simply denote these maps by $\partial ^{j} \colon [n-1] \to [n]$ and $\xi ^{j} \colon [n+1] \to [n]$ .

The category of simplicial sets $\mathsf {Fun}(\triangle ^{\mathrm {op}}, \mathsf {Set})$ is denoted by $\mathsf {sSet}$ and the standard n-simplex $\triangle (-, [n])$ by $\triangle ^n$ . For any simplicial set X we write, as usual, $X_n$ instead of $X[n]$ and identify the elements of $\triangle ^n_m$ with increasing tuples $[v_0, \dots , v_m]$ where $v_i \in \{0, \dots , n\}$ .

If X is such that $X_0$ is a singleton set, we say that X is reduced. We denote the full subcategory of reduced simplicial sets by $\mathsf {sSet}^0$ .

We consider the topological n-simplex $\mathbb {\Delta }^n$ embedded in $\mathbb {R}^{n+1}$ as

$$\begin{align*}\mathbb{\Delta}^n = \{ (x_0, \dots, x_n) \in \mathbb{R}^{n+1} | x_0^2 + \dots + x_n^2=1\} \end{align*}$$

so that the i-th vertex is given by the standard basis vector $v_i=(0,\ldots ,1, \ldots ,0) \in \mathbb {R}^{n+1}$ with $1$ in the i-th entry and $0$ in all other entries. Consider the usual adjunction pair formed by the geometric realization and singular complex

determined by the spaces $\mathbb {\Delta }^n$ and usual coface inclusions $\delta ^i \colon \mathbb {\Delta }^{n-1} \to \mathbb {\Delta }^n$ and codegeneracy projections $\sigma ^j \colon \mathbb {\Delta }^n \to \mathbb {\Delta }^{n-1}$ .

The functor of (normalized) simplicial chains $\operatorname {N^{\triangle }} \colon \mathsf {sSet} \to \mathsf {Ch}$ is defined on any simplicial set X by first considering the chain complex $(\Bbbk [X], \partial )$ , given on degree n by the free $\Bbbk $ -module generated by the n-simplices of X with differential given by the alternating sum of face maps, and then modding out by the subchain complex of degenerate elements. We remark that this functor is naturally equivalent to the composition of the geometric realization functor and that of cellular chains with respect to the standard CW structure. When no confusion arises from doing so, we write $\operatorname {\mathrm {N}}$ instead of $\operatorname {N^{\triangle }}$ and refer to it simply as the functor of chains. We will denote the functor of (simplicial) singular chains $\operatorname {N^{\triangle }} \circ \operatorname {Sing^{\triangle }} \colon \mathsf {Top} \to \mathsf {Ch}$ by $\operatorname {S^{\triangle }}$ , where $\operatorname {Sing^{\triangle }} \colon \mathsf {Top} \to \mathsf {sSet}$ denotes the singular complex functor. We modify this construction for a pointed topological space $(\mathfrak {X},x)$ by only considering maps $\mathbb {\Delta }^n \to \mathfrak {X}$ sending all vertices to x. This produces a reduced simplicial set $\operatorname {Sing^{\triangle }}(\mathfrak {X},x)$ whose normalized chains we denote by $\operatorname {S^{\triangle }}(\mathfrak {X},x)$ .

We now recall a classical chain approximation of the diagonal map on the chains of a simplicial set, that is, a lift of the functor of chains to coalgebras:

It suffices to define this functor $\operatorname {N^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}$ on standard simplices. For any $n \in \mathbb {N}$ , define $\Delta \colon \operatorname {\mathrm {N}}(\triangle ^n) \to \operatorname {\mathrm {N}}(\triangle ^n)^{\otimes 2}$ by

$$\begin{align*}\Delta \big( [v_0, \dots, v_q] \big) = \sum_{i=0}^q \ [v_0, \dots, v_i] \otimes [v_i, \dots, v_q] \end{align*}$$

and $\epsilon \colon \operatorname {\mathrm {N}}(\triangle ^n) \to \Bbbk $ by

$$\begin{align*}\epsilon \big( [v_0, \dots, v_q] \big) = \begin{cases} 1 & \text{ if } q = 0, \\ 0 & \text{ if } q>0. \end{cases} \end{align*}$$

We referred to this lift as the Alexander–Whitney coalgebra structure on simplicial chains.

If X is a reduced simplicial set, then $\operatorname {N^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(X)$ becomes a connected coalgebra with the coaugmentation $\nu \colon \Bbbk \to \operatorname {N^{\triangle }}(X)$ induced by sending $1$ to the basis element represented by the unique $0$ -simplex of X. Hence, the functor $\operatorname {N^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}$ restricts to a functor

$$\begin{align*}\operatorname{N^{\triangle}_{\!{\mathcal{A}\mathsf{s}}}} \colon \mathsf{sSet}^0 \to \mathsf{coAlg}^0. \end{align*}$$

2.4. Cubical theory

We include an abridged presentation of cubical sets parallel to the our treatment of simplicial sets in the previous section. For more details, we refer the reader to any of [Reference Friedman, Medina-Mardones and Sinha13, Reference Grandis and Mauri15, Reference Kaufmann and Medina-Mardones21].

The cube category (with a connection) $\square $ is the subcategory of $\mathsf {Set}$ whose objects are $2^n = \{0, 1\}^n$ for $n \in \mathbb {N}$ with $2^0 = \{0\}$ . We will denote $2^1$ by $2$ and $2^0$ by $1$ , which is the unit of the monoidal structure of $\square $ given by $2^n \times 2^m = 2^{n+m}$ . The morphisms of $\square $ are monoidally generated by

(2.1)

defined by

$$ \begin{gather*} \delta_0(0) = 0, \qquad \delta_1(0) = 1, \qquad \sigma(p) = 0, \qquad \gamma(p,q) = \max(p,q), \end{gather*} $$

for $p,q \in \{0,1\}$ . We remark that the cube category without connections will not be considered in this work.

The category of cubical sets $\mathsf {Fun}(\square ^{\mathrm {op}}, \mathsf {Set})$ is denoted by $\mathsf {cSet}$ and the standard n-cube $\square (-, 2^n)$ by $\square ^n$ . For any cubical set Y we write, as usual, $Y_n$ instead of $Y(2^n)$ . The monoidal structure on $\square $ induces one on $\mathsf {cSet}$ . More explicitly, for two cubical sets we have

$$\begin{align*}(Y \times Y')_n = \bigsqcup_{i+j=n} Y_i \times Y'_j. \end{align*}$$

Consider the monoidal functor $\square \to \mathsf {Top}$ defined by assigning $2$ to the usual interval $\mathbb {I}^1$ and Equation (2.1) to the continuous maps

where $\delta _i$ and $\sigma $ are the canonical cubical coface and codegeneracy maps, respectively, and $\gamma (s,t)=\text {max}(s,t)$ .

From this functor, we obtain the usual adjunction pair formed by the geometric realization and singular complex

Notice that $\operatorname {Sing^{\square }}$ is lax monoidal

$$\begin{align*}(\mathbb{I}^n \to \mathfrak{X}) \times (\mathbb{I}^m \to \mathfrak{X}') \mapsto (\mathbb{I}^{n+m} \to \mathfrak{X} \times \mathfrak{X}'). \end{align*}$$

The functor of (normalized) cubical chains $\operatorname {N^{\square }} \colon \mathsf {cSet} \to \mathsf {Ch}$ is the unique monoidal functor defined by assigning to $\square ^1$ the usual cellular chains of the interval $\mathbb {I}^1$ . Explicitly, it is defined on any cubical set Y by first considering the chain complex $(\Bbbk [Y], \delta )$ , given on degree n by the free $\Bbbk $ -module generated by the n-cubes of Y with differential $\delta $ given on any n-cube $y \in Y_n$ by

$$\begin{align*}\delta(y) = \sum_{i=1}^n (-1)^i \big( Y(\text{id}_2^{i-1} \times \delta_1 \times \text{id}_2^{n-i})(y) - Y(\text{id}_2^{i-1} \times \delta_0 \times \text{id}_2^{n-i})(y) \big) \end{align*}$$

and then modding out by the subcomplex of degenerate cubes. When no confusion arises from doing so, we write $\operatorname {\mathrm {N}}$ instead of $\operatorname {N^{\square }}$ and refer to it simply as the functor of chains. We remark that $\operatorname {N^{\square }}$ is naturally equivalent to the composition of the geometric realization and cellular chains functors with respect to the canonical CW structure. We will denote the functor of (cubical) singular chains $\operatorname {N^{\square }} \circ \operatorname {Sing^{\square }} \colon \mathsf {Top} \to \mathsf {Ch}$ by $\operatorname {S^{\square }}$ .

Since $\operatorname {\mathrm {N}}(\square ^1)$ is a coalgebra and the category of coalgebras is monoidal, there is a unique monoidal functor $\operatorname {N^{\square }_{\!{\mathcal {A}\mathsf {s}}}}$ lifting the functor of chains

We refer to this lift as the Serre coalgebra structure on cubical chains and to the coproduct $\Delta $ as the Serre diagonal.

3.1. The based loop space

The path space functor

$$\begin{align*}P \colon \mathsf{Top} \to \mathsf{Top} \end{align*}$$

assigns to $\mathfrak {X}$ the space

$$\begin{align*}P(\mathfrak{X}) = \big\{\alpha \colon [0,r] \to \mathfrak{X} \mid \alpha\text{ continuous and } r \in [0,\infty)\big\} \end{align*}$$

equipped with the compact-open topology. For any $x,x' \in \mathfrak {X}$ , the subspace $P(\mathfrak {X};x,x')$ consists of those paths starting and ending at x and $x'$ , respectively. The points of $P(\mathfrak {X})$ are often referred to as Moore paths on $\mathfrak {X}$ . We will think of this construction as a functor on bipointed spaces in the obvious way. There is a composition structure

(3.1) $$ \begin{align} P(\mathfrak{X};x,x') \times P(\mathfrak{X};x',x") &\to P(\mathfrak{X};x,x") \nonumber\\ (\alpha, \beta) &\mapsto \alpha \cdot \beta \end{align} $$

given by concatenation of paths with addition of parameters, whose identities are constant paths with $r=0$ . We may assemble these data into a topologically enriched category $\mathcal {P}(\mathfrak {X})$ that has the points of $\mathfrak {X}$ as objects, the spaces $P(\mathfrak {X};x,x')$ as morphisms from x to $x'$ , concatenation of paths as composition and constant paths with $r=0$ as identity morphisms.

Denote by $\mathsf {Top}^\ast $ the category of pointed topological spaces, which we think of as a full subcategory of that of bipointed spaces in the obvious way. The based loop space functor

$$\begin{align*}{\Omega} \colon \mathsf{Top}^\ast \to \mathsf{Mon}_{\mathsf{Top}} \end{align*}$$

associates to a pointed space $(\mathfrak {X},x)$ the space

$$\begin{align*}{\Omega}_x \mathfrak{X} = P(\mathfrak{X};x,x) \end{align*}$$

of loops in $\mathfrak {X}$ based at x with the monoid structure induced from the composition structure (3.1).

Since $\operatorname {Sing^{\square }}$ is lax monoidal, $\operatorname {Sing^{\square }}({\Omega }_x \mathfrak {X})$ is a monoid in $\mathsf {cSet}$ , and, since $\operatorname {N^{\square }}$ is monoidal, $\operatorname {S^{\square }}(\mathfrak {X},x)$ is a monoid in $\mathsf {Ch}$ .

3.2. The cobar construction

We now describe an algebraic analogue of the based loop space introduced by Adams [Reference Adams1]. The cobar construction is the functor

$$\begin{align*}\operatorname{\mathbf{\Omega}} \colon \mathsf{coAlg}^\ast \to \mathsf{Mon}_{\mathsf{Ch}} \end{align*}$$

defined on objects as follows. Let $(C, \Delta , \varepsilon , \nu )$ be a coaugmented coalgebra. Denote by $\overline {C}$ the cokernel of the coaugmentation $\nu \colon \Bbbk \to C$ , and recall that is the suspension functor. The cobar construction $\operatorname {\mathbf {\Omega }} C$ of this coaugmented coalgebra is the graded module

regarded as a monoid in $\mathsf {Ch}$ with free associative product given by concatenation, unit map by the obvious inclusion, and differential constructed by extending the linear map

as a derivation using the freeness of the underlying graded monoid. On morphisms, the functor $\operatorname {\mathbf {\Omega }}$ is defined using the functoriality of the free graded monoid construction. For any $x_1, \dots , x_k \in \overline {C}$ , we denote

3.3. Adams’s map

Adams made precise the sense in which the cobar functor may be understood as an algebraic analogue of the based loop space functor. This was achieved by constructing a natural monoidal chain map

(3.2) $$ \begin{align} \theta_{\mathfrak{X}} \colon \operatorname{\mathbf{\Omega}} \operatorname{S^{\triangle}_{\!{\mathcal{A}\mathsf{s}}}}(\mathfrak{X},x) \to \operatorname{S^{\square}}({\Omega}_x \mathfrak{X}), \end{align} $$

for any pointed topological space $(\mathfrak {X},x)$ and showing it to be a quasi-isomorphism for simply connected spaces in [Reference Adams1], a hypothesis removed in [Reference Rivera and Zeinalian31] and, using a different argument, in [Reference Rivera and Saneblidze30]. Here, $\operatorname {S^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(\mathfrak {X},x)$ denotes the coalgebra $\operatorname {N^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(\operatorname {Sing^{\triangle }}(\mathfrak {X},x))$ (using the notation introduced in 2.3) of (pointed) normalized singular chains.

We now describe the construction of Adams’ map, whose combinatorial essence is the observation that the set of ascending chains in $0 < \dots < n$ containing $0$ and n, with the inclusion order, is isomorphic to the $(n-1)^{\mathrm {th}}$ cubical lattice. To define Adams’ map one uses a collection of continuous maps

$$\begin{align*}\{\theta_n \colon \mathbb{I}^{n-1} \to P(\mathbb{\Delta}^n;v_0,v_n)\}_{n\in\mathbb{N}}, \end{align*}$$

satisfying for each j the following conditions:

1. 1. $\theta _1(0)\colon [0,\sqrt {2}] \to \mathbb {\Delta }^1$ is the path $\theta _1(0)(s) = v_0 + \frac {s}{\sqrt {2}}(v_1-v_0)$ ,

2. 2. $\theta _n \circ \delta _0^j = P(\delta ^j) \circ \theta _{n-1}$ and

3. 3. $\theta _n \circ \delta _1^j = \big (P(\delta ^{f^n_j}) \circ \theta _j\big ) \cdot \big (P(\delta ^{\ell ^n_{n-j}}) \circ \theta _{n-j}\big )$ ,

where, for $\epsilon \in \{0,1\}$ ,

$$\begin{align*}\delta_\epsilon^j = \mathsf{id}_{\mathbb{I}^{j-1}} \times \delta_\epsilon \times \mathsf{id}_{\mathbb{I}^{n-j-2}} \,\colon\ \mathbb{I}^{n-2} \to \mathbb{I}^{n-1}, \end{align*}$$

and

$$ \begin{align*} \delta^{f^n_j} = \delta^n \circ \dotsb \circ \delta^{n-j+1} &\,\colon\ \mathbb{\Delta}^j \to \mathbb{\Delta}^n, \\ \delta^{\ell^n_{n-j}} = \delta^{n-j-1} \circ \dotsb \circ \delta^{0} &\, \colon\ \mathbb{\Delta}^{n-j} \to \mathbb{\Delta}^n, \end{align*} $$

are, respectively, the inclusions into the first and last faces of $\mathbb {\Delta }^n$ .

Adams showed the existence of a (nonunique) collection of such maps by induction, using the contractibility of $P(\mathbb {\Delta }^n;v_0,v_n)$ . He then defined $\theta _{\mathfrak {X}}$ as follows. For any singular $1$ -simplex $\sigma \in \operatorname {S^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(\mathfrak {X},x)$ , let

$$\begin{align*}\theta_{\mathfrak{X}}[\sigma] = P(\sigma) \circ \theta_1 - c_x, \end{align*}$$

where $(c_x \colon \mathbb {I}^0 \to {\Omega }_x \mathfrak {X}) \in \operatorname {S^{\square }}({\Omega }_x \mathfrak {X})$ is the singular $0$ -cube determined by the constant loop (with $r=0$ ) at $x \in \mathfrak {X}$ . For any singular n-simplex $\sigma \in \operatorname {S^{\triangle }}(\mathfrak {X},x)$ with $n>1$ , let

$$\begin{align*}\theta_{\mathfrak{X}}[\sigma] = P(\sigma) \circ \theta_n. \end{align*}$$

Since the underlying graded monoid structure of $\operatorname {\mathbf {\Omega }} \operatorname {S^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(\mathfrak {X},x)$ is free, we may extend the above to a monoidal chain map $\theta _{\mathfrak {X}} \colon \operatorname {\mathbf {\Omega }} \operatorname {S^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(\mathfrak {X},x) \to \operatorname {S^{\square }}({\Omega }_x \mathfrak {X})$ . Conditions (1), (2) and (3) then imply that $\theta _{\mathfrak {X}}$ is a chain map.

3.4. An explicit choice

We now construct an explicit collection of maps

$$\begin{align*}\{\theta_n \colon \mathbb{I}^{n-1} \to P(\mathbb{\Delta}^n;v_0,v_n)\}_{n\in\mathbb{N}} \end{align*}$$

satisfying the conditions in §3.3.

Given $v,w \in \mathbb {R}^{n+1}$ denote by

$$\begin{align*}\gamma(v,w) \colon \big[0, \left\lvert {w-v} \right\rvert \big] \to \mathbb{R}^{n+1} \end{align*}$$

the straight line path from v to w parameterized by arc length, that is,

$$\begin{align*}\gamma(v,w)(s) = v + \frac{s}{|w-v|}(w-v). \end{align*}$$

For any $\mathbf {t}=(t_1, \dots , t_{n-1}) \in \mathbb {I}^{n-1}$ , we define $p_1(\mathbf {t}), \dots , p_{n-1}(\mathbf {t})$ in $\mathbb {\Delta }^n$ inductively by

$$ \begin{align*} p_1(\mathbf{t}) &= v_0+ t_1(v_1-v_0), \\ p_j(\mathbf{t}) &= p_{j-1}(\mathbf{t}) + t_j(v_j-p_{j-1}(\mathbf{t})). \end{align*} $$

We may now define

$$\begin{align*}\theta_n(\mathbf{t}) \colon [0, r_{\mathbf{t}}] \to \mathbb{\Delta}^n, \end{align*}$$

where

$$\begin{align*}r_{\mathbf{t}} = \left\lvert {p_1(\mathbf{t})-v_0} \right\rvert + \left\lvert {p_2(\mathbf{t})-p_1(\mathbf{t})} \right\rvert + \dots + \left\lvert {v_n-p_{n-1}(\mathbf{t})} \right\rvert , \end{align*}$$

as the piecewise linear path given by concatenating the straight line segments connecting the ordered sequence of points $v_0, p_1(\mathbf {t}), \dots , p_n(\mathbf {t}), v_n$ , that is,

$$\begin{align*}\theta_n(\mathbf{t}) = \gamma(v_0,p_1(\mathbf{t})) \cdot \gamma(p_1(\mathbf{t}), p_2(\mathbf{t})) \cdot\ \dotsb\ \cdot \gamma(p_{n-1}(\mathbf{t}),v_n). \end{align*}$$

Please consult Figure 1 for an example illustrating this construction.

Figure 1 In red, the path $\theta _3(\mathbf {t}) \in P(\mathbb {\Delta }^3, v_0, v_3)$ associated to a $\mathbf {t} \in \mathbb {I}^2$ .

Figure 2 The faces of the $2$ -cube in blue, labeled by a red element in their corresponding family of paths.

A straightforward computation proves that the conditions in §3.3 are satisfied by this collection. A diagrammatical verification in low dimensions is provided in Figure 2. We can extend this collection $\{\theta _n \colon \mathbb {I}^{n-1} \to P(\mathbb {\Delta }^n;v_0,v_n)\}$ to one of the form

$$\begin{align*}\{\theta_{(n_1, \dots, n_k)} \colon \mathbb{I}^{n_1 + \dots + n_k-k} \to P(\mathbb{\Delta}^{n_1} \vee \cdots \vee \mathbb{\Delta}^{n_k})\}, \end{align*}$$

where the topological necklace $\mathbb {\Delta }^{n_1} \vee \cdots \vee \mathbb {\Delta }^{n_k}$ is obtained by identifying the last vertex of $\mathbb {\Delta }^{n_i}$ with the first vertex of $\mathbb {\Delta }^{n_{i+1}}$ for each $i = 1, \dots , k-1$ assuming each $n_i> 0$ . We do so by setting

$$ \begin{align*} \theta_{(n_1, \ldots, n_k)}(t_1, \ldots, t_{n_1+ \cdots + n_k-k})& \\ &\quad = \theta_{n_1}(t_1, \ldots, t_{n_1-1}) \cdot\ \dots\ \cdot \theta_{n_k}(t_{n_1+ \cdots n_{k-1}-k},\ldots,t_{n_1 + \cdots +n_k-k}). \end{align*} $$

Thus, we can think of the topological necklace $\mathbb {\Delta }^{n_1} \vee \cdots \vee \mathbb {\Delta }^{n_k}$ as a space parameterizing a $(n_1+ \cdots + n_k-k)$ -dimensional family of paths between the first and last vertices. Please consult Figure 3 for an example illustrating this construction.

Figure 3 In red, the path $\theta _{(2,1,3,2)}(\mathbf {t})$ in $P(\mathbb {\Delta }^2 \vee \mathbb {\Delta }^1 \vee \mathbb {\Delta }^3 \vee \mathbb {\Delta }^2)$ associated to an element $\mathbf {t}$ in $\mathbb {I}^{4}$ .

3.5. Necklaces

We now present a categorical viewpoint on the necklaces encountered in the previous subsection and their intimate relationship with Adams’ cobar construction. For any pointed space $(\mathfrak {X},x)$ , the underlying graded $\Bbbk $ -module of $\operatorname {\mathbf {\Omega }} \operatorname {S^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(\mathfrak {X},x)$ can be described as the graded $\Bbbk $ -module freely generated by finite ordered sequences

$$\begin{align*}T = (\sigma_1, \dots, \sigma_k) \end{align*}$$

of simplices $\sigma _i \in \operatorname {Sing^{\triangle }}(\mathfrak {X},x)$ , with degree $ \left \lvert {T} \right \rvert = \left \lvert {\sigma _1} \right \rvert +\dots + \left \lvert {\sigma _k} \right \rvert - k$ , modulo the sub $\Bbbk $ -module generated by those sequences with at least one degenerate simplex. The differential of $\operatorname {\mathbf {\Omega }} \operatorname {S^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(\mathfrak {X},x)$ on a generator T of degree n can be expressed as a signed signed sum of all generators $T'$ of degree $n-1$ . The simplices in the ordered sequence corresponding to each of these $T'$ are all faces of simplices in the ordered sequence corresponding to T. Note that two types of generators $T'$ appear in this differential: Those that have the same length as T and those that have exactly one more simplex. The first type corresponds to terms in the differential of $\operatorname {S^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(\mathfrak {X},x)$ and the second type to terms in the Alexander–Whitney coproduct. Furthermore, the monoid structure of $\operatorname {\mathbf {\Omega }} \operatorname {S^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(\mathfrak {X},x)$ is induced by simply concatenating these ordered sequences of simplices. This perspective suggests that $\operatorname {\mathbf {\Omega }} \operatorname {S^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(\mathfrak {X},x)$ may be obtained by applying a normalized chains functor to certain cellular monoid, naturally associated to $(\mathfrak {X},x)$ , with cells labeled by finite ordered sequences of simplices in $\operatorname {Sing^{\triangle }}(\mathfrak {X},x)$ . A geometric construction reflecting this idea was described in [Reference Baues3]. We instead take a categorical approach and make this discussion precise through the framework of necklaces and necklical sets as we now discuss.

Consider the subcategory $\triangle _{*,*}$ of the simplex category $\triangle $ with the same objects and morphisms given by functors $f \colon [n] \to [m]$ satisfying $f(0) = 0$ and $f(n) = m$ . It is a strict monoidal category when equipped with the monoidal structure $[n] \otimes [m] = [n+m]$ , thought of as identifying the elements $n \in [n]$ and $0 \in [m]$ , and unit given by $[0]$ . Heuristically, we may think of $\triangle _{*,*}$ as a category of models for cells parameterizing families of paths with fixed endpoints inside a simplex.

The necklace category $\mathsf {Nec}$ is obtained from $\triangle _{*,*}$ as follows. Thinking of $\triangle _{*,*}$ as a monoid in $\mathsf {Cat}$ , we first apply the bar construction to it and produce a simplicial object in $\mathsf {Mon}_{\mathsf {Cat}}$ which, after realization, defines the strict monoidal category $\mathsf {Nec}$ . We denote the monoidal structure by

$$\begin{align*}\vee \colon \mathsf{Nec} \times \mathsf{Nec} \to \mathsf{Nec}. \end{align*}$$

We describe $(\mathsf {Nec}, \vee )$ in more explicit terms. The objects of $\mathsf {Nec}$ , called necklaces, are freely generated by the objects of $\triangle _{*,*}$ through the monoidal structure $\vee $ . Namely, the set of objects of $\mathsf {Nec}$ is the set of monomials

$$\begin{align*}\big\{ [n_1] \vee \dots \vee[n_k] \mid n_i, k \in \mathbb{N}_{>0} \big\} \end{align*}$$

together with $[0]$ serving as the monoidal unit.

The morphisms of $\mathsf {Nec}$ are generated through the monoidal structure by the following four types of morphisms for all $n \in \mathbb {N}_{>0}$

1. 1. $\partial ^j \colon [n-1] \to [n]$ for $j = 1, \dots , n-1$ ,

2. 2. $\Delta _{[j], [n-j]} \colon [j] \vee [n-j] \to [n]$ for $j = 1, \dots , n-1$ ,

3. 3. $\xi ^j \colon [n+1] \to [n]$ for $j = 0, \dots , n$ and $n>0$ and

4. 4. $\xi ^0 \colon [1] \to [0]$ .

We may identify $\mathsf {Nec}$ with a full subcategory of the category of double pointed simplicial sets $\mathsf {sSet}_{*,*}$ as follows. Consider the functor

$$\begin{align*}\mathcal{S} \colon \mathsf{Nec} \to \mathsf{sSet}_{*,*} \end{align*}$$

induced by sending any necklace $T = [n_1] \vee \dots \vee [n_k]$ to the simplicial set

$$\begin{align*}\mathcal{S}(T) = \triangle^{n_1} \vee \dots \vee \triangle^{n_k}, \end{align*}$$

where the wedge symbol now means we identify the last vertex of $\triangle ^{n_i}$ with the first vertex of $\triangle ^{n_{i+1}}$ for $i = 1, \dots , k-1$ and the two base points are given by the first vertex of $\triangle ^{n_1}$ and the last vertex of $\triangle ^{n_k}$ . Then $\mathcal {S}$ is a fully faithful functor, so $\mathsf {Nec}$ may be identified with the full subcategory of $\mathsf {sSet}_{*,*}$ having as objects those double pointed simplicial sets of the form $\triangle ^{n_1} \vee \dots \vee \triangle ^{n_k}$ . The dimension of a necklace $T = [n_1] \vee \dots \vee [n_k]$ is defined by $\dim (T) = n_1 + \dots + n_k-k$ . Note there is a canonical homeomorphism

$$\begin{align*}\mathbb{\Delta}^{n_1} \vee \cdots \vee \mathbb{\Delta}^{n_k} \cong | \triangle^{n_1} \vee \cdots \vee \triangle^{n_k}|.\end{align*}$$

The category of necklical sets $\mathsf {Fun}(\mathsf {Nec}^{\mathrm {op}}, \mathsf {Set})$ becomes a (nonsymmetric) monoidal category with monoidal structure induced from $(\mathsf {Nec}, \vee )$ . We denote the monoidal category of necklical sets by $\mathsf {nSet}$ .

3.6. From necklaces to cubes

In §3.4, we described an explicit way of decomposing a topological necklace $\mathbb {\Delta }^{n_1} \vee \cdots \vee \mathbb {\Delta }^{n_k}$ into a family of paths connecting the first and last vertices parameterized by an $(n_1 + \cdots + n_k-k)$ -dimensional cube. This construction satisfies conditions (1), (2) and (3) in §3.3. In particular, conditions (2) and (3) may be interpreted as saying that each face in the codimension $1$ boundary of such a cube of paths is in one-to-one correspondence with codimension $1$ ‘subnecklaces’ inside $\mathbb {\Delta }^{n_1} \vee \cdots \vee \mathbb {\Delta }^{n_k}$ connecting the first and last vertices. Furthermore, subnecklaces in $\mathbb {\Delta }^{n_1} \vee \cdots \vee \mathbb {\Delta }^{n_k}$ are in one-to-one correspondence with the poset

$$\begin{align*}\{J \subseteq \{0,\dots,n_1+\cdots+n_k-k\} \mid 0,n_1+\cdots+n_k-k \in J\} \end{align*}$$

ordered by inclusion, which has a canonical cubical structure. We build upon this observation to describe a functorial relation between necklaces and cubes. This will be used to explain how Adams’ map is induced by a deeper categorical construction.

We begin by defining a monoidal functor

$$\begin{align*}\mathcal{P} \colon \mathsf{Nec} \to \square \end{align*}$$

as follows. First, define $\mathcal {P}[0] = 2^0$ . On any other necklace $T \in \mathsf {Nec}$ , define $\mathcal {P}(T) = 2^{\dim (T)}$ . In order to define $\mathcal {P}$ on morphisms, it is sufficient to consider the following cases.

1. 1. For any coface map $\partial ^j \colon [n] \to [n+1]$ such that $0< j<{n+1}$ , define $\mathcal {P}(\partial ^j) \colon 2^{n-1}\to 2^{n}$ to be the cubical coface functor $\mathcal {P}(f)= \delta _0^{j}$ .

2. 2. For any $\Delta _{[j], [n+1-j]} \colon [j] \vee [n+1-j] \to [n+1]$ such that $0<j<n+1$ , define

   $$\begin{align*}\mathcal{P}(\Delta_{[j], [n+1-j]}) \colon 2^{n-1}\to 2^{n} \end{align*}$$
   to be the cubical coface functor $\mathcal {P}(f)=\delta _1^{j}$ .

3. 3. We now consider codegeneracy maps of the form $\xi ^j \colon [n+1] \to [n]$ for $n>0$ . If $j=0$ or $j=n$ , define $\mathcal {P}(f) \colon 2^n \to 2^{n-1}$ to be the cubical codegeneracy functor $\mathcal {P}(s^j)= \varepsilon ^{j}$ . If $0<j<n$ , define $\mathcal {P}(s^j) \colon 2^n \to 2^{n-1}$ to be the cubical coconnection functor $\gamma ^{j}$ .

4. 4. For $\xi ^0 \colon [1] \to [0]$ define $\mathcal {P}(\xi ^0) \colon 2^0 \to 2^0$ to be the identity functor.

The functor $\mathcal {P} \colon \mathsf {Nec} \to \square $ induces an adjunction between $\mathsf {cSet}$ and $\mathsf {nSet}$ with right and left adjoint functors given, respectively, by

$$\begin{align*}\mathcal{P}^\ast \colon \mathsf{cSet} \to \mathsf{nSet}, \qquad \text{and} \qquad \mathcal{P}_{!} \colon \mathsf{nSet} \to \mathsf{cSet}. \end{align*}$$

Explicitly, for a cubical set $Y \colon \square ^{\mathrm {op}} \to \mathsf {Set}$ ,

$$\begin{align*}\mathcal{P}^\ast(Y) = Y \circ \mathcal{P}^{\mathrm{op}}, \end{align*}$$

and for a necklical set $K \colon \mathsf {Nec}^{\mathrm {op}} \to \mathsf {Set}$ ,

$$\begin{align*}\mathcal{P}_{!}(K) \ = \operatorname*{\mathrm{colim}}_{\operatorname{\mathcal{Y}}(T) \to K} \mathcal{P}(T) \ \cong \operatorname*{\mathrm{colim}}_{\operatorname{\mathcal{Y}}(T) \to K} \square^{\dim(T)}, \end{align*}$$

where $\operatorname {\mathcal {Y}} \colon \mathsf {Nec} \to \mathsf {nSet}$ is the Yoneda embedding. Since $\mathcal {P}$ is a monoidal functor, $\mathcal {P}_{!} \colon \mathsf {nSet} \to \mathsf {cSet}$ is monoidal as well.

3.7. Cubical cobar construction

Using the framework of necklical sets, we may reinterpret Baues’ geometric cobar construction [Reference Baues3] as a functor

$$\begin{align*}{\mathbb{\Omega}^{\mathrm{nec}}} \colon \mathsf{sSet}^0 \to \mathsf{Mon}_{\mathsf{nSet}}, \end{align*}$$

which we now define.

For any reduced simplicial set X, we define a necklical set ${\mathbb {\Omega }^{\mathrm {nec}}}(X) \colon \mathsf {Nec}^{\mathrm {op}} \to \mathsf {Set}$ having as necklical cells all necklaces inside X; namely,

$$\begin{align*}{\mathbb{\Omega}^{\mathrm{nec}}}(X) \, = \! \operatorname*{\mathrm{colim}}_{\mathcal{S}(T) \to X} \operatorname{\mathcal{Y}}(T). \end{align*}$$

The monoidal structure $\vee \colon \mathsf {Nec} \times \mathsf {Nec} \to \mathsf {Nec}$ given by concatenation of necklaces induces a natural product

$$\begin{align*}{\mathbb{\Omega}^{\mathrm{nec}}}(X) \otimes {\mathbb{\Omega}^{\mathrm{nec}}}(X) \to {\mathbb{\Omega}^{\mathrm{nec}}}(X) \end{align*}$$

making ${\mathbb {\Omega }^{\mathrm {nec}}}(X)$ into a monoid in $\mathsf {nSet}$ .

We may now define the cubical cobar construction

$$\begin{align*}\operatorname{\mathbb{\Omega}} \colon \mathsf{sSet}^0 \to \mathsf{Mon}_{\mathsf{cSet}} \end{align*}$$

as the composition

$$\begin{align*}\operatorname{\mathbb{\Omega}} = \mathcal{P}_! \, \circ {\mathbb{\Omega}^{\mathrm{nec}}}. \end{align*}$$

Since $\mathcal {P}_!$ is monoidal, $\operatorname {\mathbb {\Omega }}(X)$ is a monoid in $\mathsf {cSet}$ .

3.8. Relation to the cobar construction

We now relate the cubical cobar functor $\operatorname {\mathbb {\Omega }} \colon \mathsf {sSet}^0 \to \mathsf {Mon}_{\mathsf {cSet}}$ to the cobar construction $\operatorname {\mathbf {\Omega }} \colon \mathsf {coAlg}^\ast \to \mathsf {Mon}_{\mathsf {Ch}}$ (§3.2).

Theorem 3.3. There is a natural isomorphisms of functors

$$\begin{align*}\operatorname{N^{\square}} \operatorname{\mathbb{\Omega}} \cong \operatorname{\mathbf{\Omega}} \operatorname{N^{\triangle}_{\!{\mathcal{A}\mathsf{s}}}} \colon \mathsf{sSet}^0 \to \mathsf{Mon}_{\mathsf{Ch}}. \end{align*}$$

Proof. Denote by $\iota _n \in (\square ^n)_n$ the top dimensional nondegenerate element of the standard n-cube $\square ^n$ . Note that for a reduced simplicial set X, we may represent any nondegenerate n-cube $\alpha \in (\mathcal {P}_!({\mathbb {\Omega }^{\mathrm {nec}}}(X)))_n$ as a pair $\alpha = [\sigma \colon \operatorname {\mathcal {Y}}(T) \to X, \iota _n]$ for some $T = [n_1] \vee \dots \vee [n_k] \in \mathsf {Nec}$ with $\dim (T) = n_1 + \dots + n_k - k = n$ .

To define a monoidal chain map

$$\begin{align*}\varphi_X \colon \operatorname{N^{\square}}(\mathcal{P}_!({\mathbb{\Omega}^{\mathrm{nec}}}(X))) \xrightarrow{\cong} \operatorname{\mathbf{\Omega}} \operatorname{N^{\triangle}_{\!{\mathcal{A}\mathsf{s}}}}(X), \end{align*}$$

it suffices to define it on any generator of the form $\alpha =[\sigma \colon \triangle ^{n+1} \to X, \iota _{n}]$ , that is, when T is of the form $T = [n+1]$ , for some $n \geq 0$ . If $n = 0$ , let $\varphi _X(\alpha )= [\overline {\sigma }] + 1_\Bbbk $ , where denotes the (length $1$ ) generator in the cobar construction of $\operatorname {N^{\triangle }}(X)$ determined by $\sigma \in X_{n+1}$ and $1_\Bbbk $ denotes the unit of the underlying ring $\Bbbk $ . If $n> 0$ , we let $\varphi _X(\alpha )=[\overline {\sigma }]$ . A straightforward computation yields that this gives rise to a well-defined isomorphism of algebras, which is compatible with the differentials, and natural with respect to maps of simplicial sets.

A similar result to Theorem 3.3 was observed in the case of $1$ -reduced simplicial sets in [5, Section 3.5].

3.9. Factorization of Adams’s map

Adams’s comparison map can be factored as a composition

(3.3) $$ \begin{align} \theta_{\mathfrak{X}} \colon \operatorname{\mathbf{\Omega}} \operatorname{S^{\triangle}_{\!{\mathcal{A}\mathsf{s}}}}(\mathfrak{X},x) \xrightarrow{\cong} \operatorname{N^{\square}} \operatorname{\mathbb{\Omega}}(\operatorname{Sing^{\triangle}}(\mathfrak{X},x)) \xrightarrow{\operatorname{N^{\square}}(\Theta)} \operatorname{S^{\square}}({\Omega}_x \mathfrak{X}). \end{align} $$

The first map is the monoidal isomorphism induced by Theorem 3.3. The second map is given by applying chains $\operatorname {N^{\square }} \colon \mathsf {Mon}_{\mathsf {cSet}} \to \mathsf {Mon}_{\mathsf {Ch}}$ to the map of monoidal cubical sets

$$\begin{align*}\Theta \colon \operatorname{\mathbb{\Omega}}(\operatorname{Sing^{\triangle}}(\mathfrak{X},x)) \to \operatorname{Sing^{\square}}({\Omega}_x \mathfrak{X}) \end{align*}$$

determined through the monoid structure by sending an n-simplex $(\sigma \colon \mathbb {\Delta }^n \to \mathfrak {X})$ to the singular $(n-1)$ -cube

$$\begin{align*}P(\sigma) \circ \theta_n \colon \mathbb{I}^{n-1} \to {\Omega}_x \mathfrak{X}, \end{align*}$$

where the maps $\theta _n \colon \mathbb {I}^{n-1} \to P(\mathbb {\Delta }^n;0,n)$ are discussed in §3.3.

3.10. A monoidal coalgebra structure on the cobar construction

We follow [Reference Baues4] to construct a coalgebra structure on $\operatorname {\mathbf {\Omega }} \operatorname {S^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(\mathfrak {X},x)$ , compatible with both the differential and monoid structure, such that Adams’s map becomes a map of monoids in the category of coalgebras.

Recall the Serre coalgebra lift $\operatorname {N^{\square }_{\!{\mathcal {A}\mathsf {s}}}} \colon \mathsf {cSet} \to \mathsf {coAlg}$ of $\operatorname {N^{\square }} \colon \mathsf {cSet} \to \mathsf {Ch}$ , the unique monoidal functor defined by the coalgebra structure on $\operatorname {\mathrm {N}}(\square ^1)$ . Since $\operatorname {N^{\square }_{\!{\mathcal {A}\mathsf {s}}}}$ is monoidal and $\operatorname {\mathbb {\Omega }}(\operatorname {Sing^{\triangle }}(\mathfrak {X},x))$ is a monoid, $\operatorname {N^{\square }_{\!{\mathcal {A}\mathsf {s}}}}(\operatorname {\mathbb {\Omega }}(\operatorname {Sing^{\triangle }}(\mathfrak {X},x)))$ is a monoid in $\mathsf {coAlg}$ . Similarly, the lift $\operatorname {N^{\square }_{\!{\mathcal {A}\mathsf {s}}}}$ equips $\operatorname {Sing^{\square }}({\Omega }_x \mathfrak {X})$ with a natural monoidal coalgebra structure as well. Consequently, the isomorphism in (3.3) endows $\operatorname {\mathbf {\Omega }} \operatorname {S^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}(\mathfrak {X},x)$ with a natural monoidal coalgebra structure making $\theta _{\mathfrak {X}}$ into natural map of monoidal coalgebras.

4.1. ${E_\infty }$ -operads

Recall that operads control algebraic structures with either one input and multiple outputs or vice versa. In this article, we work with dg operads, that is, operads in the monoidal category $(\mathsf {Ch}, \otimes , \mathbb{1})$ ; we refer to [Reference Loday and Vallette22] for more details.

For example, coalgebras, as defined in §2.1, are controlled by the operad ${\mathcal {A}\mathsf {s}}$ generated by two elements in degree $0$

modulo the relations

Let $\sigma \in \mathbb {S}_2$ be the nonidentity transposition. The operad $\mathcal {C}{om}$ controlling cocommutative coalgebras is obtained by adding the relation

to this presentation. We are interested in compatibly resolving the (trivial) symmetric group actions on $\mathcal {C}{om}$ associated to the permutation of factors. An $E_\infty $ -operad is an operad $\mathcal {O}$ quasi-isomorphic to $\mathcal {C}{om}$ for which the action of $\mathbb {S}_r$ on $\mathcal {O}(r)$ is free for each $r \in \mathbb {N}$ . An $E_\infty $ -coalgebra structure on a chain complex C is an operad morphism

$$\begin{align*}\mathcal{O} \to \mathrm{coEnd}(C) = \{\mathrm{Hom}(C, C^{\otimes r})\}_{r \in \mathbb{N}}. \end{align*}$$

4.2. A finitely presented ${E_\infty }$ -prop

A prop is an object controlling algebraic structures with multiple inputs and outputs. We refer to [Reference Markl24] for a detailed exposition. We recall the following construction from [Reference Medina-Mardones27]. The prop $\mathcal {M}$ is generated by adding to the presentation of ${\mathcal {A}\mathsf {s}}$ a generator and a relation. More specifically, a generator in degree $1$ with boundary

and the relation

The importance of this construction is that the operad ${\operatorname {\mathrm {U}}(\mathcal {M})} = \{\mathcal {M}(1,r)\}_{r\in \mathbb {N}}$ obtained by restriction of structure is an $E_\infty $ -operad.

4.3. ${\mathcal {M}}$ -bialgebras

An $\mathcal {M}$ -bialgebra structure on C is a prop morphism

$$\begin{align*}\mathcal{M} \to \mathrm{biEnd}(C) = \{\mathrm{Hom}(C^{\otimes s}, C^{\otimes r})\}_{r,s\in\mathbb{N}}. \end{align*}$$

More explicitly, is a coalgebra $(C, \Delta , \operatorname {\mathrm {\varepsilon }})$ together with a degree 1 product satisfying for any $a,b \in C$ that:

(4.1) $$ \begin{align} \operatorname{\mathrm{\varepsilon}}(a \ast b) =\ & 0, \end{align} $$

(4.2) $$ \begin{align} \operatorname{\mathrm{\partial}} (a \ast b) =\ & \operatorname{\mathrm{\partial}} a \ast b - (-1)^{a} a \ast \operatorname{\mathrm{\partial}} b + \operatorname{\mathrm{\varepsilon}}(a) b - (-1)^{a} a \operatorname{\mathrm{\varepsilon}}(b). \end{align} $$

The signs appearing in identity (4.2) are a result of the Koszul sign convention.

Any $\mathcal {M}$ -bialgebra structure induces an $E_\infty $ -coalgebra structure. More explicitly, let $(C, \Delta , \operatorname {\mathrm {\varepsilon }}, \ast )$ be an $\mathcal {M}$ -bialgebra. The collection of all maps $\{C \to C^{\otimes r}\}_{r \in \mathbb {N}}$ generated by $\Delta $ , $\operatorname {\mathrm {\varepsilon }}$ and $\ast $ makes C into an $E_\infty $ -coalgebra, specifically into an ${\operatorname {\mathrm {U}}(\mathcal {M})}$ -coalgebra.

4.4. ${E_\infty }$ -structure on simplicial chains

We recall the construction of an $E_\infty $ -extension of the Alexander–Whitney coalgebra structure on simplicial chains introduced in [Reference Medina-Mardones27]. Let us start by considering the representable simplicial sets. The coalgebra $\operatorname {\mathrm {N}}(\triangle ^n)$ can be made into a natural $\mathcal {M}$ -bialgebra considering an algebraic version of the join product defined by

$$ \begin{align*} \left[v_0, \dots, v_p \right] \ast \left[v_{p+1}, \dots, v_q\right] = \begin{cases} (-1)^{p} \operatorname{\mathrm{sign}}(\pi) \left[v_{\pi(0)}, \dots, v_{\pi(q)}\right] & \text{ if } v_i \neq v_j \text{ for } i \neq j, \\ \hfil 0 & \text{ if not}, \end{cases} \end{align*} $$

where $\pi $ is the permutation that orders the vertices. A Kan extension of the induced ${\operatorname {\mathrm {U}}(\mathcal {M})}$ -coalgebra structure on $\operatorname {\mathrm {N}}(\triangle ^n)$ defines a lift $\operatorname {N^{\triangle }_{{\operatorname {\mathrm {U}}(\mathcal {M})}}}$ of the Alexander–Whitney coalgebra to the category of $E_\infty $ -coalgebras defined by ${\operatorname {\mathrm {U}}(\mathcal {M})}$ . That is to say, these functors fit in a commutative diagram

where the vertical arrow is the obvious forgetful functor.

We remark that this $E_\infty $ -structure generalizes those previously introduced in [Reference Berger and Fresse6,Reference McClure and Smith25] in the sense that any cooperation $\operatorname {\mathrm {N}} \to \operatorname {\mathrm {N}}^{\otimes r}$ arising from the action of the Barratt-Eccles or surjection operad can be expressed as a cooperation arising from the action of ${\operatorname {\mathrm {U}}(\mathcal {M})}$ .

4.5. ${E_\infty }$ -structure on cubical chains

We recall the construction of an $E_\infty $ -extension of the Serre coalgebra structure on cubical chains introduced in [Reference Kaufmann and Medina-Mardones21]. Let us first consider representable cubical sets. The coalgebra structure on $\operatorname {\mathrm {N}}(\square ^n)$ can be made into a natural $\mathcal {M}$ -bialgebra considering a product defined using the following notation. For a basis element $x = x_1 \otimes \dotsb \otimes x_n$ of $\operatorname {\mathrm {N}}(\square ^n)$ and an integer $\ell \in \{1,\dots ,n\}$ , we write

$$ \begin{align*} x_{<\ell} & = x_1 \otimes \dotsb \otimes x_{\ell-1}, \\ x_{>\ell} & = x_{\ell+1} \otimes \dotsb \otimes x_n, \end{align*} $$

with the convention $x_{<1} = x_{>n} = 1 \in \mathbb {Z}$ . Then, for two such basis elements x and y we set

(4.3) $$ \begin{align} (x_1 \otimes \dotsb \otimes x_n) \ast (y_1 \otimes \dotsb \otimes y_n) = \sum_{i=1}^n x_{<i}\, \epsilon(y_{<i}) \otimes (x_i \ast y_i) \otimes \epsilon(x_{>i}) \, y_{>i}, \end{align} $$

where the only nonzero values of $x_i \ast y_i$ are

$$\begin{align*}[0] \ast [1] = [0, 1], \qquad [1] \ast [0] = -[0, 1]. \end{align*}$$

A Kan extension of the induced ${\operatorname {\mathrm {U}}(\mathcal {M})}$ -coalgebra structure on $\operatorname {\mathrm {N}}(\square ^n)$ defines a lift $\operatorname {N^{\square }_{{\operatorname {\mathrm {U}}(\mathcal {M})}}}$ of the Serre coalgebra structure to the category of $E_\infty $ -coalgebras defined by ${\operatorname {\mathrm {U}}(\mathcal {M})}$ . That is, a commutative diagram

(4.4)

4.6. Monoidal structure

In this subsection, we describe an extension of the tensor product of coalgebras to $\mathcal {M}$ -bialgebras and ${\operatorname {\mathrm {U}}(\mathcal {M})}$ -coalgebras.

Lemma 4.1. Let C and $C'$ be $\mathcal {M}$ -bialgebras. The coalgebra $C \otimes C'$ is a natural $\mathcal {M}$ -bialgebra with

(4.5) $$ \begin{align} (a \otimes b) \ast (c \otimes d) = a \operatorname{\mathrm{\varepsilon}}(c) \otimes (b \ast d) + (a \ast c) \otimes \operatorname{\mathrm{\varepsilon}}(b) d, \end{align} $$

for any $a,c \in C$ and $b,d \in C'$ .

Proof. We verify identity (4.1) using that $\operatorname {\mathrm {\varepsilon }}(b \ast d) = \operatorname {\mathrm {\varepsilon }}(a \ast c) = 0$ ,

$$ \begin{align*} \operatorname{\mathrm{\varepsilon}}\big((a \otimes b) \ast (c \otimes d)\big) =& \operatorname{\mathrm{\varepsilon}}\big(a \operatorname{\mathrm{\varepsilon}}(c) \otimes (b \ast d)\big) + \operatorname{\mathrm{\varepsilon}}\big((a \ast c) \otimes \operatorname{\mathrm{\varepsilon}}(b) d\big) \\ =& \operatorname{\mathrm{\varepsilon}}(a) \operatorname{\mathrm{\varepsilon}}(c) \otimes \operatorname{\mathrm{\varepsilon}}(b \ast d) + \operatorname{\mathrm{\varepsilon}}(a \ast c) \otimes \operatorname{\mathrm{\varepsilon}}(b) \operatorname{\mathrm{\varepsilon}}(d) \\ =& \ 0. \end{align*} $$

To verify identity (4.2), we need to show that

(4.6) $$ \begin{align} \operatorname{\mathrm{\partial}} \big((a \otimes b) \ast (c \otimes d)\big) & = \operatorname{\mathrm{\partial}} (a \otimes b) \ast (c \otimes d) - (-1)^{a+b} (a \otimes b) \ast \operatorname{\mathrm{\partial}} (c \otimes d) \nonumber\\ &\quad + \operatorname{\mathrm{\varepsilon}}(a \otimes b) (c \otimes b) - (-1)^{a+b} (a \otimes b) \operatorname{\mathrm{\varepsilon}}(c \otimes d). \end{align} $$

Let us start computing the left-hand side of the above expression.

$$ \begin{align*} \operatorname{\mathrm{\partial}}\big((a \otimes b) \ast (c \otimes d)\big) & = \operatorname{\mathrm{\partial}}\big(a \operatorname{\mathrm{\varepsilon}}(c) \otimes (b \ast d) + (a \ast c) \otimes \operatorname{\mathrm{\varepsilon}}(b) d\big) \\ & = \operatorname{\mathrm{\partial}} a \operatorname{\mathrm{\varepsilon}}(c) \otimes (b \ast d) \,+\, (-1)^{a} a \operatorname{\mathrm{\varepsilon}}(c) \otimes \operatorname{\mathrm{\partial}}(b \ast d) \\ &\quad + \operatorname{\mathrm{\partial}} (a \ast c) \otimes \operatorname{\mathrm{\varepsilon}}(b) d \,+\, (-1)^{a+c+1} (a \ast c) \otimes \operatorname{\mathrm{\varepsilon}}(b) \operatorname{\mathrm{\partial}} d. \end{align*} $$

Using that C and $C'$ satisfy identity (4.2), we have that

$$ \begin{align*} \operatorname{\mathrm{\partial}}\big((a \otimes b) \ast (c \otimes d)\big) & = \operatorname{\mathrm{\partial}} a \operatorname{\mathrm{\varepsilon}}(c) \otimes (b \ast d) \\ &\quad + (-1)^a a \operatorname{\mathrm{\varepsilon}}(c) \otimes \big(\operatorname{\mathrm{\partial}} b \ast d + (-1)^{b+1} b \ast \operatorname{\mathrm{\partial}} d + \operatorname{\mathrm{\varepsilon}}(b) d + (-1)^{b+1} b \operatorname{\mathrm{\varepsilon}}(d)\big) \\ &\quad + \big(\operatorname{\mathrm{\partial}} a \ast c + (-1)^{a+1} a \ast \operatorname{\mathrm{\partial}} c + \operatorname{\mathrm{\varepsilon}}(a) c + (-1)^{a+1} a \operatorname{\mathrm{\varepsilon}}(c)\big) \otimes \operatorname{\mathrm{\varepsilon}}(b) d \\ &\quad + (-1)^{a+c+1} (a \ast c) \otimes \operatorname{\mathrm{\varepsilon}}(b) \operatorname{\mathrm{\partial}} d. \end{align*} $$

Inspecting this expression, we label terms which sum to $\operatorname {\mathrm {\partial }}\big ((a \otimes b) \ast (c \otimes d)\big )$ :

Additionally, since

$$ \begin{align*} \operatorname{\mathrm{\partial}} (a \otimes b) \ast (c \otimes d) \,=\, (\operatorname{\mathrm{\partial}} a \otimes b) \ast (c \otimes d) \,+\, (-1)^a (a \otimes \operatorname{\mathrm{\partial}} b) \ast (c \otimes d), \end{align*} $$

the following labeled terms sum to $\operatorname {\mathrm {\partial }} (a \otimes b) \ast (c \otimes d)$ :

(4.17) $$ \begin{align}& \kern-2pt (-1)^0 \operatorname{\mathrm{\partial}} a \operatorname{\mathrm{\varepsilon}}(c) \otimes (b \ast d) \end{align} $$

(4.18) $$ \begin{align}& \kern-4pt (-1)^0 (\operatorname{\mathrm{\partial}} a \ast c) \otimes \operatorname{\mathrm{\varepsilon}}(b) d \end{align} $$

(4.19) $$ \begin{align}& (-1)^{a} a \operatorname{\mathrm{\varepsilon}}(c) \otimes (\operatorname{\mathrm{\partial}} b \ast d). \end{align} $$

Similarly, since

$$ \begin{align*} (a \otimes b) \ast \operatorname{\mathrm{\partial}} (c \otimes d) \,=\, (a \otimes b) \ast (\operatorname{\mathrm{\partial}} c \otimes d) \,+\, (-1)^{c} (a \otimes b) \ast (c \otimes \operatorname{\mathrm{\partial}} d), \end{align*} $$

the following labeled terms sum to $(-1)^{a+b+1}(a \otimes b) \ast \operatorname {\mathrm {\partial }} (c \otimes d)$ :

(4.20) $$ \begin{align}& \kern-14pt(-1)^{a+1} (a \ast \operatorname{\mathrm{\partial}} c) \otimes \operatorname{\mathrm{\varepsilon}}(b) d \end{align} $$

(4.21) $$ \begin{align}& \kern-4pt (-1)^{a+b+1} a \operatorname{\mathrm{\varepsilon}}(c) \otimes (b \ast \operatorname{\mathrm{\partial}} d) \end{align} $$

(4.22) $$ \begin{align}& (-1)^{a+c+1} (a \ast c) \otimes \operatorname{\mathrm{\varepsilon}}(b) \operatorname{\mathrm{\partial}} d. \end{align} $$

We have the following matching pairs of labeled summands.

(4.7)-(4.17) : (4.8)-(4.19) : (4.9)-(4.21) : (4.10)-(4.15) : (4.12)-(4.18) : (4.13)-(4.20) : (4.16)-(4.22).

Additionally, the sum of the unmatched terms (4.14) and (4.11) correspond to

$$\begin{align*}\operatorname{\mathrm{\varepsilon}}(a \otimes b) (c \otimes b) - (-1)^{a+b} (a \otimes b) \operatorname{\mathrm{\varepsilon}}(c \otimes d), \end{align*}$$

which concludes the verification of both identity (4.6) and the lemma.

We now give a more conceptual description of the monoidal structure on $\mathsf {biAlg}_{\mathcal {M}}$ , which generalizes to $\mathsf {coAlg}_{\operatorname {\mathrm {U}}(\mathcal {M})}$ . The prop $\mathcal {M}$ is obtained by applying the functor of cellular chains to a CW prop [Reference Medina-Mardones29]. These cells are in fact cubical, generated through compositions by the generators

The Serre diagonal defines a diagonal $\Delta _{\mathcal {M}}$ on $\mathcal {M}$ compatible with its prop structure. More specifically, for any basis element $\mu $ in $\mathcal {M}$ we have a chain map $\phi _\mu \colon \operatorname {\mathrm {N}}(\square ^n) \to \mathcal {M}$ with $\phi _\mu (2^n) = \mu $ , and $\Delta _{\mathcal {M}}(\mu )$ is set to be $\phi _D^{\otimes 2} \circ \Delta (2^n)$ . Crucially, $\Delta _{\mathcal {M}}$ acts on the generators by

which recovers the statement of Lemma 4.1. The structure preserving diagonal on the prop $\mathcal {M}$ induces one on the operad ${\operatorname {\mathrm {U}}(\mathcal {M})}$ and defines, as usual for so-called Hopf operads, a monoidal structure on $\mathsf {coAlg}_{\operatorname {\mathrm {U}}(\mathcal {M})}$ .

4.7. Revisiting the ${E_\infty }$ -coalgebra structure on cubical chains

We show that the monoidal structure on ${\operatorname {\mathrm {U}}(\mathcal {M})}$ -coalgebras recover the $E_\infty $ -structure on cubical chains defined in §4.5. In fact, we have the following stronger statement at the level of $\mathcal {M}$ -bialgebras.

Proof. Since the coalgebra part agrees by definition, we focus on the product. We will proceed by induction with the base case holding trivially. Let $x = x_1 \otimes \dotsb \otimes x_n$ and $y = y_1 \otimes \dotsb \otimes y_n$ be two elements in $\operatorname {\mathrm {N}}(\square ^n)$ . The following straightforward computation verifies that $x \ast y$ in $\operatorname {\mathrm {N}}(\square ^n)$ corresponds to $(x_{<n} \otimes x_n) \ast (y_{<n} \otimes y_n)$ in $\operatorname {\mathrm {N}}(\square ^{n-1}) \otimes \operatorname {\mathrm {N}}(\square ^1)$ :

$$ \begin{align*} x \ast y &= \sum_{i=1}^n x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i}) \, y_{>i} \\ &= \sum_{i=1}^{n-1} x_{<i}\, \epsilon(y_{<i}) \otimes x_i \ast y_i \otimes \epsilon(x_{>i}) \, y_{>i} \ +\ x_{<n}\, \epsilon(y_{<n}) \otimes x_n \ast y_n \\ &= x_{<n} \ast y_{<n} \otimes \operatorname{\mathrm{\varepsilon}}(x_n)\, y_n \ +\ x_{<n}\, \epsilon(y_{<n}) \otimes x_n \ast y_n \\ &= (x_{<n} \otimes x_n) \ast (y_{<n} \otimes y_n), \end{align*} $$

which concludes the proof.

4.8. A monoidal ${E_\infty }$ -coalgebra structure on the cobar construction

Using Theorem 4.2 and the natural equivalence of functors $\operatorname {N^{\square }} \operatorname {\mathbb {\Omega }} \cong \operatorname {\mathbf {\Omega }} \operatorname {N^{\triangle }_{\!{\mathcal {A}\mathsf {s}}}}$ proven in Theorem 3.3, we can transfer a monoidal $E_\infty $ -structure to the Adams’ cobar construction of any reduced simplicial set, extending the monoidal coalgebra structure defined by Baues. This is summarized by the following diagram commuting up to isomorphisms:

Furthermore, using the factorization of Adams’ map described in §3.9 and the naturality of our monoidal $E_\infty $ -structure, we conclude that for any pointed topological space $\mathfrak {X}$ ,

$$\begin{align*}\theta_{\mathfrak{X}} \colon \operatorname{\mathbf{\Omega}} \operatorname{S^{\triangle}_{\!{\mathcal{A}\mathsf{s}}}}(\mathfrak{X},x) \to \operatorname{S^{\square}_{{\operatorname{\mathrm{U}}(\mathcal{M})}}}({\Omega}_x \mathfrak{X}) \end{align*}$$

is a monoidal quasi-isomorphism of $E_{\infty }$ -coalgebras, as announced in the introduction. x