**Proposition 3.1.9.** If P satisfies Axioms (T0), (T1), (T2) and (T3), then, for $n \in \mathbb{N}^*$, $\epsilon \in \{-, +\}$ and $X \in \text{Cell}(P)_n$, $T_{\text{Cl}}^{\text{PC}}(\partial_{n-1}^{\epsilon}(X)) = \bar{\partial}_{n-1}^{\epsilon}(T_{\text{Cl}}^{\text{PC}}(X))$.

*Proof.* We compute that

$$ 
\begin{align*}
T_{\text{Cl}}^{\text{PC}}(\partial_{n-1}^{\epsilon}(X)) &= T_{\text{Cl}}^{\text{M}} \circ T_{\text{M}}^{\text{PC}}(\partial_{n-1}^{\epsilon}(X)) && \text{(by Proposition 3.1.6)} \\
&= T_{\text{Cl}}^{\text{M}}(\tilde{\partial}_{n-1}^{\epsilon}(T_{\text{M}}^{\text{PC}}(X))) && \text{(by Proposition 3.1.7)} \\
&= \bar{\partial}_{n-1}^{\epsilon}(T_{\text{Cl}}^{\text{M}} \circ T_{\text{M}}^{\text{PC}}(X)) && \text{(by Proposition 3.1.8)} \\
&= \bar{\partial}_{n-1}^{\epsilon}(T_{\text{Cl}}^{\text{PC}}(X)). && \square
\end{align*}
$$

**Identities and compositions** Let P be an $\omega$-hypergraph. Here, we define identity and composition operations for the graded sets $\text{Max}(P)$ and $\text{Closed}(P)$, and prove some compatibility results with the translations functions.

Given $n \in \mathbb{N}$ and a closed (resp. maximal) n-fgs $X$, we define the identity of $X$ as the closed (resp. maximal) $(n+1)$-fgs $\text{id}^{n+1}(X)$ defined by

$$ \text{id}^{n+1}(X) = (X_0, \dots, X_n, \emptyset). $$

Given $i, n \in \mathbb{N}$ with $i < n$ and two maximal n-fgs $X, Y$, we define the maximal i-composition of $X$ and $Y$ as the maximal n-fgs $X *_{i}^{M} Y$ defined by

$$ X *_{i}^{M} Y = \max(\text{R}(X) \cup \text{R}(Y)). $$

Respectively, given $i, n \in \mathbb{N}$ with $i < n$ and two closed n-fgs $X, Y$, we define the closed i-composition of $X$ and $Y$ as the closed n-fgs $X *_{i}^{\text{Cl}} Y$ defined by

$$ X *_{i}^{\text{Cl}} Y = X \cup Y. $$

For simplicity, we sometimes write $*^{\text{Cl}}$ (resp. $*^{\text{M}}$) for $*_{i}^{\text{Cl}}$ (resp. $*_{i}^{\text{M}}$). We now prove several compatibility results of the identity and composition operations with the translation functions.

**Proposition 3.1.10.** For $n \in \mathbb{N}$ and an n-cell $X \in \text{Cell}(P)$, $T_{\text{Cl}}^{\text{PC}}(\text{id}^{n+1}(X)) = \text{id}^{n+1}(T_{\text{Cl}}^{\text{PC}}(X))$.

*Proof.* It readily follows from the definitions. $\square$

**Proposition 3.1.11.** For $n \in \mathbb{N}$ and an n-cell $X \in \text{Cell}(P)$, $T_{\text{M}}^{\text{PC}}(\text{id}^{n+1}(X)) = \text{id}^{n+1}(T_{\text{M}}^{\text{PC}}(X))$.

*Proof.* It readily follows from the definitions. $\square$

**Proposition 3.1.12.** For $i, n \in \mathbb{N}$ with $i < n$, and i-composable n-cells $X$ and $Y$ in $\text{Cell}(P)$,

$$ T_{\text{Cl}}^{\text{PC}}(X *_{i} Y) = T_{\text{Cl}}^{\text{PC}}(X) *_{i}^{\text{Cl}} T_{\text{Cl}}^{\text{PC}}(Y). $$

*Proof.* Let $Z = X *_{i} Y$. We have $T_{\text{Cl}}^{\text{PC}}(X *_{i} Y) = \text{R}(\cup Z)$ and

$$ T_{\text{Cl}}^{\text{PC}}(X) *_{i}^{\text{Cl}} T_{\text{Cl}}^{\text{PC}}(Y) = \text{R}(\cup X) \cup \text{R}(\cup Y) = \text{R}((\cup X) \cup (\cup Y)). $$

By definition of composition, $\cup Z \subseteq (\cup X) \cup (\cup Y)$, so $T_{\text{Cl}}^{\text{PC}}(X *_{i} Y) \subseteq T_{\text{Cl}}^{\text{PC}}(X) *_{i}^{\text{Cl}} T_{\text{Cl}}^{\text{PC}}(Y)$. For the other inclusion, note that $X_{j,\epsilon} \subseteq Z_{j,\epsilon}$ for $j \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$ with $(j, \epsilon) \neq (i, +)$, and

$$ X_{i,+} = (X_{i,-} \cup X_{i+1,-}^{+}) \setminus X_{i+1,-}^{-} \subseteq Z_{i,-} \cup Z_{i+1,-}^{+} \subseteq \text{R}(\cup Z) $$

so $\cup X \subseteq \text{R}(\cup Z)$. Similarly, $\cup Y \subseteq \text{R}(\cup Z)$, thus $(\cup X) \cup (\cup Y) \subseteq \text{R}(\cup Z)$, which implies that

$$ T_{\text{Cl}}^{\text{PC}}(X) *_{i}^{\text{Cl}} T_{\text{Cl}}^{\text{PC}}(Y) \subseteq T_{\text{Cl}}^{\text{PC}}(X *_{i} Y). \quad \square $$
**Proposition 3.1.13.** For $i, n \in \mathbb{N}$ with $i < n$, and $X, Y \in \text{Closed}(P)_n$,

$$T_M^{\text{Cl}}(X *_i^{\text{Cl}} Y) = T_M^{\text{Cl}}(X) *_i^{\text{M}} T_M^{\text{Cl}}(Y).$$

*Proof.* We compute that

$$ \begin{align*} T_M^{\text{Cl}}(X) *_i^{\text{M}} T_M^{\text{Cl}}(Y) &= \max(\text{R}(T_M^{\text{Cl}}(X)) \cup \text{R}(T_M^{\text{Cl}}(Y))) \\ &= \max(X \cup Y) \tag*{(\text{by Proposition 3.1.3})} \\ &= T_M^{\text{Cl}}(X *_i^{\text{Cl}} Y). \end{align*} $$

□

**Proposition 3.1.14.** For $i, n \in \mathbb{N}$ with $i < n$, and $i$-composable $n$-cells $X$ and $Y$ of $P$,

$$T_M^{\text{PC}}(X *_i Y) = T_M^{\text{PC}}(X) *_i^{\text{M}} T_M^{\text{PC}}(Y).$$

*Proof.* We compute that

$$ \begin{align*} T_M^{\text{PC}}(X *_i Y) &= T_M^{\text{Cl}} \circ T_M^{\text{PC}}(X *_i Y) && \text{(by Propositions 3.1.3 and 3.1.6)} \\ &= T_M^{\text{Cl}}(T_M^{\text{PC}}(X) *_i^{\text{Cl}} T_M^{\text{PC}}(Y)) && \text{(by Proposition 3.1.9)} \\ &= T_M^{\text{Cl}} \circ T_M^{\text{PC}}(X) *_i^{\text{M}} T_M^{\text{Cl}} \circ T_M^{\text{PC}}(Y) && \text{(by Proposition 3.1.13)} \\ &= T_M^{\text{PC}}(X) *_i^{\text{M}} T_M^{\text{PC}}(Y) && \text{(by Propositions 3.1.3 and 3.1.6).} \end{align*} $$

□

**Well-formed cells** We defined above source, target, identity and composition operations for both Closed($P$) and Max($P$). However, these operations are not expected to equip the graded sets Closed($P$) and Max($P$) with a structure of $\omega$-category (in fact, not even a structure of $\omega$-globular set). In order to obtain an $\omega$-category, we need to restrict to subsets of “well-formed” elements of Closed($P$) and Max($P$). Then, we can show that the two induced $\omega$-category of cells are isomorphic to Cell($P$).

Let $P$ be an $\omega$-hypergraph. Given $n \in \mathbb{N}$ and $X \in \text{Closed}(P)_n$, we say that $X$ is closed-well-formed when

- $X_n$ is fork-free,

- $\tilde{\partial}_{n-1}^{-}(X)$ and $\tilde{\partial}_{n-1}^{+}(X)$ are closed-well-formed,

- if $n \ge 2$, $\tilde{\partial}_{n-2}^{-} \circ \tilde{\partial}_{n-1}^{-}(X) = \tilde{\partial}_{n-2}^{-} \circ \tilde{\partial}_{n-1}^{+}(X)$ and $\tilde{\partial}_{n-2}^{+} \circ \tilde{\partial}_{n-1}^{-}(X) = \tilde{\partial}_{n-2}^{+} \circ \tilde{\partial}_{n-1}^{+}(X)$.

We write Closed$_{WF}$(P) for the graded set of closed-well-formed fgs of P. Respectively, given $n \in \mathbb{N}$ and $X \in \text{Max}(P)_n$, we say that $X$ is maximal-well-formed when

- $X_n$ is fork-free,

- $\tilde{\partial}_{n-1}^{-}(X)$ and $\tilde{\partial}_{n-1}^{+}(X)$ are maximal-well-formed,

- if $n \ge 2$, $\tilde{\partial}_{n-2}^{-} \circ \tilde{\partial}_{n-1}^{-}(X) = \tilde{\partial}_{n-2}^{-} \circ \tilde{\partial}_{n-1}^{+}(X)$ and $\tilde{\partial}_{n-2}^{+} \circ \tilde{\partial}_{n-1}^{-}(X) = \tilde{\partial}_{n-2}^{+} \circ \tilde{\partial}_{n-1}^{+}(X)$.

We write Max$_{WF}$(P) for the graded set of maximal-well-formed fgs of P. We now aim at proving that both Closed$_{WF}$(P) and Max$_{WF}$(P) are $\omega$-categories isomorphic to Cell($P$) when P satisfies enough axioms of torsion-free complexes. We first show this property for Max$_{WF}$(P) after introducing several technical results.

**Lemma 3.1.15.** If P satisfies Axioms (T0), (T1), (T2) and (T3), then, for $n \in \mathbb{N}$ and $X \in \text{Cell}(P)_n$, we have $T_M^{\text{PC}}(X) \in \text{Max}_{\text{WF}}(P)_n$.
**Proposition 3.1.13.** For $i, n \in \mathbb{N}$ with $i < n$, and $X, Y \in \text{Closed}(P)_n$,

$$T_M^{\text{Cl}}(X *_i^{\text{Cl}} Y) = T_M^{\text{Cl}}(X) *_i^{\text{M}} T_M^{\text{Cl}}(Y).$$

*Proof.* We compute that

$$ \begin{aligned} T_M^{\text{Cl}}(X) *_i^{\text{M}} T_M^{\text{Cl}}(Y) &= \max(\text{R}(T_M^{\text{Cl}}(X)) \cup \text{R}(T_M^{\text{Cl}}(Y))) \\ &= \max(X \cup Y) \qquad (\text{by Proposition 3.1.3}) \\ &= T_M^{\text{Cl}}(X *_i^{\text{Cl}} Y). \qquad \square \end{aligned} $$

**Proposition 3.1.14.** For $i, n \in \mathbb{N}$ with $i < n$, and $i$-composable $n$-cells $X$ and $Y$ of $P$,

$$T_M^{\text{PC}}(X *_i Y) = T_M^{\text{PC}}(X) *_i^{\text{M}} T_M^{\text{PC}}(Y).$$

*Proof.* We compute that

$$ \begin{aligned} T_M^{\text{PC}}(X *_i Y) &= T_M^{\text{Cl}} \circ T_M^{\text{PC}}(X *_i Y) && (\text{by Propositions 3.1.3 and 3.1.6}) \\ &= T_M^{\text{Cl}}(T_M^{\text{PC}}(X) *_i^{\text{Cl}} T_M^{\text{PC}}(Y)) && (\text{by Proposition 3.1.9}) \\ &= T_M^{\text{Cl}} \circ T_M^{\text{PC}}(X) *_i^{\text{M}} T_M^{\text{Cl}} \circ T_M^{\text{PC}}(Y) && (\text{by Proposition 3.1.13}) \\ &= T_M^{\text{PC}}(X) *_i^{\text{M}} T_M^{\text{PC}}(Y) && (\text{by Propositions 3.1.3 and 3.1.6}). \end{aligned} \qquad \square $$

**Well-formed cells** We defined above source, target, identity and composition operations for both Closed($P$) and Max($P$). However, these operations are not expected to equip the graded sets Closed($P$) and Max($P$) with a structure of $\omega$-category (in fact, not even a structure of $\omega$-globular set). In order to obtain an $\omega$-category, we need to restrict to subsets of “well-formed” elements of Closed($P$) and Max($P$). Then, we can show that the two induced $\omega$-category of cells are isomorphic to Cell($P$).

Let $P$ be an $\omega$-hypergraph. Given $n \in \mathbb{N}$ and $X \in \text{Closed}(P)_n$, we say that $X$ is closed-well-formed when

- $X_n$ is fork-free,

- $\tilde{\partial}_{n-1}^{-}(X)$ and $\tilde{\partial}_{n-1}^{+}(X)$ are closed-well-formed,

- if $n \ge 2$, $\tilde{\partial}_{n-2}^{-} \circ \tilde{\partial}_{n-1}^{-}(X) = \tilde{\partial}_{n-2}^{-} \circ \tilde{\partial}_{n-1}^{+}(X)$ and $\tilde{\partial}_{n-2}^{+} \circ \tilde{\partial}_{n-1}^{-}(X) = \tilde{\partial}_{n-2}^{+} \circ \tilde{\partial}_{n-1}^{+}(X)$.

We write Closed$_{WF}$(P) for the graded set of closed-well-formed fgs of P. Respectively, given $n \in \mathbb{N}$ and $X \in \text{Max}(P)_n$, we say that $X$ is maximal-well-formed when

- $X_n$ is fork-free,

- $\tilde{\partial}_{n-1}^{-}(X)$ and $\tilde{\partial}_{n-1}^{+}(X)$ are maximal-well-formed,

- if $n \ge 2$, $\tilde{\partial}_{n-2}^{-} \circ \tilde{\partial}_{n-1}^{-}(X) = \tilde{\partial}_{n-2}^{-} \circ \tilde{\partial}_{n-1}^{+}(X)$ and $\tilde{\partial}_{n-2}^{+} \circ \tilde{\partial}_{n-1}^{-}(X) = \tilde{\partial}_{n-2}^{+} \circ \tilde{\partial}_{n-1}^{+}(X)$.

We write Max$_{WF}$(P) for the graded set of maximal-well-formed fgs of P. We now aim at proving that both Closed$_{WF}$(P) and Max$_{WF}$(P) are $\omega$-categories isomorphic to Cell($P$) when P satisfies enough axioms of torsion-free complexes. We first show this property for Max$_{WF}$(P) after introducing several technical results.

**Lemma 3.1.15.** If P satisfies Axioms (T0), (T1), (T2) and (T3), then, for $n \in \mathbb{N}$ and $X \in \text{Cell}(P)_n$, we have $T_M^{\text{PC}}(X) \in \text{Max}_{\text{WF}}(P)_n$.
Proof. We proceed by induction on $n$. If $n = 0$, the result is trivial. So suppose that $n > 0$ and let $Y = T_M^{\text{PC}}(X)$. Since $Y_n = X_n$, $Y_n$ is fork-free. Moreover, by Proposition 3.1.7, we have

$$ \tilde{\partial}_{n-1}^{\epsilon}(Y) = T_M^{\text{PC}}(\partial_{n-1}^{\epsilon}(X)) \quad \text{for } \epsilon \in \{-, +\}. $$

By the induction hypothesis, $\tilde{\partial}_{n-1}^{\epsilon}(Y)$ is maximal-well-formed. And, when $n \ge 2$, for $\eta \in \{-, +\}$, we have

$$ \begin{aligned} \tilde{\partial}_{n-2}^{\eta} \circ \tilde{\partial}_{n-1}^{-}(Y) &= T_{M}^{\text{PC}}(\partial_{n-2}^{\eta} \circ \partial_{n-1}^{-}(X)) \\ &= T_{M}^{\text{PC}}(\partial_{n-2}^{\eta} \circ \partial_{n-1}^{+}(X)) \\ &= \tilde{\partial}_{n-2}^{\eta} \circ \tilde{\partial}_{n-1}^{+}(Y). \end{aligned} \quad \text{(by Proposition 3.1.7)} $$

Hence, $Y$ is maximal-well-formed.

□

**Lemma 3.1.16.** If $P$ satisfies Axioms (T0), (T1), (T2) and (T3), then, given $n \in \mathbb{N}$ and an fgs $X \in \text{Max}_{\text{WF}}(P)_n$, there exists an $n$-cell $Y \in \text{Cell}(P)_n$ such that $T_M^{\text{PC}}(Y) = X$.

Proof. We proceed by induction on $n$. If $n = 0$, the result is trivial. So suppose that $n > 0$. By induction, let $S, T \in \text{Cell}(P)_{n-1}$ be such that $T_M^{\text{PC}}(S) = \tilde{\partial}_{n-1}^{-}(X)$ and $T_M^{\text{PC}}(T) = \tilde{\partial}_{n-1}^{+}(X)$. When $n \ge 2$, for $\epsilon \in \{-, +\}$, we have

$$ \begin{aligned} \partial_{n-2}^{\epsilon}(S) &= T_{\text{PC}}^{M} \circ T_{M}^{\text{PC}}(\partial_{n-2}^{\epsilon}(S)) && \text{(by Proposition 3.1.5)} \\ &= T_{\text{PC}}^{M}(\tilde{\partial}_{n-2}^{\epsilon}(T_{M}^{\text{PC}}(S))) && \text{(by Proposition 3.1.7)} \\ &= T_{\text{PC}}^{M}(\tilde{\partial}_{n-2}^{\epsilon} \circ \tilde{\partial}_{n-1}^{-}(X)) \\ &= T_{\text{PC}}^{M}(\tilde{\partial}_{n-2}^{\epsilon} \circ \tilde{\partial}_{n-1}^{+}(X)) && \text{(because $X$ is maximal-well-formed)} \\ &= T_{\text{PC}}^{M}(\tilde{\partial}_{n-2}^{\epsilon}(T_{M}^{\text{PC}}(T))) \\ &= T_{\text{PC}}^{M} \circ T_{M}^{\text{PC}}(\partial_{n-2}^{\epsilon}(T)) = \partial_{n-2}^{\epsilon}(T). \end{aligned} $$

Moreover,

$$ \begin{aligned} (S_{n-1} \cup X_n^+) \setminus X_n^- &= (X_{n-1} \cup X_n^{\mp} \cup X_n^+) \setminus X_n^- \\ &= X_{n-1} \cup X_n^{\pm} = T_{n-1}. \end{aligned} $$

Similarly, $(T_{n-1} \cup X_n^-) \setminus X_n^+ = S_{n-1}$ so $X_n$ moves $S_{n-1}$ to $T_{n-1}$. Thus, the $n$-pre-cell $Y$ defined by $Y_n = X_n$, $Y_{n-1, -} = S_{n-1}$, $Y_{n-1, +} = T_{n-1}$ and $Y_{i, \delta} = S_{i, \delta}$ for $i \in \mathbb{N}_{n-2}$ and $\delta \in \{-, +\}$, is an $n$-cell. Let $Z = T_M^{\text{PC}}(Y)$. We have $Z_n = X_n$ and

$$ \begin{aligned} \tilde{\partial}_{n-1}^{-}(Z) &= \tilde{\partial}_{n-1}^{-}(T_{M}^{\text{PC}}(Y)) \\ &= T_{M}^{\text{PC}}(\tilde{\partial}_{n-1}^{-}(Y)) && \text{(by Proposition 3.1.7)} \\ &= T_{M}^{\text{PC}}(S) = \tilde{\partial}_{n-1}^{-}(X). \end{aligned} $$

So, by definition of $\tilde{\partial}^{-}$, we have $Z_{n-1} \cup X_n^{\mp} = X_{n-1} \cup X_n^{\mp}$ and $Z_i = X_i$ for $i \in \mathbb{N}_{n-2}$. Since $X$ and $Z$ are maximal, we have $X_{n-1} \cap X_n^{\mp} = Z_{n-1} \cap X_n^{\mp} = \emptyset$. Hence, $X_{n-1} = Z_{n-1}$ and $X = Z = T_M^{\text{PC}}(Y)$ which concludes the proof. □

**Lemma 3.1.17.** If $P$ satisfies Axioms (T0), (T1), (T2) and (T3), then, $T_M^{\text{PC}}$ induces a bijection between $\text{Cell}(P)$ and $\text{Max}_{\text{WF}}(P)$.

Proof. By Lemma 3.1.16, $T_M^{\text{PC}}: \text{Cell}(P) \to \text{Max}_{\text{WF}}(P)$ is surjective and, by Proposition 3.1.5, it is injective, so it is bijective. □
**Proof.** We proceed by induction on $n$. If $n = 0$, the result is trivial. So suppose that $n > 0$ and let $Y = T_M^{\text{PC}}(X)$. Since $Y_n = X_n$, $Y_n$ is fork-free. Moreover, by Proposition 3.1.7, we have

$$ \tilde{\partial}_{n-1}^{\epsilon}(Y) = T_M^{\text{PC}}(\partial_{n-1}^{\epsilon}(X)) \quad \text{for } \epsilon \in \{-, +\}. $$

By the induction hypothesis, $\tilde{\partial}_{n-1}^{\epsilon}(Y)$ is maximal-well-formed. And, when $n \ge 2$, for $\eta \in \{-, +\}$, we have

$$ \begin{aligned} \tilde{\partial}_{n-2}^{\eta} \circ \tilde{\partial}_{n-1}^{-}(Y) &= T_{M}^{\text{PC}}(\partial_{n-2}^{\eta} \circ \partial_{n-1}^{-}(X)) \\ &= T_{M}^{\text{PC}}(\partial_{n-2}^{\eta} \circ \partial_{n-1}^{+}(X)) \\ &= \tilde{\partial}_{n-2}^{\eta} \circ \tilde{\partial}_{n-1}^{+}(Y). \end{aligned} \quad \text{(by Proposition 3.1.7)} $$

Hence, $Y$ is maximal-well-formed.

<figure><img src="image_16.png" /></figure>

**Lemma 3.1.16.** If $P$ satisfies Axioms (T0), (T1), (T2) and (T3), then, given $n \in \mathbb{N}$ and an fgs $X \in \text{Max}_{\text{WF}}(P)_n$, there exists an $n$-cell $Y \in \text{Cell}(P)_n$ such that $T_M^{\text{PC}}(Y) = X$.

**Proof.** We proceed by induction on $n$. If $n = 0$, the result is trivial. So suppose that $n > 0$. By induction, let $S, T \in \text{Cell}(P)_{n-1}$ be such that $T_M^{\text{PC}}(S) = \tilde{\partial}_{n-1}^{-}(X)$ and $T_M^{\text{PC}}(T) = \tilde{\partial}_{n-1}^{+}(X)$. When $n \ge 2$, for $\epsilon \in \{-, +\}$, we have

$$ \begin{aligned} \partial_{n-2}^{\epsilon}(S) &= T_{\text{PC}}^{M} \circ T_{M}^{\text{PC}}(\partial_{n-2}^{\epsilon}(S)) && \text{(by Proposition 3.1.5)} \\ &= T_{\text{PC}}^{M}(\tilde{\partial}_{n-2}^{\epsilon}(T_{M}^{\text{PC}}(S))) && \text{(by Proposition 3.1.7)} \\ &= T_{\text{PC}}^{M}(\tilde{\partial}_{n-2}^{\epsilon} \circ \tilde{\partial}_{n-1}^{-}(X)) \\ &= T_{\text{PC}}^{M}(\tilde{\partial}_{n-2}^{\epsilon} \circ \tilde{\partial}_{n-1}^{+}(X)) && \text{(because $X$ is maximal-well-formed)} \\ &= T_{\text{PC}}^{M}(\tilde{\partial}_{n-2}^{\epsilon} \circ T_{M}^{\text{PC}}(T)) \\ &= T_{\text{PC}}^{M} \circ T_{M}^{\text{PC}}(\partial_{n-2}^{\epsilon}(T)) = \partial_{n-2}^{\epsilon}(T). \end{aligned} $$

Moreover,

$$ \begin{aligned} (S_{n-1} \cup X_n^+) \setminus X_n^- &= (X_{n-1} \cup X_n^{\mp} \cup X_n^+) \setminus X_n^- \\ &= X_{n-1} \cup X_n^{\pm} = T_{n-1}. \end{aligned} $$

Similarly, $(T_{n-1} \cup X_n^-) \setminus X_n^+ = S_{n-1}$ so $X_n$ moves $S_{n-1}$ to $T_{n-1}$. Thus, the $n$-pre-cell $Y$ defined by $Y_n = X_n$, $Y_{n-1,-} = S_{n-1}$, $Y_{n-1,+} = T_{n-1}$ and $Y_{i,\delta} = S_{i,\delta}$ for $i \in \mathbb{N}_{n-2}$ and $\delta \in \{-, +\}$, is an $n$-cell. Let $Z = T_M^{\text{PC}}(Y)$. We have $Z_n = X_n$ and

$$ \begin{aligned} \tilde{\partial}_{n-1}^{-}(Z) &= \tilde{\partial}_{n-1}^{-}(T_{M}^{\text{PC}}(Y)) \\ &= T_{M}^{\text{PC}}(\tilde{\partial}_{n-1}^{-}(Y)) && \text{(by Proposition 3.1.7)} \\ &= T_{M}^{\text{PC}}(S) = \tilde{\partial}_{n-1}^{-}(X). \end{aligned} $$

So, by definition of $\tilde{\partial}^{-}$, we have $Z_{n-1} \cup X_n^{\mp} = X_{n-1} \cup X_n^{\mp}$ and $Z_i = X_i$ for $i \in \mathbb{N}_{n-2}$. Since $X$ and $Z$ are maximal, we have $X_{n-1} \cap X_n^{\mp} = Z_{n-1} \cap X_n^{\mp} = \emptyset$. Hence, $X_{n-1} = Z_{n-1}$ and $X = Z = T_M^{\text{PC}}(Y)$ which concludes the proof. $\square$

**Lemma 3.1.17.** If $P$ satisfies Axioms (T0), (T1), (T2) and (T3), then, $T_M^{\text{PC}}$ induces a bijection between $\text{Cell}(P)$ and $\text{Max}_{\text{WF}}(P)$.

**Proof.** By Lemma 3.1.16, $T_M^{\text{PC}}: \text{Cell}(P) \to \text{Max}_{\text{WF}}(P)$ is surjective and, by Proposition 3.1.5, it is injective, so it is bijective. $\square$

<figure><img src="image_17.png" /></figure>
We can now deduce that maximal-well-formed fgs's are an adequate alternative definition of cells for torsion-free complexes:

**Theorem 3.1.18.** If P satisfies Axioms (T0), (T1), (T2) and (T3), then, $\text{Max}_{\text{WF}}(P)$ is an $\omega$-category and $T_M^{\text{PC}}$ induces an isomorphism between $\text{Cell}(P)$ and $\text{Max}_{\text{WF}}(P)$.

*Proof.* By definition of $\text{Max}_{\text{WF}}(P)$, the functions $\tilde{\partial}_k^-, \tilde{\partial}_k^+$ for $k \in \mathbb{N}$ equip $\text{Max}_{\text{WF}}(P)$ with a structure of $\omega$-globular set. We first prove that the composition operation $*^M$ restricts to $\text{Max}_{\text{WF}}(P)$. Let $i, n \in \mathbb{N}$ with $i < n$, and $X, Y \in \text{Max}_{\text{WF}}(P)_n$ be such that $\tilde{\partial}_i^+(X) = \tilde{\partial}_i^-(Y)$. By Lemma 3.1.17, there exist $X', Y' \in \text{Cell}(P)_n$ such that $T_M^{\text{PC}}(X') = X$ and $T_M^{\text{PC}}(Y') = Y$. By Proposition 3.1.7, we have

$$T_M^{\text{PC}}(\tilde{\partial}_i^+(X')) = \tilde{\partial}_i^+(X) = \tilde{\partial}_i^-(Y) = T_M^{\text{PC}}(\tilde{\partial}_i^-(Y')),$$

and, by Lemma 3.1.17, $\tilde{\partial}_i^+(X') = \tilde{\partial}_i^-(Y')$ so $X'$ and $Y'$ are $i$-composable. By Lemma 3.1.17, we have $T_M^{\text{PC}}(X' *_i Y') \in \text{Max}_{\text{WF}}(P)$ and, by Proposition 3.1.14, $X *^M_i Y \in \text{Max}_{\text{WF}}(P)$.

By Propositions 3.1.7, 3.1.11 and 3.1.14, $T_M^{\text{PC}}$ commutes with the source, target, identity and composition operations and is a bijection when restricted to $\text{Max}_{\text{WF}}(P)$, so that $\text{Max}_{\text{WF}}(P)$ is an $\omega$-category since $\text{Cell}(P)$ is (by Theorem 2.2.3 and remark 2.2.4), and $T_M^{\text{PC}}$ induces an isomorphism of $\omega$-categories. $\square$

We prove a similar property for closed-well-formed fgs's after showing some technical results.

**Lemma 3.1.19.** $T_{\text{Cl}}^M$ induces a bijection between $\text{Max}_{\text{WF}}(P)$ and $\text{Closed}_{\text{WF}}(P)$.

*Proof.* We already know that $T_{\text{Cl}}^M$ is a bijection by Proposition 3.1.3. For $n \in \mathbb{N}$, we show that $T_{\text{Cl}}^M$ sends a maximal-well-formed $n$-fgs $X$ to a closed-well-formed $n$-fgs by induction on $n$. If $n = 0$, the result is trivial. So suppose that $n > 0$. Let $Y = T_{\text{Cl}}^M(X)$. Then, $Y_n = X_n$ is fork-free and, for $\epsilon \in \{-, +\}$, we have $\bar{\partial}_{n-1}^\epsilon(Y) = T_{\text{Cl}}^M(\tilde{\partial}_{n-1}^\epsilon(X))$ by Proposition 3.1.8, and it is closed-well-formed by induction. Moreover, when $n \ge 2$,

$$ \begin{aligned} \bar{\partial}_{n-2}^\epsilon \circ \bar{\partial}_{n-1}^-(Y) &= T_{\text{Cl}}^M(\tilde{\partial}_{n-2}^\epsilon \circ \bar{\partial}_{n-1}^-(X)) & \text{(by Proposition 3.1.8)} \\ &= T_{\text{Cl}}^M(\tilde{\partial}_{n-2}^\epsilon \circ \bar{\partial}_{n-1}^+(X)) \\ &= \bar{\partial}_{n-2}^\epsilon \circ \bar{\partial}_{n-1}^+(Y) \end{aligned} $$

so $Y$ is closed-well-formed. Similarly, $T_M^\text{Cl}$ sends closed-well-formed fgs to maximal-well-formed fgs, which concludes the proof. $\square$

**Lemma 3.1.20.** If P satisfies Axioms (T0), (T1), (T2) and (T3), then, $T_{\text{Cl}}^{\text{PC}}$ induces a bijection between $\text{Cell}(P)$ and $\text{Closed}_{\text{WF}}(P)$.

*Proof.* The result is a consequence of Proposition 3.1.6 and Lemmas 3.1.17 and 3.1.19. $\square$

We can now conclude that closed-well-formed fgs's are an adequate alternative definition of cells for torsion-free complexes:

**Theorem 3.1.21.** If P satisfies Axioms (T0), (T1), (T2) and (T3), then, $\text{Closed}_{\text{WF}}(P)$ is an $\omega$-category and $T_{\text{Cl}}^{\text{PC}}$ induces an isomorphism between $\text{Cell}(P)$ and $\text{Closed}_{\text{WF}}(P)$.

*Proof.* By a proof similar to the one of Theorem 3.1.18, using Propositions 3.1.9, 3.1.10 and 3.1.12 and Lemma 3.1.20. $\square$
We can now deduce that maximal-well-formed fgs's are an adequate alternative definition of cells for torsion-free complexes:

**Theorem 3.1.18.** If P satisfies Axioms (T0), (T1), (T2) and (T3), then, $\text{Max}_{\text{WF}}(P)$ is an $\omega$-category and $T_M^{\text{PC}}$ induces an isomorphism between $\text{Cell}(P)$ and $\text{Max}_{\text{WF}}(P)$.

*Proof.* By definition of $\text{Max}_{\text{WF}}(P)$, the functions $\tilde{\partial}_k^-, \tilde{\partial}_k^+$ for $k \in \mathbb{N}$ equip $\text{Max}_{\text{WF}}(P)$ with a structure of $\omega$-globular set. We first prove that the composition operation $*^M$ restricts to $\text{Max}_{\text{WF}}(P)$. Let $i, n \in \mathbb{N}$ with $i < n$, and $X, Y \in \text{Max}_{\text{WF}}(P)_n$ be such that $\tilde{\partial}_i^+(X) = \tilde{\partial}_i^-(Y)$. By Lemma 3.1.17, there exist $X', Y' \in \text{Cell}(P)_n$ such that $T_M^{\text{PC}}(X') = X$ and $T_M^{\text{PC}}(Y') = Y$. By Proposition 3.1.7, we have

$$T_M^{\text{PC}}(\tilde{\partial}_i^+(X')) = \tilde{\partial}_i^+(X) = \tilde{\partial}_i^-(Y) = T_M^{\text{PC}}(\tilde{\partial}_i^-(Y')),$$

and, by Lemma 3.1.17, $\tilde{\partial}_i^+(X') = \tilde{\partial}_i^-(Y')$ so $X'$ and $Y'$ are $i$-composable. By Lemma 3.1.17, we have $T_M^{\text{PC}}(X' *_i Y') \in \text{Max}_{\text{WF}}(P)$ and, by Proposition 3.1.14, $X *^M_i Y \in \text{Max}_{\text{WF}}(P)$.

By Propositions 3.1.7, 3.1.11 and 3.1.14, $T_M^{\text{PC}}$ commutes with the source, target, identity and composition operations and is a bijection when restricted to $\text{Max}_{\text{WF}}(P)$, so that $\text{Max}_{\text{WF}}(P)$ is an $\omega$-category since $\text{Cell}(P)$ is (by Theorem 2.2.3 and remark 2.2.4), and $T_M^{\text{PC}}$ induces an isomorphism of $\omega$-categories. $\square$

We prove a similar property for closed-well-formed fgs's after showing some technical results.

**Lemma 3.1.19.** $T_{\text{Cl}}^M$ induces a bijection between $\text{Max}_{\text{WF}}(P)$ and $\text{Closed}_{\text{WF}}(P)$.

*Proof.* We already know that $T_{\text{Cl}}^M$ is a bijection by Proposition 3.1.3. For $n \in \mathbb{N}$, we show that $T_{\text{Cl}}^M$ sends a maximal-well-formed $n$-fgs $X$ to a closed-well-formed $n$-fgs by induction on $n$. If $n = 0$, the result is trivial. So suppose that $n > 0$. Let $Y = T_{\text{Cl}}^M(X)$. Then, $Y_n = X_n$ is fork-free and, for $\epsilon \in \{-, +\}$, we have $\bar{\partial}_{n-1}^\epsilon(Y) = T_{\text{Cl}}^M(\tilde{\partial}_{n-1}^\epsilon(X))$ by Proposition 3.1.8, and it is closed-well-formed by induction. Moreover, when $n \ge 2$,

$$ \begin{aligned} \bar{\partial}_{n-2}^\epsilon \circ \bar{\partial}_{n-1}^-(Y) &= T_{\text{Cl}}^M(\tilde{\partial}_{n-2}^\epsilon \circ \bar{\partial}_{n-1}^-(X)) && (\text{by Proposition 3.1.8}) \\ &= T_{\text{Cl}}^M(\tilde{\partial}_{n-2}^\epsilon \circ \bar{\partial}_{n-1}^+(X)) \\ &= \bar{\partial}_{n-2}^\epsilon \circ \bar{\partial}_{n-1}^+(Y) \end{aligned} $$

so $Y$ is closed-well-formed. Similarly, $T_M^\text{Cl}$ sends closed-well-formed fgs to maximal-well-formed fgs, which concludes the proof. $\square$

**Lemma 3.1.20.** If P satisfies Axioms (T0), (T1), (T2) and (T3), then, $T_{\text{Cl}}^{\text{PC}}$ induces a bijection between $\text{Cell}(P)$ and $\text{Closed}_{\text{WF}}(P)$.

*Proof.* The result is a consequence of Proposition 3.1.6 and Lemmas 3.1.17 and 3.1.19. $\square$

We can now conclude that closed-well-formed fgs's are an adequate alternative definition of cells for torsion-free complexes:

**Theorem 3.1.21.** If P satisfies Axioms (T0), (T1), (T2) and (T3), then, $\text{Closed}_{\text{WF}}(P)$ is an $\omega$-category and $T_{\text{Cl}}^{\text{PC}}$ induces an isomorphism between $\text{Cell}(P)$ and $\text{Closed}_{\text{WF}}(P)$.

*Proof.* By a proof similar to the one of Theorem 3.1.18, using Propositions 3.1.9, 3.1.10 and 3.1.12 and Lemma 3.1.20. $\square$
**From polygraphs to torsion-free complexes** We saw earlier (Corollary 2.4.2) that torsion-free complexes induce free $\omega$-categories on a canonical $\omega$-polygraph. However, in practice, we are often interested in the inverse operation, i.e., representing the cells of an $\omega$-category freely generated on an $\omega$-polygraph by the cells of a torsion-free complex. Here, we define the $\omega$-hypergraph $P^H$ associated to an $\omega$-polygraph $P$ and, in the case where $P^H$ is a torsion-free complex, give conditions under which the $\omega$-category Closed$_{WF}$(P$^H$) is isomorphic to the free $\omega$-category P$^*$. In order to define P$^H$, we will use the support function introduced by Makkai for free strict categories [16]. Given a polygraph $P$, this function maps a cell of P$^*$ to the set of generators “it involves”. We first recall its definition before dealing with the other matters.

Given a set $S$, we write $\mathcal{P}_f(S)$ for the set of finite subsets of $S$. Given $n \in \mathbb{N} \cup \{\omega\}$ and an $n$-polygraph $P$, we define the support function

$$ \mathrm{supp}^{\mathbb{P}} : P^* \to \mathcal{P}_f(\bigcup_{i \in \mathbb{N}_n} P_i) $$

or simply, supp, as the unique function such that, given $u \in P^*$,

$$ - \mathrm{supp}(u) = \{g\} \text{ when } u = g \text{ for some } g \in P_0, $$

$$ - \mathrm{supp}(u) = \{g\} \cup \mathrm{supp}(\partial_{k-1}^{-}(g)) \cup \mathrm{supp}(\partial_{k-1}^{+}(u)) \text{ when } u = g \text{ for some } k \in \mathbb{N}_{n}^{*} \text{ and } g \in P_{k}, $$

$$ - \mathrm{supp}(u) = \mathrm{supp}(u') \text{ when } u = \mathrm{id}_{u'} \text{ for some } u' \in P^*, $$

$$ - \mathrm{supp}(u) = \mathrm{supp}(u_1) \cup \mathrm{supp}(u_2) \text{ when } u = u_1 *_i u_2 \text{ for some } i, k \in \mathbb{N}_n^* \text{ with } i < k \text{ and } i \text{-composable } u_1, u_2 \in P_k^* $$

The above definition completely defines supp, since the cells of P$^*$ are precisely the classes of formal composites of generators of P. It can moreover be shown well-defined (see the original proof of Makkai [16, Lemma 5] or [7, Proposition 2.4.3.2]).

Given $P \in \mathrm{Pol}_\omega$, we define an $\omega$-hypergraph $P^H$ by putting $P_n^H = P_n$ for $n \in \mathbb{N}$ and, when $n > 0$,

$$ g^{-} = \mathrm{supp}(\mathrm{d}_{n-1}^{-}(g)) \cap P_{n-1} \qquad g^{+} = \mathrm{supp}(\mathrm{d}_{n-1}^{+}(g)) \cap P_{n-1} $$

for $g \in P_n^H$. Under this definition, supp$^P$ can be seen as a function $P^* \to \mathrm{Closed}(P^H)$. We then have the following criterion to know whether $P^*$ can be faithfully represented by the closed-well-formed fgs's of $P^H$:

**Theorem 3.1.22.** Let $P \in \mathrm{Pol}_\omega$ such that $P^H$ is a torsion-free complex. Then, supp$^P$ is the underlying function of an $\omega$-functor $F: P^* \to \mathrm{Closed}_{WF}(P^H)$ if and only if, for $n \in \mathbb{N}^*$, $g \in P_n$ and $\epsilon \in \{-, +\}$, we have $\mathrm{supp}(\mathrm{d}_{n-1}^{\epsilon}(g)) = \mathrm{R}(g^{\epsilon})$. In this case, $F$ is moreover an isomorphism.

*Remark 3.1.23.* If the condition of Theorem 3.1.22 is satisfied, then $T_{\mathrm{PC}}^{\mathrm{Cl}} \circ F: P^* \to \mathrm{Cell}(P^H)$ is the unique isomorphism given by Corollary 2.4.2 which maps $g \in P$ to $\langle g \rangle \in \mathrm{Cell}(P^H)$.

*Proof.* If supp$^P$ induces an $\omega$-functor $F: P^* \to \mathrm{Closed}_{WF}(P^H)$, then we have

$$ \begin{aligned} \mathrm{supp}(\mathrm{d}_{n-1}^{\epsilon}(g)) &= F(\mathrm{d}_{n-1}^{\epsilon}(g)) \\ &= \bar{\partial}_{n-1}^{\epsilon}(F(g)) \\ &= \bar{\partial}_{n-1}^{\epsilon}(\mathrm{R}(g)) \\ &= \mathrm{R}(g^{\epsilon}) \end{aligned} \qquad (\text{by definition of } \bar{\partial}_{n-1}^{\epsilon}) $$

which proves the necessity. For sufficiency, we prove by induction on $n \in \mathbb{N}$ that supp$^P$ is the underlying function of an $n$-functor $F^n: (P^*)_{\le n} \to \mathrm{Closed}_{WF}(P^H)_{\le n}$. This is clear for $n = 0$, and, when $n > 0$, we define $F^n$ by extending $F^{n-1}$ and so that $F^n(g) = \mathrm{R}(g)$ using the universal
**From polygraphs to torsion-free complexes** We saw earlier (Corollary 2.4.2) that torsion-free complexes induce free $\omega$-categories on a canonical $\omega$-polygraph. However, in practice, we are often interested in the inverse operation, i.e., representing the cells of an $\omega$-category freely generated on an $\omega$-polygraph by the cells of a torsion-free complex. Here, we define the $\omega$-hypergraph $P^H$ associated to an $\omega$-polygraph $P$ and, in the case where $P^H$ is a torsion-free complex, give conditions under which the $\omega$-category Closed$_{WF}$(P$^H$) is isomorphic to the free $\omega$-category P$^*$. In order to define P$^H$, we will use the support function introduced by Makkai for free strict categories [16]. Given a polygraph $P$, this function maps a cell of P$^*$ to the set of generators “it involves”. We first recall its definition before dealing with the other matters.

Given a set $S$, we write $\mathcal{P}_f(S)$ for the set of finite subsets of $S$. Given $n \in \mathbb{N} \cup \{\omega\}$ and an $n$-polygraph $P$, we define the support function

$$ \mathrm{supp}^P : P^* \to \mathcal{P}_f(\bigcup_{i \in \mathbb{N}_n} P_i) $$

or simply, supp, as the unique function such that, given $u \in P^*$,

$$ - \mathrm{supp}(u) = \{g\} \text{ when } u = g \text{ for some } g \in P_0, $$

$$ - \mathrm{supp}(u) = \{g\} \cup \mathrm{supp}(\partial_{k-1}^{-}(g)) \cup \mathrm{supp}(\partial_{k-1}^{+}(u)) \text{ when } u = g \text{ for some } k \in \mathbb{N}_{n}^{*} \text{ and } g \in P_{k}, $$

$$ - \mathrm{supp}(u) = \mathrm{supp}(u') \text{ when } u = \mathrm{id}_{u'} \text{ for some } u' \in P^*, $$

$$ - \mathrm{supp}(u) = \mathrm{supp}(u_1) \cup \mathrm{supp}(u_2) \text{ when } u = u_1 *_i u_2 \text{ for some } i, k \in \mathbb{N}_n^* \text{ with } i < k \text{ and } i \text{-composable } u_1, u_2 \in P_k^* $$

The above definition completely defines supp, since the cells of P$^*$ are precisely the classes of formal composites of generators of P. It can moreover be shown well-defined (see the original proof of Makkai [16, Lemma 5] or [7, Proposition 2.4.3.2]).

Given $P \in \mathrm{Pol}_\omega$, we define an $\omega$-hypergraph $P^H$ by putting $P_n^H = P_n$ for $n \in \mathbb{N}$ and, when $n > 0$,

$$ g^{-} = \mathrm{supp}(d_{n-1}^{-}(g)) \cap P_{n-1} \qquad g^{+} = \mathrm{supp}(d_{n-1}^{+}(g)) \cap P_{n-1} $$

for $g \in P_n^H$. Under this definition, supp$^P$ can be seen as a function $P^* \to \mathrm{Closed}(P^H)$. We then have the following criterion to know whether $P^*$ can be faithfully represented by the closed-well-formed fgs's of $P^H$:

**Theorem 3.1.22.** Let $P \in \mathrm{Pol}_\omega$ such that $P^H$ is a torsion-free complex. Then, supp$^P$ is the underlying function of an $\omega$-functor $F: P^* \to \mathrm{Closed}_{WF}(P^H)$ if and only if, for $n \in \mathbb{N}^*$, $g \in P_n$ and $\epsilon \in \{-, +\}$, we have $\mathrm{supp}(d_{n-1}^{\epsilon}(g)) = \mathrm{R}(g^{\epsilon})$. In this case, $F$ is moreover an isomorphism.

*Remark 3.1.23.* If the condition of Theorem 3.1.22 is satisfied, then $T_{PC}^{\mathrm{Cl}} \circ F: P^* \to \mathrm{Cell}(P^H)$ is the unique isomorphism given by Corollary 2.4.2 which maps $g \in P$ to $\langle g \rangle \in \mathrm{Cell}(P^H)$.

*Proof.* If supp$^P$ induces an $\omega$-functor $F: P^* \to \mathrm{Closed}_{WF}(P^H)$, then we have

$$ \begin{align*} \mathrm{supp}(d_{n-1}^{\epsilon}(g)) &= F(d_{n-1}^{\epsilon}(g)) \\ &= \bar{\partial}_{n-1}^{\epsilon}(F(g)) \\ &= \bar{\partial}_{n-1}^{\epsilon}(\mathrm{R}(g)) \\ &= \mathrm{R}(g^{\epsilon}) \tag{by definition of $\bar{\partial}_{n-1}^{\epsilon}$} \end{align*} $$

which proves the necessity. For sufficiency, we prove by induction on $n \in \mathbb{N}$ that supp$^P$ is the underlying function of an $n$-functor $F^n: (P^*)_{\le n} \to \mathrm{Closed}_{WF}(P^H)_{\le n}$. This is clear for $n = 0$, and, when $n > 0$, we define $F^n$ by extending $F^{n-1}$ and so that $F^n(g) = \mathrm{R}(g)$ using the universal
property of $(\mathbb{P}^*)_{\le n} = (\mathbb{P}^*)_{\le n-1}[\mathbb{P}_n]$. This is possible since the condition of the statement implies that

$$F^{n-1}(d_{n-1}^{\epsilon}(g)) = \bar{\partial}_{n-1}^{\epsilon}(R(g))$$

for $g \in \mathbb{P}_n$ and $\epsilon \in \{-, +\}$. We then obtain an $\omega$-functor $F: \mathbb{P}^* \to \mathrm{Closed}_{\mathrm{WF}}(\mathbb{P}^H)$ using Remark 1.2.1, which satisfies that $F(g) = R(g)$ for $g \in \mathbb{P}$. Then, by Theorem 3.1.21, $T_{\mathrm{PC}}^{\mathrm{Cl}} \circ F$ is an $\omega$-functor $\mathbb{P}^* \to \mathrm{Cell}(P)$ which maps $g$ to $\langle g \rangle$. It is then an isomorphism by Corollary 2.4.2, so that $F$ is an isomorphism too. $\square$

*Example 3.1.24.* Let $\mathbb{P}$ be the $\omega$-polygraph with

$$\begin{align*}
\mathbb{P}_0 &= \{x, y, z\} & \mathbb{P}_1 &= \{f: x \to y, g, g': y \to z\} & \mathbb{P}_2 &= \{\alpha, \alpha': g \Rightarrow g'\} \\
\mathbb{P}_3 &= \{A: \mathrm{id}_f^2 *_0 \alpha \Rightarrow \mathrm{id}_f^2 *_0 \alpha'\}
\end{align*}$$

and $\mathbb{P}_k = \emptyset$ for $k \in \mathbb{N}$ with $k \ge 4$ as in

$$x \xrightarrow{f} y \alpha \downarrow \downarrow \alpha' \xrightarrow{g} z \quad \text{and} \quad x \xrightarrow{f*_{0g}} \downarrow \downarrow \mathrm{id}_f^2 *_0 \alpha \xrightarrow{f*_{0g'}} z \quad \Rightarrow \quad x \xrightarrow{f*_{0g}} \downarrow \downarrow \mathrm{id}_f^2 *_0 \alpha' \xrightarrow{f*_{0g'}} z .$$

We can verify that $\mathbb{P}^H$ is a torsion-free complex. But, by Theorem 3.1.22, the function $\mathrm{supp}^P$ does not induce an $\omega$-functor $\mathbb{P}^* \to \mathrm{Closed}_{\mathrm{WF}}(\mathbb{P}^H)$ since

$$\mathrm{supp}(d_2^-(A)) = \{x, y, z, f, g, g', \alpha\} \neq \{y, z, g, g', \alpha\} = \mathrm{R}(A^-).$$

However, by considering a modified version of $\mathbb{P}$ where $\mathbb{P}_3 = \{A: \alpha \Rightarrow \alpha'\}$ it can be verified that $\mathbb{P}^H$ is still a torsion-free complex and that, by Theorem 3.1.22, the function $\mathrm{supp}^P$ induces an $\omega$-functor $\mathbb{P}^* \to \mathrm{Closed}_{\mathrm{WF}}(\mathbb{P}^H)$ which is an isomorphism.

**3.2 Embedding parity complexes** In this section, we show that parity complexes are a particular case of torsion-free complexes, under two reasonable caveats. Firstly, since parity complexes do not require all the generators to be relevant, there are parity complexes that are not torsion-free complexes. But, by [23, Theorem 4.2], irrelevant generators of a parity complex $P$ do not play any role in the generated $\omega$-category $\mathrm{Cell}(P)$, so that, by restraining $P$ to the $\omega$-hypergraph $\bar{P}$ of relevant generators, we have $\mathrm{Cell}(P) = \mathrm{Cell}(\bar{P})$. Thus, it is reasonable to assume that all the parity complexes we are considering for embedding in torsion-free complexes have relevant generators, i.e., satisfy Axiom (T2). Secondly, as discussed in Section 1.4, general parity complexes are not freely generated by their atoms and, since the latter property is supposed to be the *raison d’être* of such structures, it is reasonable to only consider the parity complexes that satisfy this property. We believe that Axiom (T4) is the minimal additional condition to require for the $\omega$-category of cells of a parity complex to be freely generated, so we will only consider parity complexes that moreover satisfy Axiom (T4).

Under the assumptions given above, we are only left to derive Axiom (T3) from the axioms of a parity complex. We show below that it is essentially a consequence of the tightness requirements stated by Axiom (C5). First, we recall from [24] the link between tightness and the segment property:
property of $(\mathbb{P}^*)_{\le n} = (\mathbb{P}^*)_{\le n-1}[\mathbb{P}_n]$. This is possible since the condition of the statement implies that

$$F^{n-1}(d_{n-1}^{\epsilon}(g)) = \bar{\partial}_{n-1}^{\epsilon}(R(g))$$

for $g \in \mathbb{P}_n$ and $\epsilon \in \{-, +\}$. We then obtain an $\omega$-functor $F: \mathbb{P}^* \to \mathrm{Closed}_{\mathrm{WF}}(\mathbb{P}^H)$ using Remark 1.2.1, which satisfies that $F(g) = R(g)$ for $g \in \mathbb{P}$. Then, by Theorem 3.1.21, $T_{\mathrm{PC}}^{\mathrm{Cl}} \circ F$ is an $\omega$-functor $\mathbb{P}^* \to \mathrm{Cell}(\mathbb{P})$ which maps $g$ to $\langle g \rangle$. It is then an isomorphism by Corollary 2.4.2, so that $F$ is an isomorphism too. $\square$

*Example 3.1.24.* Let $\mathbb{P}$ be the $\omega$-polygraph with

$$\begin{align*}
\mathbb{P}_0 &= \{x, y, z\} & \mathbb{P}_1 &= \{f: x \to y g, g': y \to z\} & \mathbb{P}_2 &= \{\alpha, \alpha': g \Rightarrow g'\} \\
\mathbb{P}_3 &= \{A: \mathrm{id}_f^2 *_0 \alpha \Rightarrow \mathrm{id}_f^2 *_0 \alpha'\}
\end{align*}$$

and $\mathbb{P}_k = \emptyset$ for $k \in \mathbb{N}$ with $k \ge 4$ as in

$$x \xrightarrow{f} \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] 
    \node (top) at (0,1) {$g$};
    \node (bottom) at (0,0) {$g'$};
    \draw[->] (top) -- (bottom);
    \node at (0.5,0.5) {$\alpha \Downarrow \Downarrow \alpha' z$};
    \node at (-0.5,0.5) {$y$};
    \node at (0.5,-0.5) {$z$};
    \draw[->] (top) -- (bottom);
    \node at (0,0) {$\alpha$};
    \node at (0,-0.5) {$g$};
    \draw[->] (bottom) -- (top);
    \node at (0,0) {$g'$};
    \end{tikzpicture} \quad \text{and} \quad \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] 
    \node (top) at (0,1) {$f$};
    \node (bottom) at (0,0) {$f$};
    \draw[->] (top) -- (bottom);
    \node at (0.5,0.5) {$\alpha *_0 \alpha'$};
    \node at (0.5,-0.5) {$\alpha$};
    \draw[->] (top) -- (bottom);
    \node at (0,0) {$\alpha$};
    \node at (0,-0.5) {$g'$};
    \end{tikzpicture} \quad \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] 
    \node (top) at (0,1) {$f$};
    \node (bottom) at (0,0) {$f$};
    \draw[->] (top) -- (bottom);
    \node at (0.5,0.5) {$\alpha *_0 \alpha'$};
    \node at (0.5,-0.5) {$\alpha$};
    \draw[->] (top) -- (bottom);
    \node at (0,0) {$\alpha$};
    \node at (0,-0.5) {$g'$};
    \end{tikzpicture} \quad \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] 
    \node (top) at (0,1) {$f$};
    \node (bottom) at (0,0) {$f$};
    \draw[->] (top) -- (bottom);
    \node at (0.5,0.5) {$\alpha *_0 \alpha'$};
    \node at (0.5,-0.5) {$\alpha$};
    \draw[->] (top) -- (bottom);
    \node at (0,0) {$\alpha$};
    \node at (0,-0.5) {$g'$};
    \end{tikzpicture} .
$$

We can verify that $\mathbb{P}^H$ is a torsion-free complex. But, by Theorem 3.1.22, the function $\mathrm{supp}^P$ does not induce an $\omega$-functor $\mathbb{P}^* \to \mathrm{Closed}_{\mathrm{WF}}(\mathbb{P}^H)$ since

$$\mathrm{supp}(\bar{d}_2(A)) = \{x, y, z, f, g, g', \alpha\} \neq \{y, z, g, g', \alpha\} = \mathrm{R}(A^{-}).$$

However, by considering a modified version of $\mathbb{P}$ where $\mathbb{P}_3 = \{A: \alpha \Rightarrow \alpha'\}$ it can be verified that $\mathbb{P}^H$ is still a torsion-free complex and that, by Theorem 3.1.22, the function $\mathrm{supp}^P$ induces an $\omega$-functor $\mathbb{P}^* \to \mathrm{Closed}_{\mathrm{WF}}(\mathbb{P}^H)$ which is an isomorphism.

**3.2 Embedding parity complexes** In this section, we show that parity complexes are a particular case of torsion-free complexes, under two reasonable caveats. Firstly, since parity complexes do not require all the generators to be relevant, there are parity complexes that are not torsion-free complexes. But, by [23, Theorem 4.2], irrelevant generators of a parity complex $\mathbb{P}$ do not play any role in the generated $\omega$-category $\mathrm{Cell}(\mathbb{P})$, so that, by restraining $\mathbb{P}$ to the $\omega$-hypergraph $\bar{P}$ of relevant generators, we have $\mathrm{Cell}(\mathbb{P}) = \mathrm{Cell}(\bar{P})$. Thus, it is reasonable to assume that all the parity complexes we are considering for embedding in torsion-free complexes have relevant generators, i.e., satisfy Axiom (T2). Secondly, as discussed in Section 1.4, general parity complexes are not freely generated by their atoms and, since the latter property is supposed to be the *raison d’être* of such structures, it is reasonable to only consider the parity complexes that satisfy this property. We believe that Axiom (T4) is the minimal additional condition to require for the $\omega$-category of cells of a parity complex to be freely generated, so we will only consider parity complexes that moreover satisfy Axiom (T4).

Under the assumptions given above, we are only left to derive Axiom (T3) from the axioms of a parity complex. We show below that it is essentially a consequence of the tightness requirements stated by Axiom (C5). First, we recall from [24] the link between tightness and the segment property:
**Proposition 3.2.1** ([24, Proposition 1.4]). Let $P$ be an $\omega$-hypergraph. For $n \in \mathbb{N}^*$, subsets $U, V \subseteq P_n$ with $U$ tight, $V$ fork-free and $U \subseteq V$, we have that $U$ is a segment for $\triangleleft_V$.

*Proof.* Let $x, y, z \in V$ such that $x, z \in U$ and $x \triangleleft_V y \triangleleft_V z$. Then, there is $w \in x^+ \cap y^-$. By definition of tightness, since $y \triangleleft_V z$, we have $y^- \cap U^\pm = \emptyset$. So there is $\bar{y} \in U$ such that $w \in \bar{y}^-$. Since $V$ is fork-free, $y = \bar{y}$. Hence, $U$ is a segment for $\triangleleft_V$. $\square$

Then, we show how to derive the segment property from the axioms of parity complexes:

**Lemma 3.2.2.** Let $P$ be a parity complex which satisfies Axiom (T2). Given $n \in \mathbb{N}$ and $x \in P_n$, $x$ satisfies the segment condition.

*Proof.* Let $k, n \in \mathbb{N}$ with $k < n$, $x \in P_n$ and $X$ be a $k$-cell. Suppose first that $\langle x \rangle_{k,-} \subseteq X_k$. By Axiom (C5), the set $\langle x \rangle_{k,-}$ is tight, so that, by Proposition 3.2.1, $\langle x \rangle_{k,-}$ is a segment for $\triangleleft_{X_k}$.

Now suppose that $\langle x \rangle_{k,+} \subseteq X_k$. By contradiction, assume that $\langle x \rangle_{k,+}$ is not a segment for $\triangleleft_{X_k}$. By definition of $\triangleleft_{X_k}$, there exist $p > 1$ and $u_0, \dots, u_p \in X_k$ such that

$$u_0, u_p \in \langle x \rangle_{k,+}, \quad u_1, \dots, u_{p-1} \notin \langle x \rangle_{k,+} \quad \text{and} \quad u_i \triangleleft_{X_k} u_{i+1}.$$

By definition of $\triangleleft_{X_k}^1$, there exist $z_0, \dots, z_{p-1}$ such that $z_i \in u_i^+ \cap u_{i+1}^-$. Note that $z_0 \in \langle x \rangle_{k,+}^\pm$. Indeed, if $z_0 \in v^-$ for some $v \in X_k$, then, since $X_k$ is fork-free, $v = u_1$, so $v \notin \langle x \rangle_{k,+}$. Similarly, we have $z_{p-1} \in \langle x \rangle_{k,+}^\mp$. Since $x$ is relevant by Axiom (T2), we have $\langle x \rangle_{k+1,+}^\pm = \langle x \rangle_{k,+} \subseteq X_k$. By [23, Lemma 3.2] (which is the analogous for parity complexes of Theorem 2.1.1) and Axiom (T2), we have that $\langle x \rangle_{k,-} \cap X_n \subseteq \langle x \rangle_{k+1,+}^- \cap X_n = \emptyset$ and the $k$-pre-cell $Y = \overline{\mathrm{Act}}(X, \langle x \rangle_{k+1,+})$ is a $k$-cell. Moreover, by Lemma A.1.6,

$$Y_k = (X_k \cup \langle x \rangle_{k+1,+}^-) \setminus \langle x \rangle_{k+1,+}^+ = (X_k \setminus \langle x \rangle_{k,+}) \cup \langle x \rangle_{k,-}.$$

Thus, $\langle x \rangle_{k,-} \subseteq Y_k$ and, similarly as above, $\langle x \rangle_{k,-}$ is a segment for $\triangleleft_{Y_k}$. Since $\langle x \rangle_{k,-}^\mp = \langle x \rangle_{k,+}^\mp$ and $\langle x \rangle_{k,-}^\pm = \langle x \rangle_{k,+}^\pm$, there exist $\tilde{u}_0, \tilde{u}_p \in \langle x \rangle_{k,-}$ such that $z_0 \in \tilde{u}_0^+$ and $z_{p-1} \in \tilde{u}_p^-$. So

$$\tilde{u}_0 \triangleleft_{X_k}^1 u_1 \triangleleft_{X_k}^1 \dots \triangleleft_{X_k}^1 u_{p-1} \triangleleft_{X_k}^1 \tilde{u}_p$$

with $u_1, \dots, u_{p-1} \notin \langle x \rangle_{k,-}$ (since $\langle x \rangle_{k+1,+}^- \cap X_n = \emptyset$), contradicting the fact that $\langle x \rangle_{k,-}$ is a segment for $\triangleleft_{Y_k}$. Thus, $\langle x \rangle_{k,+}$ is a segment for $\triangleleft_{X_k}$. Hence, $x$ satisfies the segment condition. $\square$

We conclude that parity complexes are embedded into torsion-free complexes:

**Theorem 3.2.3.** Given a parity complex $P$ which satisfies Axiom (T2) and Axiom (T4), $P$ is a torsion-free complex.

*Proof.* Axiom (T0) is a consequence of Axiom (C0). Axiom (T1) is a consequence of Axiom (C3). And Axiom (T3) is a consequence of Lemma 3.2.2. $\square$

**Remark 3.2.4.** Given $P$ as in Theorem 3.2.3, the $\omega$-category Cell($P$) of cells of the parity complex $P$ is, of course, exactly the $\omega$-category Cell($P$) of cells of the torsion-free complex $P$.
**Proposition 3.2.1** ([24, Proposition 1.4]). Let $P$ be an $\omega$-hypergraph. For $n \in \mathbb{N}^*$, subsets $U, V \subseteq P_n$ with $U$ tight, $V$ fork-free and $U \subseteq V$, we have that $U$ is a segment for $\triangleleft_V$.

*Proof.* Let $x, y, z \in V$ such that $x, z \in U$ and $x \triangleleft_V y \triangleleft_V z$. Then, there is $w \in x^+ \cap y^-$. By definition of tightness, since $y \triangleleft_V z$, we have $y^- \cap U^\pm = \emptyset$. So there is $\bar{y} \in U$ such that $w \in \bar{y}^-$. Since $V$ is fork-free, $y = \bar{y}$. Hence, $U$ is a segment for $\triangleleft_V$. $\square$

Then, we show how to derive the segment property from the axioms of parity complexes:

**Lemma 3.2.2.** Let $P$ be a parity complex which satisfies Axiom (T2). Given $n \in \mathbb{N}$ and $x \in P_n$, $x$ satisfies the segment condition.

*Proof.* Let $k, n \in \mathbb{N}$ with $k < n$, $x \in P_n$ and $X$ be a $k$-cell. Suppose first that $\langle x \rangle_{k,-} \subseteq X_k$. By Axiom (C5), the set $\langle x \rangle_{k,-}$ is tight, so that, by Proposition 3.2.1, $\langle x \rangle_{k,-}$ is a segment for $\triangleleft_{X_k}$.

Now suppose that $\langle x \rangle_{k,+} \subseteq X_k$. By contradiction, assume that $\langle x \rangle_{k,+}$ is not a segment for $\triangleleft_{X_k}$. By definition of $\triangleleft_{X_k}$, there exist $p > 1$ and $u_0, \dots, u_p \in X_k$ such that

$$u_0, u_p \in \langle x \rangle_{k,+}, \quad u_1, \dots, u_{p-1} \notin \langle x \rangle_{k,+} \quad \text{and} \quad u_i \triangleleft_{X_k} u_{i+1}.$$

By definition of $\triangleleft_{X_k}^1$, there exist $z_0, \dots, z_{p-1}$ such that $z_i \in u_i^+ \cap u_{i+1}^-$. Note that $z_0 \in \langle x \rangle_{k,+}^\pm$. Indeed, if $z_0 \in v^-$ for some $v \in X_k$, then, since $X_k$ is fork-free, $v = u_1$, so $v \notin \langle x \rangle_{k,+}$. Similarly, we have $z_{p-1} \in \langle x \rangle_{k,+}^\mp$. Since $x$ is relevant by Axiom (T2), we have $\langle x \rangle_{k+1,+}^\pm = \langle x \rangle_{k,+} \subseteq X_k$. By [23, Lemma 3.2] (which is the analogous for parity complexes of Theorem 2.1.1) and Axiom (T2), we have that $\langle x \rangle_{k,-} \cap X_n \subseteq \langle x \rangle_{k+1,+}^- \cap X_n = \emptyset$ and the $k$-pre-cell $Y = \overline{\text{Act}}(X, \langle x \rangle_{k+1,+})$ is a $k$-cell. Moreover, by Lemma A.1.6,

$$Y_k = (X_k \cup \langle x \rangle_{k+1,+}^-) \setminus \langle x \rangle_{k+1,+}^+ = (X_k \setminus \langle x \rangle_{k,+}) \cup \langle x \rangle_{k,-}.$$

Thus, $\langle x \rangle_{k,-} \subseteq Y_k$ and, similarly as above, $\langle x \rangle_{k,-}$ is a segment for $\triangleleft_{Y_k}$. Since $\langle x \rangle_{k,-}^\mp = \langle x \rangle_{k,+}^\mp$ and $\langle x \rangle_{k,-}^\pm = \langle x \rangle_{k,+}^\pm$, there exist $\tilde{u}_0, \tilde{u}_p \in \langle x \rangle_{k,-}$ such that $z_0 \in \tilde{u}_0^+$ and $z_{p-1} \in \tilde{u}_p^-$. So

$$\tilde{u}_0 \triangleleft_{X_k}^1 u_1 \triangleleft_{X_k}^1 \dots \triangleleft_{X_k}^1 u_{p-1} \triangleleft_{X_k}^1 \tilde{u}_p$$

with $u_1, \dots, u_{p-1} \notin \langle x \rangle_{k,-}$ (since $\langle x \rangle_{k+1,+}^- \cap X_n = \emptyset$), contradicting the fact that $\langle x \rangle_{k,-}$ is a segment for $\triangleleft_{Y_k}$. Thus, $\langle x \rangle_{k,+}$ is a segment for $\triangleleft_{X_k}$. Hence, $x$ satisfies the segment condition. $\square$

We conclude that parity complexes are embedded into torsion-free complexes:

**Theorem 3.2.3.** Given a parity complex $P$ which satisfies Axiom (T2) and Axiom (T4), $P$ is a torsion-free complex.

*Proof.* Axiom (T0) is a consequence of Axiom (C0). Axiom (T1) is a consequence of Axiom (C3). And Axiom (T3) is a consequence of Lemma 3.2.2. $\square$

**Remark 3.2.4.** Given $P$ as in Theorem 3.2.3, the $\omega$-category Cell($P$) of cells of the parity complex $P$ is, of course, exactly the $\omega$-category Cell($P$) of cells of the torsion-free complex $P$.
**3.3 Embedding pasting schemes** In this section, we show that loop-free pasting schemes are a particular case of torsion-free complexes, under the caveat that we only consider loop-free pasting schemes that satisfy Axiom (T4) since, like for parity complexes, loop-free pasting schemes do not induce free $\omega$-categories in general. We think that it is a reasonable requirement since we also believe that Axiom (T4) is the minimal additional condition to add to the axioms of loop-free pasting schemes for this property to hold.

In order to embed pasting schemes into torsion-free complexes, our main concerns will be to derive Axioms (T2) and (T3) from Axioms (S3) and (S4). For this purpose, we will need to relate the cells of torsion-free complexes with the wfs's (defined in Section 1.5), using closed-well-formed fgs's (defined in Section 3.1) as an intermediate. In fact, we will prove that the latter are exactly the wfs's. First, we prove a technical result about the relations B and E:

**Lemma 3.3.1.** Let P be a pasting scheme, $k, n \in \mathbb{N}$ with $k < n$, $x \in P_n$ and $y \in P_k$. If $xB_{n-1}^n R_k^{n-1}y$ then $y \in B_k^n(x)$ or $xE_{n-1}^n R_k^{n-1}y$. Dually, if $xE_{n-1}^n R_k^{n-1}y$ then $y \in E_k^n(x)$ or $xB_{n-1}^n R_k^{n-1}y$.

*Proof.* We do an induction on $n-k$. If $k=n-1$, the result is trivial. If $k=n-2$, the result is a consequence of Axiom (S1). So suppose that $k < n-2$. We will only prove the first part, since the second is dual. So assume that $y \notin B_k^n(x)$. By the definition of B, we have

$$ \neg(x B_{n-1}^n B_k^{n-1} y) \text{ or } \neg(x B_{n-1}^n E_k^{n-1} y). $$

By symmetry, we can suppose that $\neg(x B_{n-1}^n E_k^{n-1} y)$. Let $u \in P_{n-1}$ be minimal for $\triangleleft$ such that $xB_{n-1}^n u R_k^{n-1} y$. Then, there are two possible cases: either $u B_{n-2}^{n-1} R_k^{n-2} y$ or $u E_{n-2}^{n-1} R_k^{n-2} y$.

In the first case, let $v \in P_{n-2}$ be such that $u B_{n-2}^{n-1} v R_k^{n-2} y$. By the minimality of $u$, we have $\neg(x B_{n-1}^n E_{n-2}^{n-1} v)$, so $\neg(x B_{n-2}^n v)$ by definition of B. By Axiom (S1), we have $x E_{n-1}^n E_{n-2}^{n-1} v$. So $x E_{n-1}^n R_k^{n-1} y$.

In the second case, since we supposed $\neg(x B_{n-1}^n E_k^{n-1} y)$, we have $\neg(u E_k^{n-1} y)$. By induction hypothesis, we deduce $u B_{n-2}^{n-1} R_k^{n-2} y$ and we can conclude using the first case. $\square$

Then, we prove that the source and target of wfs's computed by the operations defined for pasting schemes in Section 1.5 are the same as the ones computed with the operations defined for closed fgs's in Section 3.1:

**Lemma 3.3.2.** Let P be a loop-free pasting scheme. Given $n \in \mathbb{N}^*$, $\epsilon \in \{-, +\}$ and an n-wfs X of P, we have $\partial_{n-1}^\epsilon(X) = \bar{\partial}_{n-1}^\epsilon(X)$.

*Proof.* We only prove the case $\epsilon = -$. Recall that

$$ \partial_{n-1}^{-}(X) = X \setminus E(X) \quad \text{and} \quad \bar{\partial}_{n-1}^{-}(X) = R(X \setminus (X_n \cup R(X_n^{+}))). $$

We first prove $\bar{\partial}_{n-1}^{-}(X) \subseteq \partial_{n-1}^{-}(X)$, that is,

$$ R(X \setminus (X_n \cup R(X_n^{+}))) \subseteq X \setminus E(X). $$

Since $X \setminus E(X)$ is closed (by [13, Theorem 12]), it is equivalent to $X \setminus (X_n \cup R(X_n^{+})) \subseteq X \setminus E(X)$ which is itself equivalent to $E(X) \subseteq (X_n \cup R(X_n^{+}))$ which holds. We now prove that we have $\partial_{n-1}^{-}(X) \subseteq \bar{\partial}_{n-1}^{-}(X)$, that is,

$$ X \setminus E(X) \subseteq R(X \setminus (X_n \cup R(X_n^{+}))) = \bar{\partial}_{n-1}^{-}(X). $$
**3.3 Embedding pasting schemes** In this section, we show that loop-free pasting schemes are a particular case of torsion-free complexes, under the caveat that we only consider loop-free pasting schemes that satisfy Axiom (T4) since, like for parity complexes, loop-free pasting schemes do not induce free $\omega$-categories in general. We think that it is a reasonable requirement since we also believe that Axiom (T4) is the minimal additional condition to add to the axioms of loop-free pasting schemes for this property to hold.

In order to embed pasting schemes into torsion-free complexes, our main concerns will be to derive Axioms (T2) and (T3) from Axioms (S3) and (S4). For this purpose, we will need to relate the cells of torsion-free complexes with the wfs's (defined in Section 1.5), using closed-well-formed fgs's (defined in Section 3.1) as an intermediate. In fact, we will prove that the latter are exactly the wfs's. First, we prove a technical result about the relations B and E:

**Lemma 3.3.1.** Let $P$ be a pasting scheme, $k, n \in \mathbb{N}$ with $k < n$, $x \in P_n$ and $y \in P_k$. If $xB_{n-1}^n R_k^{n-1}y$ then $y \in B_k^n(x)$ or $xE_{n-1}^n R_k^{n-1}y$. Dually, if $xE_{n-1}^n R_k^{n-1}y$ then $y \in E_k^n(x)$ or $xB_{n-1}^n R_k^{n-1}y$.

*Proof.* We do an induction on $n-k$. If $k=n-1$, the result is trivial. If $k=n-2$, the result is a consequence of Axiom (S1). So suppose that $k < n-2$. We will only prove the first part, since the second is dual. So assume that $y \notin B_k^n(x)$. By the definition of B, we have

$$ \neg(x B_{n-1}^n B_k^{n-1} y) \text{ or } \neg(x B_{n-1}^n E_k^{n-1} y). $$

By symmetry, we can suppose that $\neg(x B_{n-1}^n E_k^{n-1} y)$. Let $u \in P_{n-1}$ be minimal for $\triangleleft$ such that $xB_{n-1}^n u R_k^{n-1} y$. Then, there are two possible cases: either $u B_{n-2}^{n-1} R_k^{n-2} y$ or $u E_{n-2}^{n-1} R_k^{n-2} y$.

In the first case, let $v \in P_{n-2}$ be such that $u B_{n-2}^{n-1} v R_k^{n-2} y$. By the minimality of $u$, we have $\neg(x B_{n-1}^n E_{n-2}^{n-1} v)$, so $\neg(x B_{n-2}^n v)$ by definition of B. By Axiom (S1), we have $x E_{n-1}^n E_{n-2}^{n-1} v$. So $x E_{n-1}^n R_k^{n-1} y$.

In the second case, since we supposed $\neg(x B_{n-1}^n E_k^{n-1} y)$, we have $\neg(u E_k^{n-1} y)$. By induction hypothesis, we deduce $u B_{n-2}^{n-1} R_k^{n-2} y$ and we can conclude using the first case. $\square$

Then, we prove that the source and target of wfs's computed by the operations defined for pasting schemes in Section 1.5 are the same as the ones computed with the operations defined for closed fgs's in Section 3.1:

**Lemma 3.3.2.** Let $P$ be a loop-free pasting scheme. Given $n \in \mathbb{N}^*$, $\epsilon \in \{-, +\}$ and an $n$-wfs $X$ of $P$, we have $\partial_{n-1}^\epsilon(X) = \bar{\partial}_{n-1}^\epsilon(X)$.

*Proof.* We only prove the case $\epsilon = -$. Recall that

$$ \partial_{n-1}^{-}(X) = X \setminus E(X) \quad \text{and} \quad \bar{\partial}_{n-1}^{-}(X) = R(X \setminus (X_n \cup R(X_n^{+}))). $$

We first prove $\bar{\partial}_{n-1}^{-}(X) \subseteq \partial_{n-1}^{-}(X)$, that is,

$$ R(X \setminus (X_n \cup R(X_n^{+}))) \subseteq X \setminus E(X). $$

Since $X \setminus E(X)$ is closed (by [13, Theorem 12]), it is equivalent to $X \setminus (X_n \cup R(X_n^{+})) \subseteq X \setminus E(X)$ which is itself equivalent to $E(X) \subseteq (X_n \cup R(X_n^{+}))$ which holds. We now prove that we have $\partial_{n-1}^{-}(X) \subseteq \bar{\partial}_{n-1}^{-}(X)$, that is,

$$ X \setminus E(X) \subseteq R(X \setminus (X_n \cup R(X_n^{+}))) = \bar{\partial}_{n-1}^{-}(X). $$
Let $k \in \mathbb{N}_{n-1}$ and $x \in (X \setminus E(X))_k$. If $x \notin R(X_n^+)$ then $x \in \bar{\partial}_{n-1}^-(X)$. So suppose that $x \in R(X_n^+)$. Since $E(X)_{n-1} = X_n^+$, it implies that $k < n-1$. By definition of $R(X_n^+)$, there exists $y \in X_n$ such that $y E_{n-1}^n R_k^{n-1} x$ and, by Axiom (S2), we can take $y$ minimal for $\triangleleft$ satisfying this property. By Lemma 3.3.1, it holds that $y B_{n-1}^n R_k^{n-1} x$. Let $z \in P_{n-1}$ be such that $y B_{n-1}^n z R_k^{n-1} x$. Then, there is no $\bar{y} \in X_n$ such that $\bar{y} E_{n-1}^n z$: otherwise, $\bar{y} E_{n-1}^n R_k^{n-1} x$ and $\bar{y} \triangleleft y$, contradicting the minimality of $y$. So $z \notin R(X_n^+)$ and $z R x$. It implies that $z \in X \setminus (X_n \cup R(X_n^+))$ and $x \in \bar{\partial}_{n-1}^-(X)$. $\square$

We can then prove the inclusion of wfs's into closed-well-formed fgs's:

**Proposition 3.3.3.** Let $P$ be a loop-free pasting scheme. Given $n \in \mathbb{N}$ and $X \in \text{WF}(P)_n$, we have $X \in \text{Closed}_{\text{WF}}(P)_n$.

*Proof.* We prove this lemma by induction on $n$. If $n=0$, the result is trivial. So suppose $n>0$. Since $X$ is well-formed, $X_n$ is fork-free. Moreover, by Lemma 3.3.2, for $\epsilon \in \{-, +\}$, we have that $\bar{\partial}_{n-1}^\epsilon(X) = \partial_{n-1}^\epsilon(X)$ is a well-formed $(n-1)$-fgs. By induction, $\bar{\partial}_{n-1}^\epsilon(X) \in \text{Closed}_{\text{WF}}(P)_{n-1}$. Moreover, when $n \ge 2$, since $\partial_{n-2}^\epsilon \circ \partial_{n-1}^\epsilon(X) = \partial_{n-2}^\epsilon \circ \partial_{n-1}^+(X)$, by Lemma 3.3.2,

$$ \bar{\partial}_{n-2}^{\epsilon} \circ \bar{\partial}_{n-1}^{-}(X) = \bar{\partial}_{n-2}^{\epsilon} \circ \bar{\partial}_{n-1}^{+}(X). $$

Hence, $X \in \text{Closed}_{\text{WF}}(P)_n$. $\square$

Next, we prove an analogue of the gluing Theorem 2.1.1 for wfs's:

**Lemma 3.3.4.** Let $P$ be a loop-free pasting scheme, $n \in \mathbb{N}$, $X$ be an $n$-wfs, $S \subseteq P_{n+1}$ be a finite subset with $S$ fork-free and $S^\mp \subseteq X$, and $Y = X \cup R(S)$. Then, $Y$ is an $(n+1)$-wfs of $P$ and $\partial_n^-(Y) = X$.

*Proof.* We show this lemma by induction on $k = |S|$. If $k=0$, the result is trivial. If $k=1$, the result is a consequence of [13, Proposition 8]. So suppose that $k > 1$. By Axiom (S2), take $x \in S$ minimal for $\triangleleft$. By minimality, we have $x^- \subseteq S^\mp \subseteq X$. Using [13, Proposition 8], $X \cup R(x)$ is well-formed. By Axiom (S5), $X \cap E(x) = \emptyset$, so we have that $\partial_n^-(X \cup R(x)) = X$. Let

$$ \bar{X} = \partial_n^+(X \cup R(x)) \quad \text{and} \quad \bar{S} = S \setminus \{x\}. $$

We have

$$ \begin{aligned} \bar{S}^{\mp} \subseteq \bar{X}_n &\Leftrightarrow \bar{S}^{-} \subseteq \bar{X}_n \cup \bar{S}^{+} &\Leftrightarrow S^{-} \subseteq \bar{X}_n \cup \bar{S}^{+} \cup x^{-} \\ &\Leftrightarrow S^{-} \subseteq (X_n \setminus x^{-}) \cup x^{+} \cup \bar{S}^{+} \cup x^{-} &\Leftrightarrow S^{-} \subseteq X_n \cup S^{+} &\Leftrightarrow S^{\mp} \subseteq X_n \end{aligned} $$

so $\bar{S}^{\mp} \subseteq \bar{X}$. By induction, $\bar{X} \cup R(\bar{S})$ is well-formed and $\partial_n^-(\bar{X} \cup R(\bar{S})) = \bar{X}$. Since WF($P$) has the structure of an $\omega$-category by [13, Theorem 12], we can compose $X \cup R(x)$ and $\bar{X} \cup R(\bar{S})$. So

$$ X \cup R(S) = X \cup R(x) \cup \bar{X} \cup R(\bar{S}) $$

is well-formed and $\partial_n^-(X \cup R(S)) = X$. $\square$

We can now prove the converse inclusion of closed-well-formed fgs's into wfs's:

**Proposition 3.3.5.** Let $P$ be a loop-free pasting scheme. Given $n \in \mathbb{N}$ and $X \in \text{Closed}_{\text{WF}}(P)_n$, we have $X \in \text{WF}(P)_n$.
Let $k \in \mathbb{N}_{n-1}$ and $x \in (X \setminus E(X))_k$. If $x \notin R(X_n^+)$ then $x \in \bar{\partial}_{n-1}^-(X)$. So suppose that $x \in R(X_n^+)$. Since $E(X)_{n-1} = X_n^+$, it implies that $k < n-1$. By definition of $R(X_n^+)$, there exists $y \in X_n$ such that $y E_{n-1}^n R_k^{n-1} x$ and, by Axiom (S2), we can take $y$ minimal for $\triangleleft$ satisfying this property. By Lemma 3.3.1, it holds that $y B_{n-1}^n R_k^{n-1} x$. Let $z \in P_{n-1}$ be such that $y B_{n-1}^n z R_k^{n-1} x$. Then, there is no $\bar{y} \in X_n$ such that $\bar{y} E_{n-1}^n z$: otherwise, $\bar{y} E_{n-1}^n R_k^{n-1} x$ and $\bar{y} \triangleleft y$, contradicting the minimality of $y$. So $z \notin R(X_n^+)$ and $z R x$. It implies that $z \in X \setminus (X_n \cup R(X_n^+))$ and $x \in \bar{\partial}_{n-1}^-(X)$. $\square$

We can then prove the inclusion of wfs's into closed-well-formed fgs's:

**Proposition 3.3.3.** Let $P$ be a loop-free pasting scheme. Given $n \in \mathbb{N}$ and $X \in \mathrm{WF}(P)_n$, we have $X \in \mathrm{Closed}_{\mathrm{WF}}(P)_n$.

*Proof.* We prove this lemma by induction on $n$. If $n=0$, the result is trivial. So suppose $n>0$. Since $X$ is well-formed, $X_n$ is fork-free. Moreover, by Lemma 3.3.2, for $\epsilon \in \{-, +\}$, we have that $\bar{\partial}_{n-1}^{\epsilon}(X) = \partial_{n-1}^{\epsilon}(X)$ is a well-formed $(n-1)$-fgs. By induction, $\bar{\partial}_{n-1}^{\epsilon}(X) \in \mathrm{Closed}_{\mathrm{WF}}(P)_{n-1}$. Moreover, when $n \ge 2$, since $\partial_{n-2}^{\epsilon} \circ \partial_{n-1}^{-}(X) = \partial_{n-2}^{\epsilon} \circ \partial_{n-1}^{+}(X)$, by Lemma 3.3.2,

$$ \bar{\partial}_{n-2}^{\epsilon} \circ \bar{\partial}_{n-1}^{-}(X) = \bar{\partial}_{n-2}^{\epsilon} \circ \bar{\partial}_{n-1}^{+}(X). $$

Hence, $X \in \mathrm{Closed}_{\mathrm{WF}}(P)_n$. $\square$

Next, we prove an analogue of the gluing Theorem 2.1.1 for wfs's:

**Lemma 3.3.4.** Let $P$ be a loop-free pasting scheme, $n \in \mathbb{N}$, $X$ be an $n$-wfs, $S \subseteq P_{n+1}$ be a finite subset with $S$ fork-free and $S^{\mp} \subseteq X$, and $Y = X \cup R(S)$. Then, $Y$ is an $(n+1)$-wfs of $P$ and $\partial_n^-(Y) = X$.

*Proof.* We show this lemma by induction on $k = |S|$. If $k=0$, the result is trivial. If $k=1$, the result is a consequence of [13, Proposition 8]. So suppose that $k > 1$. By Axiom (S2), take $x \in S$ minimal for $\triangleleft$. By minimality, we have $x^- \subseteq S^{\mp} \subseteq X$. Using [13, Proposition 8], $X \cup R(x)$ is well-formed. By Axiom (S5), $X \cap E(x) = \emptyset$, so we have that $\partial_n^-(X \cup R(x)) = X$. Let

$$ \bar{X} = \partial_n^+(X \cup R(x)) \quad \text{and} \quad \bar{S} = S \setminus \{x\}. $$

We have

$$ \begin{align*} \bar{S}^{\mp} \subseteq \bar{X}_n &\Leftrightarrow \bar{S}^{-} \subseteq \bar{X}_n \cup \bar{S}^{+} &\Leftrightarrow S^{-} \subseteq \bar{X}_n \cup \bar{S}^{+} \cup x^{-} \\ &\Leftrightarrow S^{-} \subseteq (X_n \setminus x^{-}) \cup x^{+} \cup \bar{S}^{+} \cup x^{-} &\Leftrightarrow S^{-} \subseteq X_n \cup S^{+} &\Leftrightarrow S^{\mp} \subseteq X_n \end{align*} $$

so $\bar{S}^{\mp} \subseteq \bar{X}$. By induction, $\bar{X} \cup R(\bar{S})$ is well-formed and $\partial_n^-(\bar{X} \cup R(\bar{S})) = \bar{X}$. Since $\mathrm{WF}(P)$ has the structure of an $\omega$-category by [13, Theorem 12], we can compose $X \cup R(x)$ and $\bar{X} \cup R(\bar{S})$. So

$$ X \cup R(S) = X \cup R(x) \cup \bar{X} \cup R(\bar{S}) $$

is well-formed and $\partial_n^-(X \cup R(S)) = X$. $\square$

We can now prove the converse inclusion of closed-well-formed fgs's into wfs's:

**Proposition 3.3.5.** Let $P$ be a loop-free pasting scheme. Given $n \in \mathbb{N}$ and $X \in \mathrm{Closed}_{\mathrm{WF}}(P)_n$, we have $X \in \mathrm{WF}(P)_n$.
**Proof.** We prove this lemma by induction on $n$. If $n=0$, the result is trivial. So suppose $n>0$. Let $Y = \bar{\partial}_{n-1}^{-}(X)$. By definition of Closed<sub>WF</sub>(P), $Y \in$ Closed<sub>WF</sub>(P) and, by induction, $Y \in$ WF(P). By definition of $\bar{\partial}^{-}$, we have $X_{n}^{\mp} \subseteq Y$. Moreover, by Lemma 3.3.4, $Y \cup R(X_{n})$ is well-formed. But $Y = R(X \setminus (X_{n} \cup R(X_{n}^{+})))$, so that $X = Y \cup R(X_{n})$ is well-formed. $\square$

We now give a simple form for the sources and targets of atomic wfs's:

**Lemma 3.3.6.** Let P be a loop-free pasting scheme. Given $i, n \in \mathbb{N}$ such that $i < n$, $\epsilon \in \{-, +\}$ and $x \in P_n$, we have $\partial_i^{\epsilon}(R(x)) = R(\langle x \rangle_{i, \epsilon})$.

*Proof.* By symmetry, we can suppose that $\epsilon = -$. We compute that

$$ 
\begin{align*}
\partial_i^{-}(R(x)) &= \partial_i^{-}(T_{\text{Cl}}^{\text{M}}(\{x\})) \\
&= T_{\text{Cl}}^{\text{M}}(\tilde{\partial}_i^{-}(\{x\})) \quad \text{(by Proposition 3.1.8 and Lemma 3.3.2)} \\
&= T_{\text{Cl}}^{\text{M}}(\langle x \rangle_{i,-}) = R(\langle x \rangle_{i,-}). \quad \square
\end{align*}
$$

Using the above lemma, we deduce the relevance of the generators:

**Lemma 3.3.7.** Let P be a loop-free pasting scheme. Given $n \in \mathbb{N}$ and $x \in P_n$, $x$ is relevant.

*Proof.* By Axiom (S3), $R(x)$ is well-formed. So, for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$, $\partial_i^{\epsilon}(R(x))$ is well-formed. Then, by Lemma 3.3.6, $\langle x \rangle_{i,-}$ and $\langle x \rangle_{i,+}$ are fork-free. We show that $\langle x \rangle_{i+1,-}^{\pm} = \langle x \rangle_{i,+}$ and $\langle x \rangle_{i+1,+}^{\mp} = \langle x \rangle_{i,-}$. We have

$$ \langle x \rangle_{n,-}^{\pm} = \langle x \rangle^{\pm} = x^{+} = \langle x \rangle_{n-1,+} $$

and, similarly, $\langle x \rangle_{n,+}^{\mp} = \langle x \rangle_{n-1,-}$. For $i \in \mathbb{N}_{n-1}$, we have

$$ 
\begin{align*}
\langle x \rangle_{i+1,-}^{\pm} &= (\partial_i^{+}(R(\langle x \rangle_{i+1,-})))_i && \text{(by definition of } \partial_i^{+}) \\
&= (\partial_i^{+} \circ \partial_{i+1}^{-}(R(x)))_i && \text{(by Lemma 3.3.6)} \\
&= (\partial_i^{+}(R(x)))_i && \text{(by globularity)} \\
&= (R(\langle x \rangle_{i,+}))_i && \text{(by Lemma 3.3.6)} \\
&= \langle x \rangle_{i,+}
\end{align*}
$$

and similarly, $\langle x \rangle_{i+1,+}^{\mp} = \langle x \rangle_{i,-}$. Moreover, we have

$$ 
\begin{align*}
(\langle x \rangle_{i,-} \cup \langle x \rangle_{i+1,-}^{+}) \setminus \langle x \rangle_{i+1,-}^{-} &= ((\langle x \rangle_{i+1,-}^{-} \setminus \langle x \rangle_{i+1,-}^{+}) \cup \langle x \rangle_{i+1,-}^{+}) \setminus \langle x \rangle_{i+1,-}^{-} \\
&= \langle x \rangle_{i+1,-}^{+} \setminus \langle x \rangle_{i+1,-}^{-} = \langle x \rangle_{i,+}
\end{align*}
$$

and similarly $(\langle x \rangle_{i,+} \cup \langle x \rangle_{i+1,-}^{-}) \setminus \langle x \rangle_{i+1,-}^{+} = \langle x \rangle_{i,-}$. Thus, $\langle x \rangle_{i+1,-}$ moves $\langle x \rangle_{i,-}$ to $\langle x \rangle_{i,+}$ and so does $\langle x \rangle_{i+1,+}$. Hence, $\langle x \rangle$ is a cell. $\square$

We now prove that the cells (in the sense of Section 1.4) of pasting schemes are sent to wfs's by $T_{\text{Cl}}^{\text{PC}}$, and that all the generators satisfy the segment condition:

**Lemma 3.3.8.** Let P be a loop-free pasting scheme and $n \in \mathbb{N}$. The following hold:

(i) for $x \in P_n$, $x$ satisfies the segment condition,

(ii) for $X \in \text{Cell}(P)_n$, $T_{\text{Cl}}^{\text{PC}}(X) \in \text{WF}(P)_n$.
**Proof.** We prove this lemma by induction on $n$. If $n = 0$, the result is trivial. So suppose $n > 0$. Let $Y = \bar{\partial}_{n-1}^{-}(X)$. By definition of Closed<sub>WF</sub>(P), $Y \in$ Closed<sub>WF</sub>(P) and, by induction, $Y \in$ WF(P). By definition of $\bar{\partial}^{-}$, we have $X_{n}^{\mp} \subseteq Y$. Moreover, by Lemma 3.3.4, $Y \cup R(X_{n})$ is well-formed. But $Y = R(X \setminus (X_{n} \cup R(X_{n}^{+})))$, so that $X = Y \cup R(X_{n})$ is well-formed. $\square$

We now give a simple form for the sources and targets of atomic wfs's:

**Lemma 3.3.6.** Let P be a loop-free pasting scheme. Given $i, n \in \mathbb{N}$ such that $i < n$, $\epsilon \in \{-, +\}$ and $x \in P_n$, we have $\partial_i^{\epsilon}(R(x)) = R(\langle x \rangle_{i, \epsilon})$.

*Proof.* By symmetry, we can suppose that $\epsilon = -$. We compute that

$$ 
\begin{align*}
\partial_i^{-}(R(x)) &= \partial_i^{-}(T_{\mathrm{Cl}}^{\mathrm{M}}(\{x\})) \\
&= T_{\mathrm{Cl}}^{\mathrm{M}}(\tilde{\partial}_i^{-}(\{x\})) \tag*{(by Proposition 3.1.8 and Lemma 3.3.2)} \\
&= T_{\mathrm{Cl}}^{\mathrm{M}}(\langle x \rangle_{i,-}) = R(\langle x \rangle_{i,-}). \tag*{\square}
\end{align*}
$$

Using the above lemma, we deduce the relevance of the generators:

**Lemma 3.3.7.** Let P be a loop-free pasting scheme. Given $n \in \mathbb{N}$ and $x \in P_n$, $x$ is relevant.

*Proof.* By Axiom (S3), $R(x)$ is well-formed. So, for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$, $\partial_i^{\epsilon}(R(x))$ is well-formed. Then, by Lemma 3.3.6, $\langle x \rangle_{i,-}$ and $\langle x \rangle_{i,+}$ are fork-free. We show that $\langle x \rangle_{i+1,-}^{\pm} = \langle x \rangle_{i,+}$ and $\langle x \rangle_{i+1,+}^{\mp} = \langle x \rangle_{i,-}$. We have

$$ \langle x \rangle_{n,-}^{\pm} = \langle x \rangle^{\pm} = x^{+} = \langle x \rangle_{n-1,+} $$

and, similarly, $\langle x \rangle_{n,+}^{\mp} = \langle x \rangle_{n-1,-}$. For $i \in \mathbb{N}_{n-1}$, we have

$$ 
\begin{align*}
\langle x \rangle_{i+1,-}^{\pm} &= (\partial_i^{+}(R(\langle x \rangle_{i+1,-})))_i && \text{(by definition of } \partial_i^{+}) \\
&= (\partial_i^{+} \circ \partial_{i+1}^{-}(R(x)))_i && \text{(by Lemma 3.3.6)} \\
&= (\partial_i^{+}(R(x)))_i && \text{(by globularity)} \\
&= (R(\langle x \rangle_{i,+}))_i && \text{(by Lemma 3.3.6)} \\
&= \langle x \rangle_{i,+}
\end{align*}
$$

and similarly, $\langle x \rangle_{i+1,+}^{\mp} = \langle x \rangle_{i,-}$. Moreover, we have

$$ 
\begin{align*}
(\langle x \rangle_{i,-} \cup \langle x \rangle_{i+1,-}^{+}) \setminus \langle x \rangle_{i+1,-}^{-} &= ((\langle x \rangle_{i+1,-}^{-} \setminus \langle x \rangle_{i+1,-}^{+}) \cup \langle x \rangle_{i+1,-}^{+}) \setminus \langle x \rangle_{i+1,-}^{-} \\
&= \langle x \rangle_{i+1,-}^{+} \setminus \langle x \rangle_{i+1,-}^{-} = \langle x \rangle_{i,+}
\end{align*}
$$

and similarly $(\langle x \rangle_{i,+} \cup \langle x \rangle_{i+1,-}^{-}) \setminus \langle x \rangle_{i+1,-}^{+} = \langle x \rangle_{i,-}$. Thus, $\langle x \rangle_{i+1,-}$ moves $\langle x \rangle_{i,-}$ to $\langle x \rangle_{i,+}$ and so does $\langle x \rangle_{i+1,+}$. Hence, $\langle x \rangle$ is a cell. $\square$

We now prove that the cells (in the sense of Section 1.4) of pasting schemes are sent to wfs's by $T_{\mathrm{Cl}}^{\mathrm{PC}}$, and that all the generators satisfy the segment condition:

**Lemma 3.3.8.** Let P be a loop-free pasting scheme and $n \in \mathbb{N}$. The following hold:

(i) for $x \in P_n$, $x$ satisfies the segment condition,

(ii) for $X \in \mathrm{Cell}(P)_n$, $T_{\mathrm{Cl}}^{\mathrm{PC}}(X) \in \mathrm{WF}(P)_n$.
*Proof.* We prove this lemma by an induction on $n$. If $n = 0$, the result is trivial. So suppose that $n > 0$.

We start with the proof of (i). Let $k \in \mathbb{N}_{n-1}$, $x \in P_n$, $X$ be a $k$-cell such that $\langle x \rangle_{k,-} \subseteq X_k$, and $Y = T_{\mathrm{Cl}}^{\mathrm{PC}}(X)$. By induction, $Y \in \mathrm{WF}(P)$. Moreover, by Lemma 3.3.6,

$$ \partial_k^-(R(x)) = R(\langle x \rangle_{k,-}) \subseteq Y. $$

So, by Axiom (S4), $\langle x \rangle_{k,-}$ is a segment for $\langle x_k \rangle = \langle x_k \rangle$. Hence, $x$ satisfies the segment condition.

We now prove (ii). Let $X \in \mathrm{Cell}(P)_n$. By Proposition 3.3.5, it is enough to show that $T_{\mathrm{Cl}}^{\mathrm{PC}}(X)$ is closed-well-formed. This latter property can be obtained from Theorem 3.1.21 which requires the full segment axiom. But we can consider the restriction of $P$ to an $\omega$-hypergraph $\bar{P}$ where $\bar{P}_i = P_i$ for $i \le n$ and $\bar{P}_i = \emptyset$ for $i > n$. By (i), $\bar{P}$ satisfies Axiom (T3). Then, using Theorem 3.1.21, $T_{\mathrm{Cl}}^{\mathrm{PC}}(X)$ is closed-well-formed and is still closed-well-formed in $P$. Hence, by Proposition 3.3.5, $T_{\mathrm{Cl}}^{\mathrm{PC}}(X) \in \mathrm{WF}(P)$. $\square$

We can conclude the embedding of pasting schemes into torsion-free complexes:

**Theorem 3.3.9.** Let $P$ be a loop-free pasting scheme. Then, $P$ satisfies Axioms (T0), (T1), (T2) and (T3). In particular, if $P$ satisfies Axiom (T4), then $P$ is a torsion-free complex.

*Proof.* The different axioms of torsion-free complexes can be deduced as follows: Axiom (T0) is a consequence of Axiom (S0), Axiom (T1) is a consequence of Axiom (S2), Axiom (T2) is a consequence of Lemma 3.3.7 and Axiom (T3) is a consequence of Lemma 3.3.8. $\square$

Moreover, one translates the cells of the pasting scheme to the wfs's using the operation $T_{\mathrm{Cl}}^{\mathrm{PC}}$:

**Theorem 3.3.10.** Let $P$ be a loop-free pasting scheme. $T_{\mathrm{Cl}}^{\mathrm{PC}}$ is an isomorphism between the $\omega$-categories $\mathrm{Cell}(P)$ and $\mathrm{WF}(P)$. Moreover, for all $x \in P$, $T_{\mathrm{Cl}}^{\mathrm{PC}}(\langle x \rangle) = R(x)$.

*Proof.* By Propositions 3.3.3 and 3.3.5, we have $\mathrm{Closed}_{\mathrm{WF}}(P) = \mathrm{WF}(P)$ as graded sets and, by Lemma 3.3.2 and the definition of id, $\ast^{\mathrm{Cl}}$ and $\ast$, the two have the same structure of $\omega$-category. Thus, by Theorems 3.3.9 and 3.1.21, $T_{\mathrm{Cl}}^{\mathrm{PC}}: \mathrm{Cell}(P) \to \mathrm{WF}(P)$ is an isomorphism. Moreover, by Proposition 3.1.6, for $x \in P$, we have

$$ T_{\mathrm{Cl}}^{\mathrm{PC}}(\langle x \rangle) = T_{\mathrm{Cl}}^{\mathrm{M}} \circ T_{\mathrm{M}}^{\mathrm{PC}}(\langle x \rangle) = T_{\mathrm{Cl}}^{\mathrm{M}}(\{x\}) = R(x). \quad \square $$

**3.4 Embedding augmented directed complexes** In this section, we embed augmented directed complexes with loop-free unital basis into torsion-free complexes. More precisely, given an $\mathrm{adc}$ with a loop-free unital basis, we prove that the basis induces an $\omega$-hypergraph which is a torsion-free complex such that the $\omega$-category of cells of the $\mathrm{adc}$ is isomorphic to the $\omega$-category of cells of this torsion-free complex. For this purpose, we relate properties defined for $\omega$-hypergraphs, like fork-freeness (Section 1.3) and movement (Section 1.4), to analogous properties in augmented directed complexes, and define translation functions between the cells of augmented directed complexes and the ones of the associated $\omega$-hypergraphs.

**Adc's as $\omega$-hypergraphs** Here, dually to the translation given in Section 1.6, we associate a canonical $\omega$-hypergraph to an $\mathrm{adc}$ with basis. Let $(K, d, e)$ be an $\mathrm{adc}$ with a basis $P$. Note that $P$ is canonically a graded set and, in the following, given $n \in \mathbb{N}$ and $x \in P_n$, we write $\bar{x}$ to refer to $x$ as an element of the graded set $P$ whereas $x$ alone refers to $x$ as an element of the monoid $K_n^*$. Given $n \in \mathbb{N}$,
Proof. We prove this lemma by an induction on $n$. If $n = 0$, the result is trivial. So suppose that $n > 0$.

We start with the proof of (i). Let $k \in \mathbb{N}_{n-1}$, $x \in P_n$, $X$ be a $k$-cell such that $\langle x \rangle_{k,-} \subseteq X_k$, and $Y = T_{\mathrm{Cl}}^{\mathrm{PC}}(X)$. By induction, $Y \in \mathrm{WF}(P)$. Moreover, by Lemma 3.3.6,

$$ \partial_k^-(R(x)) = R(\langle x \rangle_{k,-}) \subseteq Y. $$

So, by Axiom (S4), $\langle x \rangle_{k,-}$ is a segment for $\langle x_k \rangle = \langle x_k \rangle$. Hence, $x$ satisfies the segment condition.

We now prove (ii). Let $X \in \mathrm{Cell}(P)_n$. By Proposition 3.3.5, it is enough to show that $T_{\mathrm{Cl}}^{\mathrm{PC}}(X)$ is closed-well-formed. This latter property can be obtained from Theorem 3.1.21 which requires the full segment axiom. But we can consider the restriction of $P$ to an $\omega$-hypergraph $\bar{P}$ where $\bar{P}_i = P_i$ for $i \le n$ and $\bar{P}_i = \emptyset$ for $i > n$. By (i), $\bar{P}$ satisfies Axiom (T3). Then, using Theorem 3.1.21, $T_{\mathrm{Cl}}^{\mathrm{PC}}(X)$ is closed-well-formed and is still closed-well-formed in $P$. Hence, by Proposition 3.3.5, $T_{\mathrm{Cl}}^{\mathrm{PC}}(X) \in \mathrm{WF}(P)$. $\square$

We can conclude the embedding of pasting schemes into torsion-free complexes:

**Theorem 3.3.9.** Let $P$ be a loop-free pasting scheme. Then, $P$ satisfies Axioms (T0), (T1), (T2) and (T3). In particular, if $P$ satisfies Axiom (T4), then $P$ is a torsion-free complex.

*Proof.* The different axioms of torsion-free complexes can be deduced as follows: Axiom (T0) is a consequence of Axiom (S0), Axiom (T1) is a consequence of Axiom (S2), Axiom (T2) is a consequence of Lemma 3.3.7 and Axiom (T3) is a consequence of Lemma 3.3.8. $\square$

Moreover, one translates the cells of the pasting scheme to the wfs's using the operation $T_{\mathrm{Cl}}^{\mathrm{PC}}$:

**Theorem 3.3.10.** Let $P$ be a loop-free pasting scheme. $T_{\mathrm{Cl}}^{\mathrm{PC}}$ is an isomorphism between the $\omega$-categories $\mathrm{Cell}(P)$ and $\mathrm{WF}(P)$. Moreover, for all $x \in P$, $T_{\mathrm{Cl}}^{\mathrm{PC}}(\langle x \rangle) = R(x)$.

*Proof.* By Propositions 3.3.3 and 3.3.5, we have $\mathrm{Closed}_{\mathrm{WF}}(P) = \mathrm{WF}(P)$ as graded sets and, by Lemma 3.3.2 and the definition of id, $*^{\mathrm{Cl}}$ and $*$, the two have the same structure of $\omega$-category. Thus, by Theorems 3.3.9 and 3.1.21, $T_{\mathrm{Cl}}^{\mathrm{PC}}: \mathrm{Cell}(P) \to \mathrm{WF}(P)$ is an isomorphism. Moreover, by Proposition 3.1.6, for $x \in P$, we have

$$ T_{\mathrm{Cl}}^{\mathrm{PC}}(\langle x \rangle) = T_{\mathrm{Cl}}^{\mathrm{M}} \circ T_{\mathrm{M}}^{\mathrm{PC}}(\langle x \rangle) = T_{\mathrm{Cl}}^{\mathrm{M}}(\{x\}) = R(x). \quad \square $$

**3.4 Embedding augmented directed complexes** In this section, we embed augmented directed complexes with loop-free unital basis into torsion-free complexes. More precisely, given an $\mathrm{adc}$ with a loop-free unital basis, we prove that the basis induces an $\omega$-hypergraph which is a torsion-free complex such that the $\omega$-category of cells of the $\mathrm{adc}$ is isomorphic to the $\omega$-category of cells of this torsion-free complex. For this purpose, we relate properties defined for $\omega$-hypergraphs, like fork-freeness (Section 1.3) and movement (Section 1.4), to analogous properties in augmented directed complexes, and define translation functions between the cells of augmented directed complexes and the ones of the associated $\omega$-hypergraphs.

**Adc's as $\omega$-hypergraphs** Here, dually to the translation given in Section 1.6, we associate a canonical $\omega$-hypergraph to an $\mathrm{adc}$ with basis. Let $(K, d, e)$ be an $\mathrm{adc}$ with a basis $P$. Note that $P$ is canonically a graded set and, in the following, given $n \in \mathbb{N}$ and $x \in P_n$, we write $\bar{x}$ to refer to $x$ as an element of the graded set $P$ whereas $x$ alone refers to $x$ as an element of the monoid $K_n^*$. Given $n \in \mathbb{N}$,
- for $s \in K_n^*$, we write $S_n(s)$ for $\{\bar{x} \in P_n \mid x \le s\}$,

- for a finite subset $S \subseteq P_n$, we write $\bar{\Sigma}_n(S)$ for $\sum_{x \in S} x$.

From these definitions, we readily have:

**Lemma 3.4.1.** For all $n \in \mathbb{N}$, $S_n \circ \bar{\Sigma}_n = 1_{P_f(P_n)}$.

For $n \in \mathbb{N}^*$ and $\bar{x} \in P_{n+1}$, we define subsets $\bar{x}^-$, $\bar{x}^+ \subseteq P_n$ such that

$$ \bar{x}^- = S_n(x^-) \quad \text{and} \quad \bar{x}^+ = S_n(x^+) $$

where $x^-$, $x^+$ are the elements of $K_{n-1}$ defined in Section 1.6. We thus obtain an $\omega$-hypergraph $(P, (-)^-, (-)^+)$ that we call the $\omega$-hypergraph associated to $K$. In the following, we prove that, when $P$ is a unital loop-free basis of $K$, $P$ is a torsion-free complex. We already have:

**Lemma 3.4.2.** If $P$ is a unital basis of $K$, given $n \in \mathbb{N}^*$ and $\bar{x} \in P_n$, we have $\bar{x}^- \neq \emptyset$ and $\bar{x}^+ \neq \emptyset$. That is, $P$ satisfies Axiom (T0).

*Proof.* By contradiction, if $\bar{x}^- = \emptyset$, it implies that $[x]_{n-1, -} = 0$. Hence, $[x]_{i, -} = 0$ for $i \in \mathbb{N}_{n-1}$. In particular, $e([x]_{0, -}) = 0$, contradicting the fact that the basis is unital. Hence, $\bar{x}^- \neq \emptyset$ and similarly, $\bar{x}^+ \neq \emptyset$. $\square$

**Fork-freeness and radicality** We now define an analogue for adc's of the notion of fork-freeness defined for $\omega$-hypergraphs, and relate the notions between the two settings.

Let $(K, d, e)$ be an adc with a loop-free unital basis $P$. Given $n \in \mathbb{N}^*$, an element $s \in K_n^*$ is said *fork-free* when for all $x, y \in P_n$ such that $x+y \le s$, it holds that $\bar{x}^\epsilon \cap \bar{y}^\epsilon = \emptyset$ for $\epsilon \in \{-, +\}$. Moreover, in dimension 0, $s \in K_0^*$ is said to be fork-free when $e(s) = 1$. We extend the notion of fork-freeness to cells: given $n \in \mathbb{N}$ and $X \in \text{Cell}^*(K)$, $X$ is said *fork-free* when, for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$, $X_{i, \epsilon}$ is fork-free.

Contrary to subsets of the $\omega$-hypergraph $P$, an element of $P$ can appear in an element of $K_n^*$ with a multiplicity greater than one (since $K_n^*$ is the free monoid on $P_n$). It is then useful to distinguish the elements of $K_n^*$ where generators appear with multiplicity at most one: given $n \in \mathbb{N}$ and $s \in K_n^*$, $s$ is said *radical* when for all $z \in K_n^*$ such that $2z \le s$, we have $z = 0$. We then readily have:

**Lemma 3.4.3.** For all $n \in \mathbb{N}$ and $s \in K_n^*$, $\bar{\Sigma}_n \circ S_n(s) = s$

Moreover, fork-freeness implies radicality:

**Lemma 3.4.4.** Given $n \in \mathbb{N}$ and $s \in K_n^*$, if $s$ is fork-free, then $s$ is radical.

*Proof.* If $n = 0$, $s \in K_n^*$ can be written $s = \sum_{1 \le i \le k} x_i$ for some $k \in \mathbb{N}$ and $x_i \in P_0$ for $i \in \mathbb{N}_k^*$. So $e(s) = k$, and, by fork-freeness, $k = 1$. Hence, $s$ is radical.

Otherwise, assume that $n > 0$. By contradiction, suppose that there is $\bar{x} \in P_n$ such that $2x \le s$. By Lemma 3.4.2, it means that $\bar{x}^- \cap \bar{x}^- \neq \emptyset$, contradicting the fact that $s$ is fork-free. Hence, $s$ is radical. $\square$

Like for cells of torsion-free complexes, cells of adc's with loop-free basis are fork-free:

**Lemma 3.4.5.** Given $n \in \mathbb{N}$ and $X \in \text{Cell}^*(K)_n$, $X$ is fork-free.
- for $s \in K_n^*$, we write $S_n(s)$ for $\{\bar{x} \in P_n | x \le s\}$,

- for a finite subset $S \subseteq P_n$, we write $\bar{\Sigma}_n(S)$ for $\sum_{x \in S} x$.

From these definitions, we readily have:

**Lemma 3.4.1.** For all $n \in \mathbb{N}$, $S_n \circ \bar{\Sigma}_n = 1_{P_f(P_n)}$.

For $n \in \mathbb{N}^*$ and $\bar{x} \in P_{n+1}$, we define subsets $\bar{x}^-, \bar{x}^+ \subseteq P_n$ such that

$$ \bar{x}^{-} = S_{n}(x^{-}) \quad \text{and} \quad \bar{x}^{+} = S_{n}(x^{+}) $$

where $x^-, x^+$ are the elements of $K_{n-1}$ defined in Section 1.6. We thus obtain an $\omega$-hypergraph $(P, (-)^-, (-)^+)$ that we call the $\omega$-hypergraph associated to $K$. In the following, we prove that, when $P$ is a unital loop-free basis of $K$, $P$ is a torsion-free complex. We already have:

**Lemma 3.4.2.** If $P$ is a unital basis of $K$, given $n \in \mathbb{N}^*$ and $\bar{x} \in P_n$, we have $\bar{x}^- \neq \emptyset$ and $\bar{x}^+ \neq \emptyset$. That is, $P$ satisfies Axiom (T0).

*Proof.* By contradiction, if $\bar{x}^- = \emptyset$, it implies that $[x]_{n-1,-} = 0$. Hence, $[x]_{i,-} = 0$ for $i \in \mathbb{N}_{n-1}$. In particular, $e([x]_{0,-}) = 0$, contradicting the fact that the basis is unital. Hence, $\bar{x}^- \neq \emptyset$ and, similarly, $\bar{x}^+ \neq \emptyset$. $\square$

**Fork-freeness and radicality** We now define an analogue for adc's of the notion of fork-freeness defined for $\omega$-hypergraphs, and relate the notions between the two settings.

Let $(K, d, e)$ be an adc with a loop-free unital basis $P$. Given $n \in \mathbb{N}^*$, an element $s \in K_n^*$ is said *fork-free* when for all $x, y \in P_n$ such that $x+y \le s$, it holds that $\bar{x}^e \cap \bar{y}^e = \emptyset$ for $\epsilon \in \{-, +\}$. Moreover, in dimension 0, $s \in K_0^*$ is said to be fork-free when $e(s) = 1$. We extend the notion of fork-freeness to cells: given $n \in \mathbb{N}$ and $X \in \text{Cell}^*(K)$, $X$ is said *fork-free* when, for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$, $X_{i,\epsilon}$ is fork-free.

Contrary to subsets of the $\omega$-hypergraph $P$, an element of $P$ can appear in an element of $K_n^*$ with a multiplicity greater than one (since $K_n^*$ is the free monoid on $P_n$). It is then useful to distinguish the elements of $K_n^*$ where generators appear with multiplicity at most one: given $n \in \mathbb{N}$ and $s \in K_n^*$, $s$ is said *radical* when for all $z \in K_n^*$ such that $2z \le s$, we have $z = 0$. We then readily have:

**Lemma 3.4.3.** For all $n \in \mathbb{N}$ and $s \in K_n^*$, $\bar{\Sigma}_n \circ S_n(s) = s$

Moreover, fork-freeness implies radicality:

**Lemma 3.4.4.** Given $n \in \mathbb{N}$ and $s \in K_n^*$, if $s$ is fork-free, then $s$ is radical.

*Proof.* If $n = 0$, $s \in K_n^*$ can be written $s = \sum_{1 \le i \le k} x_i$ for some $k \in \mathbb{N}$ and $x_i \in P_0$ for $i \in \mathbb{N}_k^*$. So $e(s) = k$, and, by fork-freeness, $k = 1$. Hence, $s$ is radical.

Otherwise, assume that $n > 0$. By contradiction, suppose that there is $\bar{x} \in P_n$ such that $2x \le s$. By Lemma 3.4.2, it means that $\bar{x}^- \cap \bar{x}^- \neq \emptyset$, contradicting the fact that $s$ is fork-free. Hence, $s$ is radical. $\square$

Like for cells of torsion-free complexes, cells of adc's with loop-free basis are fork-free:

**Lemma 3.4.5.** Given $n \in \mathbb{N}$ and $X \in \text{Cell}^*(K)_n$, $X$ is fork-free.
**Proof.** We prove this lemma using an induction on $n$. If $n = 0$, since $e(X_0) = 1$, $X$ is fork-free by definition.

Otherwise, suppose that $n > 0$. By induction, $\partial_{n-1}^-(X)$ and $\partial_{n-1}^+(X)$ are fork-free, so $X_{i,\epsilon}$ is fork-free for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-,+\}$. Let $\bar{x}, \bar{y} \in P_n$ be such that $x+y \le X_n$. By contradiction, suppose that there is $\bar{z} \in P_{n-1}$ such that $\bar{z} \in \bar{x}^- \cap \bar{y}^-$. By [21, Proposition 5.4], there are

$$k \ge 1, \quad \bar{x}_1, \dots, \bar{x}_k \in P_n \quad \text{and} \quad X^1, \dots, X^k \in \mathrm{Cell}^*(K)$$

with $X_n^i = \bar{x}_i$ for $i \in \mathbb{N}_k^*$ and such that $X = X^1 *_{n-1} \dots *_{n-1} X^k$, so $X_n = x_1 + \dots + x_k$. Hence, there are $1 \le i_1, i_2 \le k$ with $i_1 \ne i_2$ such that $x_{i_1} = x$ and $x_{i_2} = y$. By symmetry, we can suppose that $i_1 < i_2$. If there is some $i$ such that $\bar{z} \in \bar{x}_i^+$, by [21, Proposition 5.4], we have $i < i_1$. So, for $i_1 \le i \le i_2$, it holds that $\bar{z} \notin \bar{x}_i^+$. Let $Y = X^{i_1} *_{n-1} X^{i_1+1} *_{n-1} \dots *_{n-1} X^{i_2}$. We have that $Y \in \mathrm{Cell}^*(K)$ and

$$Y_{n-1,-} = \sum_{i_1 \le i \le i_2} [x_i]_{n-1,-} - \sum_{i_1 \le i \le i_2} [x_i]_{n-1,+} + Y_{n-1,+}$$

with

$$2z \le \sum_{i_1 \le i \le i_2} [x_i]_{n-1,-} \quad \text{and} \quad \neg(z \le \sum_{i_1 \le i \le i_2} [x_i]_{n-1,+}) \quad \text{and} \quad Y_{n-1,+} \ge 0$$

so $2z \le Y_{n-1,-}$, contradicting the fact that $\partial_{n-1}^-(Y)$ is radical by Lemma 3.4.4. Thus $\bar{x}^- \cap \bar{y}^- = \emptyset$ and, similarly, $\bar{x}^+ \cap \bar{y}^+ = \emptyset$. Hence, $X$ is fork-free. $\square$

We now give several compatibility results for the operations $\bar{\Sigma}_n$ with sets and the structure of $\omega$-hypergraph on $P$:

**Lemma 3.4.6.** Let $n \in \mathbb{N}$, $U, V \subseteq P_n$ be finite subsets and $x \in P_n$. The following hold:

(i) if $U \cap V = \emptyset$, then $\bar{\Sigma}_n(U) \wedge \bar{\Sigma}_n(V) = 0$ and $\bar{\Sigma}_n(U \cup V) = \bar{\Sigma}_n(U) + \bar{\Sigma}_n(V)$,

(ii) if $U \subseteq V$, then $\bar{\Sigma}_n(U) \le \bar{\Sigma}_n(V)$ and $\bar{\Sigma}_n(V \setminus U) = \bar{\Sigma}_n(V) - \bar{\Sigma}_n(U)$,

(iii) if $n > 0$, then $\bar{\Sigma}_{n-1}(\bar{x}^\epsilon) = x^\epsilon$,

(iv) Suppose that $U$ is fork-free. Then $\bar{\Sigma}_n(U)$ is fork-free. Moreover, in the case where $n > 0$, we have $\bar{\Sigma}_{n-1}(U^\epsilon) = (\bar{\Sigma}_n(U))^\epsilon$.

**Proof.** (i) and (ii) are direct consequences of the definitions. For (iii), note that $\bar{x}^\epsilon = S_{n-1}(x^\epsilon)$. By Lemma 3.4.5, $[x]_{n-1,\epsilon}$ is fork-free and, by Lemma 3.4.4, it is radical. So, by Lemma 3.4.3, we have $\bar{\Sigma}_{n-1}(\bar{x}^\epsilon) = x^\epsilon$.

For (iv), suppose that $U \subseteq P_n$ is fork-free. If $n = 0$, the result is trivial. So suppose that $n > 0$. Given $x, y \in P_n$ with $x \le \bar{\Sigma}_n(U)$ and $y \le \bar{\Sigma}_n(U)$, $\bar{z} \in P_{n-1}$ and $\epsilon \in \{-,+\}$ such that $z \le x^\epsilon$ and $z \le y^\epsilon$, we have $\bar{z} \in \bar{x}^\epsilon$ and $\bar{z} \in \bar{y}^\epsilon$. Since $U$ is fork-free, $x = y$. Also, $\bar{\Sigma}_n(U)$ is radical by definition of $\bar{\Sigma}_n$, so that $\neg(x + y \le \bar{\Sigma}_n(U))$. Hence, $\bar{\Sigma}_n(U)$ is fork-free. For the second part, note that, for $x, y \in U$ with $x \ne y$, we have $\bar{x}^\epsilon \cap \bar{y}^\epsilon = \emptyset$. Hence, by (i) and (iii),

$$\bar{\Sigma}_{n-1}(U^\epsilon) = \bar{\Sigma}_{n-1}(\cup_{\bar{x} \in U} \bar{x}^\epsilon) = \sum_{\bar{x} \in U} \bar{\Sigma}_{n-1}(\bar{x}^\epsilon) = \sum_{\bar{x} \in U} x^\epsilon = (\bar{\Sigma}_n(U))^\epsilon. \quad \square$$

We give analogous compatibility results for the operations $S_n$ with the group structure of $K_n$ and the operations $(-)^-$ and $(-)^+$ defined on $K_n$:

**Lemma 3.4.7.** Let $n \in \mathbb{N}$, $u, v \in K_n^*$ be such that $u, v$ are radical and $z \in P_n$. The following hold:
**Proof.** We prove this lemma using an induction on $n$. If $n = 0$, since $e(X_0) = 1$, $X$ is fork-free by definition.

Otherwise, suppose that $n > 0$. By induction, $\partial_{n-1}^-(X)$ and $\partial_{n-1}^+(X)$ are fork-free, so $X_{i,\epsilon}$ is fork-free for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-,+\}$. Let $\bar{x}, \bar{y} \in P_n$ be such that $x + y \le X_n$. By contradiction, suppose that there is $\bar{z} \in P_{n-1}$ such that $\bar{z} \in \bar{x}^- \cap \bar{y}^-$. By [21, Proposition 5.4], there are

$$k \ge 1, \quad \bar{x}_1, \dots, \bar{x}_k \in P_n \quad \text{and} \quad X^1, \dots, X^k \in \mathrm{Cell}^*(K)$$

with $X_n^i = \bar{x}_i$ for $i \in \mathbb{N}_k^*$ and such that $X = X^1 *_{n-1} \dots *_{n-1} X^k$, so $X_n = x_1 + \dots + x_k$. Hence, there are $1 \le i_1, i_2 \le k$ with $i_1 \ne i_2$ such that $x_{i_1} = x$ and $x_{i_2} = y$. By symmetry, we can suppose that $i_1 < i_2$. If there is some $i$ such that $\bar{z} \in \bar{x}_i^+$, by [21, Proposition 5.4], we have $i < i_1$. So, for $i_1 \le i \le i_2$, it holds that $\bar{z} \notin \bar{x}_i^+$. Let $Y = X^{i_1} *_{n-1} X^{i_1+1} *_{n-1} \dots *_{n-1} X^{i_2}$. We have that $Y \in \mathrm{Cell}^*(K)$ and

$$Y_{n-1,-} = \sum_{i_1 \le i \le i_2} [x_i]_{n-1,-} - \sum_{i_1 \le i \le i_2} [x_i]_{n-1,+} + Y_{n-1,+}$$

with

$$2z \le \sum_{i_1 \le i \le i_2} [x_i]_{n-1,-} \quad \text{and} \quad \neg(z \le \sum_{i_1 \le i \le i_2} [x_i]_{n-1,+}) \quad \text{and} \quad Y_{n-1,+} \ge 0$$

so $2z \le Y_{n-1,-}$, contradicting the fact that $\partial_{n-1}^-(Y)$ is radical by Lemma 3.4.4. Thus $\bar{x}^- \cap \bar{y}^- = \emptyset$ and, similarly, $\bar{x}^+ \cap \bar{y}^+ = \emptyset$. Hence, $X$ is fork-free. $\square$

We now give several compatibility results for the operations $\bar{\Sigma}_n$ with sets and the structure of $\omega$-hypergraph on $P$:

**Lemma 3.4.6.** Let $n \in \mathbb{N}$, $U, V \subseteq P_n$ be finite subsets and $x \in P_n$. The following hold:

(i) if $U \cap V = \emptyset$, then $\bar{\Sigma}_n(U) \wedge \bar{\Sigma}_n(V) = 0$ and $\bar{\Sigma}_n(U \cup V) = \bar{\Sigma}_n(U) + \bar{\Sigma}_n(V)$,

(ii) if $U \subseteq V$, then $\bar{\Sigma}_n(U) \le \bar{\Sigma}_n(V)$ and $\bar{\Sigma}_n(V \setminus U) = \bar{\Sigma}_n(V) - \bar{\Sigma}_n(U)$,

(iii) if $n > 0$, then $\bar{\Sigma}_{n-1}(\bar{x}^\epsilon) = x^\epsilon$,

(iv) Suppose that $U$ is fork-free. Then $\bar{\Sigma}_n(U)$ is fork-free. Moreover, in the case where $n > 0$, we have $\bar{\Sigma}_{n-1}(U^\epsilon) = (\bar{\Sigma}_n(U))^\epsilon$.

**Proof.** (i) and (ii) are direct consequences of the definitions. For (iii), note that $\bar{x}^\epsilon = S_{n-1}(x^\epsilon)$. By Lemma 3.4.5, $[x]_{n-1,\epsilon}$ is fork-free and, by Lemma 3.4.4, it is radical. So, by Lemma 3.4.3, we have $\bar{\Sigma}_{n-1}(\bar{x}^\epsilon) = x^\epsilon$.

For (iv), suppose that $U \subseteq P_n$ is fork-free. If $n = 0$, the result is trivial. So suppose that $n > 0$. Given $x, y \in P_n$ with $x \le \bar{\Sigma}_n(U)$ and $y \le \bar{\Sigma}_n(U)$, $\bar{z} \in P_{n-1}$ and $\epsilon \in \{-,+\}$ such that $z \le x^\epsilon$ and $z \le y^\epsilon$, we have $\bar{z} \in \bar{x}^\epsilon$ and $\bar{z} \in \bar{y}^\epsilon$. Since $U$ is fork-free, $x = y$. Also, $\bar{\Sigma}_n(U)$ is radical by definition of $\bar{\Sigma}_n$, so that $\neg(x + y \le \bar{\Sigma}_n(U))$. Hence, $\bar{\Sigma}_n(U)$ is fork-free. For the second part, note that, for $x, y \in U$ with $x \ne y$, we have $\bar{x}^\epsilon \cap \bar{y}^\epsilon = \emptyset$. Hence, by (i) and (iii),

$$\bar{\Sigma}_{n-1}(U^\epsilon) = \bar{\Sigma}_{n-1}(\cup_{\bar{x} \in U} \bar{x}^\epsilon) = \sum_{\bar{x} \in U} \bar{\Sigma}_{n-1}(\bar{x}^\epsilon) = \sum_{\bar{x} \in U} x^\epsilon = (\bar{\Sigma}_n(U))^\epsilon. \quad \square$$

We give analogous compatibility results for the operations $S_n$ with the group structure of $K_n$ and the operations $(-)^-$ and $(-)^+$ defined on $K_n$:

**Lemma 3.4.7.** Let $n \in \mathbb{N}$, $u, v \in K_n^*$ be such that $u, v$ are radical and $z \in P_n$. The following hold:
(i) if $u \wedge v = 0$, then $S_n(u) \cap S_n(v) = \emptyset$ and $S_n(u + v) = S_n(u) \cup S_n(v)$,

(ii) if $u \le v$, then $S_n(u) \subseteq S_n(v)$ and $S_n(v - u) = (S_n(v)) \setminus (S_n(u))$,

(iii) if $n > 0$, then $S_{n-1}(z^\epsilon) = \bar{z}^\epsilon$,

(iv) Suppose that $u$ is fork-free. Then, $S_n(u)$ is fork-free. Moreover, in the case where $n > 0$, we have $S_{n-1}(u^\epsilon) = (S_n(u))^\epsilon$.

**Proof.** (i), (ii) and (iii) are direct consequences of the definitions. For (iv), suppose that $u$ is fork-free. If $n = 0$, the result is trivial, so suppose that $n > 0$. Given $\bar{x}, \bar{y} \in S_n(u)$, $\bar{z} \in P_{n-1}$ and $\epsilon \in \{-, +\}$ such that $\bar{z} \in \bar{x}^\epsilon \cap \bar{y}^\epsilon$, we have $z \le x^\epsilon$ and $z \le y^\epsilon$. By fork-freeness, $\neg(x + y \le u)$. But $x \le u$ and $y \le u$, so that $x = y$. Thus, $S_n(u)$ is fork-free. For the second part, note that, for $x, y \in P_n$ with $x \ne y$, $x \le u$ and $y \le u$, we have $x^\epsilon \wedge y^\epsilon = 0$. Hence, by (i) and (iii),

$$S_{n-1}(u^{\epsilon}) = S_{n-1}\left(\sum_{x \in P_n, x \le u} x^{\epsilon}\right) = \bigcup_{x \in P_n, x \le u} S_{n-1}(x^{\epsilon}) = \bigcup_{x \in P_n, x \le u} \bar{x}^{\epsilon} = (S_n(u))^{\epsilon}. \quad \square$$

**Movement properties** We now relate the movement properties of $\omega$-hypergraphs (as defined in Section 1.4) to properties of augmented directed complexes. Let $(K, d, e)$ be an adc with a loop-free unital basis $P$. We first prove a compatibility result of the functions $\bar{\Sigma}_n$ with the operations $(-)^\mp$ and $(-)^\pm$ on $\omega$-hypergraphs and adc's:

**Lemma 3.4.8.** Let $n \in \mathbb{N}^*$, $u \in K_n^*$ fork-free and $U = S_n(u)$. We have

$$u^{\mp} = \bar{\Sigma}_{n-1}(U^{\mp}) \quad \text{and} \quad u^{\pm} = \bar{\Sigma}_{n-1}(U^{\pm}).$$

*Proof.* We compute that

$$d(u) = u^{\pm} - u^{\mp} = u^{+} - u^{-} \\ = \bar{\Sigma}_{n-1}(U^{+}) - \bar{\Sigma}_{n-1}(U^{-}) \qquad (\text{by Lemma 3.4.6})$$

$$= (\bar{\Sigma}_{n-1}(U^{+}) + \bar{\Sigma}_{n-1}(U^{+} \cap U^{-})) - (\bar{\Sigma}_{n-1}(U^{\mp}) + \bar{\Sigma}_{n-1}(U^{+} \cap U^{-})) \quad (\text{by Lemma 3.4.6}) \\ = \bar{\Sigma}_{n-1}(U^{\pm}) - \bar{\Sigma}_{n-1}(U^{\mp}).$$

Since $U^{\pm} \cap U^{\mp} = \emptyset$, we have $\bar{\Sigma}_{n-1}(U^{\pm}) \wedge \bar{\Sigma}_{n-1}(U^{\mp}) = \emptyset$. By uniqueness of the decomposition, we have $u^{\mp} = \bar{\Sigma}_{n-1}(U^{\mp})$ and $u^{\pm} = \bar{\Sigma}_{n-1}(U^{\pm})$. $\square$

Now, we show a compatibility of the operations $\bar{\Sigma}_n$ with movement:

**Lemma 3.4.9.** Let $n \in \mathbb{N}$, $S \subseteq P_{n+1}$ be a finite and fork-free set and $U, V \subseteq P_n$ be finite sets such that $S$ moves $U$ to $V$. Then, $d(\bar{\Sigma}_{n+1}(S)) = \bar{\Sigma}_n(V) - \bar{\Sigma}_n(U)$.

*Proof.* By definition of movement, $V = (U \cup S^+) \setminus S^-$. Hence,

$$\bar{\Sigma}_n(V) = \bar{\Sigma}_n((U \cup S^+) \setminus S^-) \\ = \bar{\Sigma}_n(U \cup S^+) - \bar{\Sigma}_n(S^-) \qquad (\text{by Lemma 3.4.6, since } S^- \subseteq U \cup S^+)$$

$$= \bar{\Sigma}_n(U) + \bar{\Sigma}_n(S^+) - \bar{\Sigma}_n(S^-) \qquad (\text{since } U \cap S^+ = \emptyset \text{ by Lemma A.1.1})$$

$$= \bar{\Sigma}_n(U) + (\bar{\Sigma}_{n+1}(S))^+ - (\bar{\Sigma}_{n+1}(S))^- \qquad (\text{by Lemma 3.4.6})$$

$$= \bar{\Sigma}_n(U) + d(\bar{\Sigma}_{n+1}(S)). \qquad \square$$

Conversely, we prove sufficient conditions for the operations $S_n$ to induce movement:
(i) if $u \wedge v = 0$, then $S_n(u) \cap S_n(v) = \emptyset$ and $S_n(u + v) = S_n(u) \cup S_n(v)$,

(ii) if $u \le v$, then $S_n(u) \subseteq S_n(v)$ and $S_n(v - u) = (S_n(v)) \setminus (S_n(u))$,

(iii) if $n > 0$, then $S_{n-1}(z^\epsilon) = \bar{z}^\epsilon$,

(iv) Suppose that $u$ is fork-free. Then, $S_n(u)$ is fork-free. Moreover, in the case where $n > 0$, we have $S_{n-1}(u^\epsilon) = (S_n(u))^\epsilon$.

**Proof.** (i), (ii) and (iii) are direct consequences of the definitions. For (iv), suppose that $u$ is fork-free. If $n = 0$, the result is trivial, so suppose that $n > 0$. Given $\bar{x}, \bar{y} \in S_n(u)$, $\bar{z} \in P_{n-1}$ and $\epsilon \in \{-, +\}$ such that $\bar{z} \in \bar{x}^\epsilon \cap \bar{y}^\epsilon$, we have $z \le x^\epsilon$ and $z \le y^\epsilon$. By fork-freeness, $\neg(x + y \le u)$. But $x \le u$ and $y \le u$, so that $x = y$. Thus, $S_n(u)$ is fork-free. For the second part, note that, for $x, y \in P_n$ with $x \ne y$, $x \le u$ and $y \le u$, we have $x^\epsilon \wedge y^\epsilon = 0$. Hence, by (i) and (iii),

$$S_{n-1}(u^\epsilon) = S_{n-1}\left(\sum_{x \in P_n, x \le u} x^\epsilon\right) = \bigcup_{x \in P_n, x \le u} S_{n-1}(x^\epsilon) = \bigcup_{x \in P_n, x \le u} \bar{x}^\epsilon = (S_n(u))^\epsilon. \quad \square$$

**Movement properties** We now relate the movement properties of $\omega$-hypergraphs (as defined in Section 1.4) to properties of augmented directed complexes. Let $(K, d, e)$ be an adc with a loop-free unital basis $P$. We first prove a compatibility result of the functions $\bar{\Sigma}_n$ with the operations $(-)^\mp$ and $(-)^\pm$ on $\omega$-hypergraphs and adc's:

**Lemma 3.4.8.** Let $n \in \mathbb{N}^*$, $u \in K_n^*$ fork-free and $U = S_n(u)$. We have

$$u^{\mp} = \bar{\Sigma}_{n-1}(U^{\mp}) \quad \text{and} \quad u^{\pm} = \bar{\Sigma}_{n-1}(U^{\pm}).$$

*Proof.* We compute that

$$
\begin{align*}
d(u) &= u^{\pm} - u^{\mp} = u^{+} - u^{-} \\
&= \bar{\Sigma}_{n-1}(U^{+}) - \bar{\Sigma}_{n-1}(U^{-}) \tag*{\text{(by Lemma 3.4.6)}} \\
&= (\bar{\Sigma}_{n-1}(U^{+}) + \bar{\Sigma}_{n-1}(U^{+} \cap U^{-})) - (\bar{\Sigma}_{n-1}(U^{\mp}) + \bar{\Sigma}_{n-1}(U^{+} \cap U^{-})) \tag*{\text{(by Lemma 3.4.6)}} \\
&= \bar{\Sigma}_{n-1}(U^{\pm}) - \bar{\Sigma}_{n-1}(U^{\mp}).
\end{align*}
$$

Since $U^{\pm} \cap U^{\mp} = \emptyset$, we have $\bar{\Sigma}_{n-1}(U^{\pm}) \wedge \bar{\Sigma}_{n-1}(U^{\mp}) = \emptyset$. By uniqueness of the decomposition, we have $u^{\mp} = \bar{\Sigma}_{n-1}(U^{\mp})$ and $u^{\pm} = \bar{\Sigma}_{n-1}(U^{\pm})$. $\square$

Now, we show a compatibility of the operations $\bar{\Sigma}_n$ with movement:

**Lemma 3.4.9.** Let $n \in \mathbb{N}$, $S \subseteq P_{n+1}$ be a finite and fork-free set and $U, V \subseteq P_n$ be finite sets such that $S$ moves $U$ to $V$. Then, $d(\bar{\Sigma}_{n+1}(S)) = \bar{\Sigma}_n(V) - \bar{\Sigma}_n(U)$.

*Proof.* By definition of movement, $V = (U \cup S^+) \setminus S^-$. Hence,

$$
\begin{align*}
\bar{\Sigma}_n(V) &= \bar{\Sigma}_n((U \cup S^+) \setminus S^-) \\
&= \bar{\Sigma}_n(U \cup S^+) - \bar{\Sigma}_n(S^-) \tag*{\text{(by Lemma 3.4.6, since } S^- \subseteq U \cup S^+) \\
&= \bar{\Sigma}_n(U) + \bar{\Sigma}_n(S^+) - \bar{\Sigma}_n(S^-) \tag*{\text{(since } U \cap S^+ = \emptyset \text{ by Lemma A.1.1)} \\
&= \bar{\Sigma}_n(U) + (\bar{\Sigma}_{n+1}(S))^+ - (\bar{\Sigma}_{n+1}(S))^- \tag*{\text{(by Lemma 3.4.6)} \\
&= \bar{\Sigma}_n(U) + d(\bar{\Sigma}_{n+1}(S)). \tag*{\square}
\end{align*}
$$

Conversely, we prove sufficient conditions for the operations $S_n$ to induce movement:
**Lemma 3.4.10.** Let $n \in \mathbb{N}$, $s \in K_{n+1}^*$ fork-free, $u, v \in K_n^*$ with $u, v$ radical, such that

$$d(s) = v - u, \quad u \wedge s^+ = 0 \quad \text{and} \quad s^- \wedge v = 0.$$

Then, $S_{n+1}(s)$ moves $S_n(u)$ to $S_n(v)$.

*Proof.* Let $S = S_{n+1}(s)$, $U = S_n(u)$ and $V = S_n(v)$. Since $d(s) = v - u$, we have

$$s^- \leq s^- + v = u + s^+$$

so $S^- = S_n(s^-) \subseteq S_n(u + s^+) = U \cup S^+$. We compute that

$$\begin{align*}
\bar{\Sigma}_n((U \cup S^+) \setminus S^-) &= \bar{\Sigma}_n(U \cup S^+) - \bar{\Sigma}_n(S^-) \\
&= \bar{\Sigma}_n \circ S_n(u + s^+) - s^- \tag*{\text{(by Lemma 3.4.6)}} \\
&= u + s^+ - s^- \\
&= u + d(s) = v = \bar{\Sigma}_n(V).
\end{align*}$$

By Lemma 3.4.1, $V = (U \cup S^+) \setminus S^-$. Similarly, $U = (V \cup S^-) \setminus S^+$. So, $S$ moves $U$ to $V$. $\square$

Finally, we show empty intersection results for cells of $\text{Cell}^*(K)$, analogous to the ones for $\text{Cell}(P)$:

**Lemma 3.4.11.** Let $n \in \mathbb{N}^*$ and $X \in \text{Cell}^*(K)_n$. Then, for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$, we have

$$X_{i,-} \wedge X_{i+1,\epsilon}^{+} = 0 \quad \text{and} \quad X_{i+1,\epsilon} \wedge X_{i,+} = 0.$$

*Proof.* By contradiction, suppose given $n \in \mathbb{N}^*$, $X \in \text{Cell}^*(K)_n$, $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$ that give a counter-example for this property. By applying $\partial^-$, $\partial^+$ sufficiently, we can suppose that $i = n - 1$. Also, by symmetry, we only need to handle the first case, that is, when there is $z \in P_{n-1}$ such that $z \leq X_{n-1,-} \wedge X_n^+$. So there is $x \in P_n$ such that $x \leq X_n$ and $z \leq x^+$. By the definition of a cell, we have $d(X_n) = X_{n-1,+} - X_{n-1,-}$, thus

$$X_{n-1,+} + \sum_{u \in P_n, u \leq X_n} u^{-} = X_{n-1,-} + \sum_{u \in P_n, u \leq X_n} u^{+} \geq 2z$$

and, since $X_{n-1,+}$ is radical, there is $y \in P_n$ with $y \leq X_n$ such that $z \leq y^-$. By [21, Proposition 5.1], there are $k \in \mathbb{N}^*$, $x_1, \dots, x_k \in P_n$ with $x_1 + \dots + x_k = X_n$, $i_1, i_2 \in \mathbb{N}_k^*$ with $i_1 < i_2$, $x_{i_1} = x$ and $x_{i_2} = y$, and $X^1, \dots, X^k \in \text{Cell}^*(K)$ with $X_n^i = x_i$ for $i \in \mathbb{N}_k^*$ such that we have the decomposition $X = X^1 *_{n-1} \dots *_{n-1} X^k$. Let $Y = X^1 *_{n-1} \dots *_{n-1} X^{i_1}$. Since $Y$ is a cell, we have

$$Y_{n-1,+} + \sum_{1 \leq i \leq k} x_i^{-} = Y_{n-1,-} + \sum_{1 \leq i \leq k} x_i^{+} = X_{n-1,-} + \sum_{1 \leq i \leq k} x_i^{+} \geq 2z.$$

Moreover, since $X$ is fork-free and $z \leq x_{i_2}^-$, we have $\neg(z \leq x_i^-)$ for $i \in \mathbb{N}_{i_1}$. So $2z \leq Y_{n-1,+}$, contradicting the fact that $Y_{n-1,+}$ is radical derived from Lemmas 3.4.5 and 3.4.4. Hence, we have $X_{i,-} \wedge X_n^+ = 0$. $\square$

**The translation operations** We now introduce translation functions between the cells of augmented directed complexes and the cells of their associated $\omega$-hypergraphs, and show that these translations are bijective.

Let $(K, d, e)$ be an adc with a loop-free unital basis $P$. We extend the operations $\bar{\Sigma}_n$ and $S_n$ to translation functions between the pre-cells of $P$ and the pre-cells of $K$. Given $n \in \mathbb{N}$
**Lemma 3.4.10.** Let $n \in \mathbb{N}$, $s \in K_{n+1}^*$ fork-free, $u, v \in K_n^*$ with $u, v$ radical, such that

$$d(s) = v - u, \quad u \wedge s^+ = 0 \quad \text{and} \quad s^- \wedge v = 0.$$

Then, $S_{n+1}(s)$ moves $S_n(u)$ to $S_n(v)$.

*Proof.* Let $S = S_{n+1}(s)$, $U = S_n(u)$ and $V = S_n(v)$. Since $d(s) = v - u$, we have

$$s^{-} \leq s^{-} + v = u + s^{+}$$

so $S^{-} = S_n(s^{-}) \subseteq S_n(u + s^{+}) = U \cup S^{+}$. We compute that

$$\begin{align*}
\bar{\Sigma}_n((U \cup S^+) \setminus S^-) &= \bar{\Sigma}_n(U \cup S^+) - \bar{\Sigma}_n(S^-) \\
&= \bar{\Sigma}_n \circ S_n(u + s^+) - s^- \tag*{\text{(by Lemma 3.4.6)}} \\
&= u + s^+ - s^- \\
&= u + d(s) = v = \bar{\Sigma}_n(V).
\end{align*}$$

By Lemma 3.4.1, $V = (U \cup S^+) \setminus S^-$. Similarly, $U = (V \cup S^-) \setminus S^+$. So, $S$ moves $U$ to $V$. $\square$

Finally, we show empty intersection results for cells of $\text{Cell}^*(K)$, analogous to the ones for $\text{Cell}(P)$:

**Lemma 3.4.11.** Let $n \in \mathbb{N}^*$ and $X \in \text{Cell}^*(K)_n$. Then, for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$, we have

$$X_{i,-} \wedge X_{i+1,\epsilon}^{+} = 0 \quad \text{and} \quad X_{i+1,\epsilon}^{-} \wedge X_{i,+} = 0.$$

*Proof.* By contradiction, suppose given $n \in \mathbb{N}^*$, $X \in \text{Cell}^*(K)_n$, $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$ that give a counter-example for this property. By applying $\partial^-$, $\partial^+$ sufficiently, we can suppose that $i = n - 1$. Also, by symmetry, we only need to handle the first case, that is, when there is $z \in P_{n-1}$ such that $z \leq X_{n-1,-} \wedge X_n^+$. So there is $x \in P_n$ such that $x \leq X_n$ and $z \leq x^+$. By the definition of a cell, we have $d(X_n) = X_{n-1,+} - X_{n-1,-}$, thus

$$X_{n-1,+} + \sum_{u \in P_n, u \leq X_n} u^{-} = X_{n-1,-} + \sum_{u \in P_n, u \leq X_n} u^{+} \geq 2z$$

and, since $X_{n-1,+}$ is radical, there is $y \in P_n$ with $y \leq X_n$ such that $z \leq y^-$. By [21, Proposition 5.1], there are $k \in \mathbb{N}^*$, $x_1, \dots, x_k \in P_n$ with $x_1 + \dots + x_k = X_n$, $i_1, i_2 \in \mathbb{N}_k^*$ with $i_1 < i_2$, $x_{i_1} = x$ and $x_{i_2} = y$, and $X^1, \dots, X^k \in \text{Cell}^*(K)$ with $X_n^i = x_i$ for $i \in \mathbb{N}_k^*$ such that we have the decomposition $X = X^1 *_{n-1} \dots *_{n-1} X^k$. Let $Y = X^1 *_{n-1} \dots *_{n-1} X^{i_1}$. Since $Y$ is a cell, we have

$$Y_{n-1,+} + \sum_{1 \leq i \leq k} x_i^{-} = Y_{n-1,-} + \sum_{1 \leq i \leq k} x_i^{+} = X_{n-1,-} + \sum_{1 \leq i \leq k} x_i^{+} \geq 2z.$$

Moreover, since $X$ is fork-free and $z \leq x_{i_2}^-$, we have $\neg(z \leq x_i^-)$ for $i \in \mathbb{N}_{i_1}$. So $2z \leq Y_{n-1,+}$, contradicting the fact that $Y_{n-1,+}$ is radical derived from Lemmas 3.4.5 and 3.4.4. Hence, we have $X_{i,-} \wedge X_n^+ = 0$. $\square$

**The translation operations** We now introduce translation functions between the cells of augmented directed complexes and the cells of their associated $\omega$-hypergraphs, and show that these translations are bijective.

Let $(K, d, e)$ be an adc with a loop-free unital basis $P$. We extend the operations $\bar{\Sigma}_n$ and $S_n$ to translation functions between the pre-cells of $P$ and the pre-cells of $K$. Given $n \in \mathbb{N}$
and an $n$-pre-cell $X \in \mathrm{PCell}(P)_n$, we define $\bar{\Sigma}(X) \in \mathrm{Cell}^*(K)$ as the $n$-pre-cell $Y$ such that $Y_{i,\epsilon} = \bar{\Sigma}_i(X_{i,\epsilon})$ for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$. Similarly, given an $n$-pre-cell $X \in \mathrm{PCell}^*(K)$, we define $S(X) \in \mathrm{PCell}(P)$ as the $n$-pre-cell $Y$ such that $Y_{i,\epsilon} = S_i(X_{i,\epsilon})$ for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$. We then have:

**Proposition 3.4.12.** $\bar{\Sigma}$ induces a bijection with inverse $S$ from $\mathrm{Cell}(P)$ to $\mathrm{Cell}^*(K)$. Moreover, given $x \in P$, we have $S([x]) = \langle \bar{x} \rangle$.

*Proof.* Let $n \in \mathbb{N}$ and $X \in \mathrm{Cell}(P)_n$. Then, by Lemma 3.4.6, for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$, $\bar{\Sigma}_i(X_{i,\epsilon})$ is fork-free. Plus, when $i < n$, by Lemma 3.4.9, we have $d(\bar{\Sigma}_{i+1}(X_{i+1,\epsilon})) = \bar{\Sigma}_i(X_{i,+}) - \bar{\Sigma}_i(X_{i,-})$ so $\bar{\Sigma}(X) \in \mathrm{Cell}^*(K)$. Conversely, let $n \in \mathbb{N}$ and $X \in \mathrm{Cell}^*(K)_n$. By Lemma 3.4.7, given $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$, $S_i(X_{i,\epsilon})$ is fork-free. Moreover, when $i < n$, by Lemmas 3.4.10 and 3.4.11, we have that $S_{i+1}(X_{i+1,\epsilon})$ moves $S_i(X_{i,-})$ to $S_i(X_{i,+})$ so $S(X) \in \mathrm{Cell}(P)$. By Lemma 3.4.1, for $X \in \mathrm{Cell}(P)$, $S \circ \bar{\Sigma}(X) = X$, and, by Lemmas 3.4.5, 3.4.4 and 3.4.3, for $X \in \mathrm{Cell}^*(K)$, $\bar{\Sigma} \circ S(X) = X$. Hence, $\bar{\Sigma}$ and $S$ induce bijections between $\mathrm{Cell}(P)$ and $\mathrm{Cell}^*(K)$ and are inverse of each other.

Now let $n \in \mathbb{N}$, $x \in P_n$ and $X = S([x])$. We have $X_n = S_n([x]_n) = \{x\}$. We show by a decreasing induction on $i$ that $X_{i,\epsilon} = \langle x \rangle_{i,\epsilon}$ for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$. We have $[x]_{i,-} = [x]_{i+1,-}^\perp$ so, by Lemmas 3.4.5 and 3.4.8, $X_{i,-} = S_i([x]_{i+1,-}^\perp) = X_{i+1,-}^\perp$. Thus, $X_{i,-} = \langle x \rangle_{i,-}$. Similarly, $X_{i,+} = \langle x \rangle_{i,+}$. Hence, $S([x]) = \langle \bar{x} \rangle$. $\square$

**Adc's are torsion-free complexes** We now prove that the $\omega$-hypergraphs associated to adc's equipped with loop-free unital bases are torsion-free complexes. In fact, we will show that they moreover satisfy the stronger Axioms (T3') and (T4').

Let $(K, d, e)$ be an adc with a loop-free unital basis $P$. We have already shown how to derive Axiom (T0) for $P$ in Lemma 3.4.2, and we now derive the other ones in the following lemmas.

**Lemma 3.4.13.** *P satisfies Axiom (T1).*

*Proof.* Note that, for $n \in \mathbb{N}^*$ and $\bar{x}, \bar{y} \in P_n$, $\bar{x} \triangleleft_P^n \bar{y}$ implies $\bar{x} <_{n-1} \bar{y}$. So, by transitivity, we have $\triangleleft_{P_n} \subseteq <_{n-1}$. Since the basis $P$ is loop-free, $<_{n-1}$ is irreflexive and so is $\triangleleft_{P_n}$. Hence, $\triangleleft$ is irreflexive. $\square$

**Lemma 3.4.14.** *P satisfies Axiom (T2).*

*Proof.* Given $\bar{x} \in P$, we have $S([x]) = \langle \bar{x} \rangle$ By Proposition 3.4.12. Moreover, by Proposition 3.4.12, we have $S([x]) \in \mathrm{Cell}(P)$. Hence, $\bar{x}$ is relevant. $\square$

**Lemma 3.4.15.** *P satisfies Axiom (T3').*

*Proof.* By contradiction, suppose that there are $i, n \in \mathbb{N}$ with $i < n$ and an element $\bar{x} \in P_n$ such that $\langle \bar{x} \rangle_{i,+} \not\sim^* \langle \bar{x} \rangle_{i,-}$. So there are $k \ge 1$, $\bar{y}_1, \dots, \bar{y}_k \in P_i$ such that $\bar{y}_1 \in \langle \bar{x} \rangle_{i,+}$, $\bar{y}_k \in \langle \bar{x} \rangle_{i,-}$ and $\bar{y}_j \not\sim \bar{y}_{j+1}$ for $1 \le j < k$. By definition of $\not\sim$, it gives $\bar{z}_1, \dots, \bar{z}_{k-1} \in P_{i+1}$ with $\bar{y}_j \in \bar{z}_j^-$ and $\bar{y}_{j+1} \in \bar{z}_j^+$ for $1 \le j < k$. So we have $x <_i z_1 <_i \dots <_i z_{k-1} <_i x$, contradicting the loop-freeness of the basis $P$. Hence, $P$ satisfies Axiom (T3'). $\square$

**Lemma 3.4.16.** *P satisfies Axiom (T4').*
and an $n$-pre-cell $X \in \mathrm{PCell}(P)_n$, we define $\bar{\Sigma}(X) \in \mathrm{Cell}^*(K)$ as the $n$-pre-cell $Y$ such that $Y_{i,\epsilon} = \bar{\Sigma}_i(X_{i,\epsilon})$ for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$. Similarly, given an $n$-pre-cell $X \in \mathrm{PCell}^*(K)$, we define $S(X) \in \mathrm{PCell}(P)$ as the $n$-pre-cell $Y$ such that $Y_{i,\epsilon} = S_i(X_{i,\epsilon})$ for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$. We then have:

**Proposition 3.4.12.** $\bar{\Sigma}$ induces a bijection with inverse $S$ from $\mathrm{Cell}(P)$ to $\mathrm{Cell}^*(K)$. Moreover, given $x \in P$, we have $S([x]) = \langle \bar{x} \rangle$.

*Proof.* Let $n \in \mathbb{N}$ and $X \in \mathrm{Cell}(P)_n$. Then, by Lemma 3.4.6, for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$, $\bar{\Sigma}_i(X_{i,\epsilon})$ is fork-free. Plus, when $i < n$, by Lemma 3.4.9, we have $d(\bar{\Sigma}_{i+1}(X_{i+1,\epsilon})) = \bar{\Sigma}_i(X_{i,+}) - \bar{\Sigma}_i(X_{i,-})$ so $\bar{\Sigma}(X) \in \mathrm{Cell}^*(K)$. Conversely, let $n \in \mathbb{N}$ and $X \in \mathrm{Cell}^*(K)_n$. By Lemma 3.4.7, given $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$, $S_i(X_{i,\epsilon})$ is fork-free. Moreover, when $i < n$, by Lemmas 3.4.10 and 3.4.11, we have that $S_{i+1}(X_{i+1,\epsilon})$ moves $S_i(X_{i,-})$ to $S_i(X_{i,+})$ so $S(X) \in \mathrm{Cell}(P)$. By Lemma 3.4.1, for $X \in \mathrm{Cell}(P)$, $S \circ \bar{\Sigma}(X) = X$, and, by Lemmas 3.4.5, 3.4.4 and 3.4.3, for $X \in \mathrm{Cell}^*(K)$, $\bar{\Sigma} \circ S(X) = X$. Hence, $\bar{\Sigma}$ and $S$ induce bijections between $\mathrm{Cell}(P)$ and $\mathrm{Cell}^*(K)$ and are inverse of each other.

Now let $n \in \mathbb{N}$, $x \in P_n$ and $X = S([x])$. We have $X_n = S_n([x]_n) = \{x\}$. We show by a decreasing induction on $i$ that $X_{i,\epsilon} = \langle x \rangle_{i,\epsilon}$ for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$. We have $[x]_{i,-} = [x]_{i+1,-}^\perp$ so, by Lemmas 3.4.5 and 3.4.8, $X_{i,-} = S_i([x]_{i+1,-}^\perp) = X_{i+1,-}^\perp$. Thus, $X_{i,-} = \langle x \rangle_{i,-}$. Similarly, $X_{i,+} = \langle x \rangle_{i,+}$. Hence, $S([x]) = \langle \bar{x} \rangle$. $\square$

**Adc's are torsion-free complexes** We now prove that the $\omega$-hypergraphs associated to adc's equipped with loop-free unital bases are torsion-free complexes. In fact, we will show that they moreover satisfy the stronger Axioms (T3') and (T4').

Let $(K, \mathrm{d}, \mathrm{e})$ be an adc with a loop-free unital basis $P$. We have already shown how to derive Axiom (T0) for $P$ in Lemma 3.4.2, and we now derive the other ones in the following lemmas.

**Lemma 3.4.13.** *P satisfies Axiom (T1).*

*Proof.* Note that, for $n \in \mathbb{N}^*$ and $\bar{x}, \bar{y} \in P_n$, $\bar{x} \triangleleft_P^n \bar{y}$ implies $\bar{x} <_{n-1} \bar{y}$. So, by transitivity, we have $\triangleleft_{P_n} \subseteq <_{n-1}$. Since the basis $P$ is loop-free, $<_{n-1}$ is irreflexive and so is $\triangleleft_{P_n}$. Hence, $\triangleleft$ is irreflexive. $\square$

**Lemma 3.4.14.** *P satisfies Axiom (T2).*

*Proof.* Given $\bar{x} \in P$, we have $S([x]) = \langle \bar{x} \rangle$ By Proposition 3.4.12. Moreover, by Proposition 3.4.12, we have $S([x]) \in \mathrm{Cell}(P)$. Hence, $\bar{x}$ is relevant. $\square$

**Lemma 3.4.15.** *P satisfies Axiom (T3').*

*Proof.* By contradiction, suppose that there are $i, n \in \mathbb{N}$ with $i < n$ and an element $\bar{x} \in P_n$ such that $\langle \bar{x} \rangle_{i,+} \not\sim^* \langle \bar{x} \rangle_{i,-}$. So there are $k \ge 1$, $\bar{y}_1, \dots, \bar{y}_k \in P_i$ such that $\bar{y}_1 \in \langle \bar{x} \rangle_{i,+}$, $\bar{y}_k \in \langle \bar{x} \rangle_{i,-}$ and $\bar{y}_j \not\sim \bar{y}_{j+1}$ for $1 \le j < k$. By definition of $\not\sim$, it gives $\bar{z}_1, \dots, \bar{z}_{k-1} \in P_{i+1}$ with $\bar{y}_j \in \bar{z}_j^-$ and $\bar{y}_{j+1} \in \bar{z}_j^+$ for $1 \le j < k$. So we have $x <_i z_1 <_i \dots <_i z_{k-1} <_i x$, contradicting the loop-freeness of the basis $P$. Hence, $P$ satisfies Axiom (T3'). $\square$

**Lemma 3.4.16.** *P satisfies Axiom (T4').*
Proof. By contradiction, suppose that there are $i \in \mathbb{N}^*$, $m, n \in \mathbb{N}$ with $m > i$ and $n > i$, $\bar{x} \in P_m$ and $\bar{y} \in P_n$ such that $\langle \bar{x} \rangle_{i,+} \cap \langle \bar{y} \rangle_{i,-} = \emptyset$, $\langle \bar{x} \rangle_{i-1,+} \frown^* \langle \bar{y} \rangle_{i-1,-}$ and $\langle \bar{y} \rangle_{i-1,+} \frown^* \langle \bar{x} \rangle_{i-1,-}$. By the same method as for Lemma 3.4.15, we get $r, s \in \mathbb{N}$, $u_1, \dots, u_r \in P_i$, $v_1, \dots, v_s \in P_i$ such that

$$x <_i u_1 <_i \dots <_i u_r <_i y <_i v_1 <_i \dots <_i v_s <_i x,$$

contradicting the loop-freeness of the basis P. Hence, P satisfies Axiom (T4'). □

We can conclude that:

**Theorem 3.4.17.** The $\omega$-hypergraph P associated to K is a torsion-free complex.

*Proof.* This follows from Lemmas 3.4.2 and 3.4.13 to 3.4.16 and Propositions 1.7.1 and 1.7.2. □

Finally, we show that $\bar{\Sigma}$ exhibits an isomorphism between the two $\omega$-categories of cells:

**Theorem 3.4.18.** $\bar{\Sigma}$ induces an isomorphism of $\omega$-categories between $\text{Cell}(P)$ and $\text{Cell}^*(K)$. Moreover, for $\bar{x} \in P$, we have $\bar{\Sigma}(\langle \bar{x} \rangle) = [x]$.

*Proof.* By definition, $\bar{\Sigma}$ commutes with the source, target and identity operations defined on the $\omega$-categories $\text{Cell}(P)$ and $\text{Cell}^*(K)$. We show that it commutes with the composition operations. Given $i, n \in \mathbb{N}$ with $i < n$, $i$-composable cells $X, Y \in \text{Cell}(P)_n$, by Lemma 2.2.2, we have $X_{j,\epsilon} \cap Y_{j,\epsilon} = \emptyset$ for $j \in \mathbb{N}$ with $i < j \le n$ and $\epsilon \in \{-, +\}$. Thus, by Lemma 3.4.6, it follows readily that $\bar{\Sigma}_n(X_*_i Y) = \bar{\Sigma}_n(X)_*_i \bar{\Sigma}_n(Y)$. Thus, $\bar{\Sigma}$ is a morphism of $\omega$-categories. We conclude with Proposition 3.4.12. □

**3.5 Absence of other embeddings** We conclude our comparison of the pasting diagram formalisms by showing that there are no embeddings between the four formalisms except the ones already proved, that is, that parity complexes, pasting scheme and augmented directed complexes are particular cases of torsion-free complexes (under the caveats stated for parity complexes and pasting schemes). We show these nonexistence results by simply exhibiting counter-examples to the other embeddings.

Since adc's are not exactly $\omega$-hypergraphs, we should make the following precision. When we say that “there is no embedding of adc's with loop-free unital bases into the formalism X”, we mean that, in general, the $\omega$-hypergraph obtained from an adc with loop-free unital basis (as described in Section 3.4) is not an instance of X. Conversely, when we say that “there is no embedding of the formalism X into adc's with loop-free unital bases”, we mean that, in general, the pre-adc with basis obtained from an $\omega$-hypergraph which is an instance of X (as described in Section 1.6) is not an adc with loop-free unital basis.

**No embedding in parity complexes** We show that there are no embeddings into parity complexes of the other formalisms. Considering the axioms of parity complexes, Axiom (C4) is relatively strong, and it has no real equivalent in the other formalisms, so it can be used to build a counter-example to embeddings. The $\omega$-hypergraph (9) is a pasting scheme satisfying Axiom (T4) (and thus is a torsion-free complex) and is an adc with loop-free unital basis. But it is not a parity complex as we have seen in Section 1.4, because it does not satisfy Axiom (C4). So pasting schemes, augmented directed complexes and torsion-free complexes are not parity complexes in general.
**Proof.** By contradiction, suppose that there are $i \in \mathbb{N}^*$, $m, n \in \mathbb{N}$ with $m > i$ and $n > i$, $\bar{x} \in P_m$ and $\bar{y} \in P_n$ such that $\langle \bar{x} \rangle_{i,+} \cap \langle \bar{y} \rangle_{i,-} = \emptyset$, $\langle \bar{x} \rangle_{i-1,+} \frown^* \langle \bar{y} \rangle_{i-1,-}$ and $\langle \bar{y} \rangle_{i-1,+} \frown^* \langle \bar{x} \rangle_{i-1,-}$. By the same method as for Lemma 3.4.15, we get $r, s \in \mathbb{N}$, $u_1, \dots, u_r \in P_i$, $v_1, \dots, v_s \in P_i$ such that

$$x <_i u_1 <_i \dots <_i u_r <_i y <_i v_1 <_i \dots <_i v_s <_i x,$$

contradicting the loop-freeness of the basis $P$. Hence, $P$ satisfies Axiom (T4'). □

We can conclude that:

**Theorem 3.4.17.** The $\omega$-hypergraph $P$ associated to $K$ is a torsion-free complex.

**Proof.** This follows from Lemmas 3.4.2 and 3.4.13 to 3.4.16 and Propositions 1.7.1 and 1.7.2. □

Finally, we show that $\bar{\Sigma}$ exhibits an isomorphism between the two $\omega$-categories of cells:

**Theorem 3.4.18.** $\bar{\Sigma}$ induces an isomorphism of $\omega$-categories between $\text{Cell}(P)$ and $\text{Cell}^*(K)$. Moreover, for $\bar{x} \in P$, we have $\bar{\Sigma}(\langle \bar{x} \rangle) = [x]$.

**Proof.** By definition, $\bar{\Sigma}$ commutes with the source, target and identity operations defined on the $\omega$-categories $\text{Cell}(P)$ and $\text{Cell}^*(K)$. We show that it commutes with the composition operations. Given $i, n \in \mathbb{N}$ with $i < n$, $i$-composable cells $X, Y \in \text{Cell}(P)_n$, by Lemma 2.2.2, we have $X_{j,\epsilon} \cap Y_{j,\epsilon} = \emptyset$ for $j \in \mathbb{N}$ with $i < j \le n$ and $\epsilon \in \{-, +\}$. Thus, by Lemma 3.4.6, it follows readily that $\bar{\Sigma}_n(X *_i Y) = \bar{\Sigma}_n(X) *_i \bar{\Sigma}_n(Y)$. Thus, $\bar{\Sigma}$ is a morphism of $\omega$-categories. We conclude with Proposition 3.4.12. □

**3.5 Absence of other embeddings** We conclude our comparison of the pasting diagram formalisms by showing that there are no embeddings between the four formalisms except the ones already proved, that is, that parity complexes, pasting scheme and augmented directed complexes are particular cases of torsion-free complexes (under the caveats stated for parity complexes and pasting schemes). We show these nonexistence results by simply exhibiting counter-examples to the other embeddings.

Since adc's are not exactly $\omega$-hypergraphs, we should make the following precision. When we say that “there is no embedding of adc's with loop-free unital bases into the formalism X”, we mean that, in general, the $\omega$-hypergraph obtained from an adc with loop-free unital basis (as described in Section 3.4) is not an instance of X. Conversely, when we say that “there is no embedding of the formalism X into adc's with loop-free unital bases”, we mean that, in general, the pre-adc with basis obtained from an $\omega$-hypergraph which is an instance of X (as described in Section 1.6) is not an adc with loop-free unital basis.

**No embedding in parity complexes** We show that there are no embeddings into parity complexes of the other formalisms. Considering the axioms of parity complexes, Axiom (C4) is relatively strong, and it has no real equivalent in the other formalisms, so it can be used to build a counter-example to embeddings. The $\omega$-hypergraph (9) is a pasting scheme satisfying Axiom (T4) (and thus is a torsion-free complex) and is an adc with loop-free unital basis. But it is not a parity complex as we have seen in Section 1.4, because it does not satisfy Axiom (C4). So pasting schemes, augmented directed complexes and torsion-free complexes are not parity complexes in general.
**No embedding in pasting schemes** We now show that there are no embeddings into pasting schemes of the other formalisms. We use the relatively strong Axiom (S2) to build a counter-example to the embeddings. The following $\omega$-hypergraph is a parity complex satisfying Axiom (T4) (and thus it is a torsion-free complex) and is an adc with loop-free unital basis but it is not a pasting scheme:

$$ \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.8, every node/.style={scale=0.8}] 
% Define vertices
\node[circle, fill, inner sep=1.2pt] (d) at (0,0) {};
\node[circle, fill, inner sep=1.2pt] (e) at (0,1) {};
\node[circle, fill, inner sep=1.2pt] (a) at (1,0) {};
\node[circle, fill, inner sep=1.2pt] (b) at (1,1) {};
\node[circle, fill, inner sep=1.2pt] (c) at (2,0) {};
\node[circle, fill, inner sep=1.2pt] (x) at (2,1) {};
\node[circle, fill, inner sep=1.2pt] (y) at (3,0) {};
\node[circle, fill, inner sep=1.2pt] (z) at (3,1) {};

% Define edges
\draw (d) -- (e) node[left] {$d$};
\draw (e) -- (a) node[left] {$\alpha_4$};
\draw (a) -- (b) node[right] {$b$};
\draw (b) -- (c) node[right] {$w$};
\draw (c) -- (x) node[right] {$\alpha_1$};
\draw (x) -- (y) node[right] {$y$};
\draw (y) -- (a') node[right] {$\alpha_2$};
\draw (a') -- (b') node[right] {$b'$};
\draw (c) -- (z) node[right] {$\alpha_3$};
\draw (z) -- (e) node[left] {$e$};

% Labels
\node[above] at (d) {$z$};
\node[above] at (e) {$y$};
\node[above] at (a) {$\alpha_4$};
\node[above] at (b) {$b$};
\node[above] at (c) {$\alpha_2$};
\node[above] at (x) {$\alpha_1$};
\node[above] at (y) {$y$};
\node[above] at (z) {$\alpha_3$};

% Arrows
\draw[->] (d) -- (e);
\draw[->] (a) -- (b);
\draw[->] (c) -- (x);
\draw[->] (y) -- (a');
\draw[->] (c) -- (z);

\end{tikzpicture} \quad (16) $$

Indeed, Axiom (S2) is not satisfied because $\alpha_2 \triangleleft \alpha_3$ and $y \in B(\alpha_2) \cap E(\alpha_3) \neq \emptyset$. Note that (16) is essentially the $\omega$-hypergraph (14) without the 3-generator $A$ and the 2-generators $\alpha'_1$ and $\alpha'_4$.

**No embedding in augmented directed complexes** Finally, we prove that there are no embeddings into augmented directed complexes with loop-free unital basis of the other formalisms. As shown in Section 3.4, such adc's satisfy Axiom (T4'), which is a stronger version of Axiom (T4). Thus, we can find a counter-example to embedding into adc's with loop-free unital basis by considering an adequate $\omega$-hypergraph which satisfies Axiom (T4) but not Axiom (T4'). Consider the $\omega$-hypergraph $P$ from Figure 4 where the 3-generators $A, B, C$ are such that

$$ \begin{align*} A^{-} &= \{\beta, \gamma\}, & B^{-} &= \{\delta, \epsilon\}, & C^{-} &= \{\alpha, \gamma', \delta', \zeta\}, \\ A^{+} &= \{\beta', \gamma'\}, & B^{+} &= \{\delta', \epsilon'\}, & C^{+} &= \{\alpha', \gamma'', \zeta'\}. \end{align*} $$

It can be shown that it is a parity complex and a pasting scheme. It moreover satisfies Axiom (T4) so that it is a torsion-free complex by Theorem 3.3.9. But its associated pre-adc is an adc with a basis which is not loop-free unital. Indeed, we have $e \le [A]_{1,+} \wedge [B]_{1,-}$, $h \le [B]_{1,+} \wedge [C]_{1,-}$ and $b \le [C]_{1,-} \wedge [A]_{1,+}$, so that $A <_1 B <_1 C <_1 A$. Hence, the basis of the associated augmented directed complex is not loop-free.
**No embedding in pasting schemes** We now show that there are no embeddings into pasting schemes of the other formalisms. We use the relatively strong Axiom (S2) to build a counter-example to the embeddings. The following $\omega$-hypergraph is a parity complex satisfying Axiom (T4) (and thus it is a torsion-free complex) and is an adc with loop-free unital basis but it is not a pasting scheme:

<figure><img src="image_18.png" /></figure>

Indeed, Axiom (S2) is not satisfied because $\alpha_2 \triangleleft \alpha_3$ and $y \in B(\alpha_2) \cap E(\alpha_3) \neq \emptyset$. Note that (16) is essentially the $\omega$-hypergraph (14) without the 3-generator A and the 2-generators $\alpha'_1$ and $\alpha'_4$.

**No embedding in augmented directed complexes** Finally, we prove that there are no embeddings into augmented directed complexes with loop-free unital basis of the other formalisms. As shown in Section 3.4, such adc's satisfy Axiom (T4'), which is a stronger version of Axiom (T4). Thus, we can find a counter-example to embedding into adc's with loop-free unital basis by considering an adequate $\omega$-hypergraph which satisfies Axiom (T4) but not Axiom (T4'). Consider the $\omega$-hypergraph P from Figure 4 where the 3-generators A, B, C are such that

$$ 
\begin{align*}
A^{-} &= \{\beta, \gamma\}, & B^{-} &= \{\delta, \epsilon\}, & C^{-} &= \{\alpha, \gamma', \delta', \zeta\}, \\
A^{+} &= \{\beta', \gamma'\}, & B^{+} &= \{\delta', \epsilon'\}, & C^{+} &= \{\alpha', \gamma'', \zeta'\}.
\end{align*}
$$

It can be shown that it is a parity complex and a pasting scheme. It moreover satisfies Axiom (T4) so that it is a torsion-free complex by Theorem 3.3.9. But its associated pre-adc is an adc with a basis which is not loop-free unital. Indeed, we have $e \le [A]_{1,+} \wedge [B]_{1,-}$, $h \le [B]_{1,+} \wedge [C]_{1,-}$ and $b \le [C]_{1,-} \wedge [A]_{1,+}$, so that $A <_1 B <_1 C <_1 A$. Hence, the basis of the associated augmented directed complex is not loop-free.
<figure><img src="image_15.png" /><figcaption>Figure 4: The $\omega$-hypergraph $P$</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Figure 4: The $\omega$-hypergraph $P$</figcaption></figure>
# Appendix A: Details on the gluing theorem

This section is devoted to the proof of Theorem 2.1.1, which allows gluing sets of generators to existing cells in order to get new cells. This result requires some technical results about movement, which appears in the definition of cells. We first introduce these results and then discuss the proof of the gluing theorem.

**A.1 Movement properties** Here, we state and prove here several such properties, some of which coming already present in [23].

In the following, we suppose given an $\omega$-hypergraph $P$. We first state a criterion for movement:

**Lemma A.1.1 ([23, Proposition 2.1])**. For $n \in \mathbb{N}$, finite subsets $U \subseteq P_n$ and $S \subseteq P_{n+1}$, there exists $V \subseteq P_n$ such that $S$ moves $U$ to $V$ if and only if $S^\pm \subseteq U$ and $U \cap S^+ = \emptyset$.

*Proof.* If $S$ moves $U$ to $V$, then, by definition, $S^\pm \subseteq (V \cup S^-) \setminus S^+ = U$ and

$$U \cap S^+ = ((V \cup S^-) \setminus S^+) \cap S^+ = \emptyset.$$

Conversely, if $S^\pm \subseteq U$ and $U \cap S^+ = \emptyset$, let $V = (U \cup S^+) \setminus S^-$. Then

$$ \begin{aligned} (V \cup S^-) \setminus S^+ &= (U \cup S^+ \cup S^-) \setminus S^+ \\ &= (U \setminus S^+) \cup (S^- \setminus S^+) \\ &= U \cup S^\pm \qquad (\text{since } U \cap S^+ = \emptyset) \\ &= U \qquad (\text{since } S^\pm \subseteq U) \end{aligned} $$

and $S$ moves $U$ to $V$.

□

The next property states that it is possible to modify a movement by adding or removing “independent” elements.

**Lemma A.1.2 ([23, Proposition 2.2])**. Let $n \in \mathbb{N}$, $U, V \subseteq P_n$ and $S \subseteq P_{n+1}$ be finite subsets such that $S$ moves $U$ to $V$. Then, given $X, Y \subseteq P_n$ such that $X \subseteq U$, $X \cap S^\pm = \emptyset$ and $Y \cap (S^- \cup S^+) = \emptyset$, we have that $S$ moves $(U \cup Y) \setminus X$ to $(V \cup Y) \setminus X$.

*Proof.* By Lemma A.1.1, we have $S^\pm \subseteq U$ and $U \cap S^+ = \emptyset$. Using the hypothesis, we can refine both equalities to $S^\pm \subseteq (U \cup Y) \setminus X$ and $((U \cup Y) \setminus X) \cap S^+ = \emptyset$. Using Lemma A.1.1 again, $S$ moves $(U \cup Y) \setminus X$ to $W$ where

$$ \begin{aligned} W &= (((U \cup Y) \setminus X) \cup S^+) \setminus S^- \\ &= ((U \cup S^+ \cup Y) \setminus X) \setminus S^- \qquad (\text{since } X \cap S^+ \subseteq U \cap S^+ = \emptyset) \\ &= (((U \cup S^+) \setminus S^-) \cup Y) \setminus X \qquad (\text{since } Y \cap S^- = \emptyset) \\ &= (V \cup Y) \setminus X. \end{aligned} $$

□

The following property gives sufficient conditions for composing movements.

**Lemma A.1.3 ([23, Proposition 2.3])**. Let $n \in \mathbb{N}$, and $U, V, W \subseteq P_n$, $S, T \subseteq P_{n+1}$ be finite subsets such that $S$ moves $U$ to $V$ and $T$ moves $V$ to $W$, if $S^- \cap T^+ = \emptyset$ then $S \cup T$ moves $U$ to $W$.
# Appendix A: Details on the gluing theorem

This section is devoted to the proof of Theorem 2.1.1, which allows gluing sets of generators to existing cells in order to get new cells. This result requires some technical results about movement, which appears in the definition of cells. We first introduce these results and then discuss the proof of the gluing theorem.

**A.1 Movement properties** Here, we state and prove here several such properties, some of which coming already present in [23].

In the following, we suppose given an $\omega$-hypergraph $P$. We first state a criterion for movement:

**Lemma A.1.1 ([23, Proposition 2.1])**. For $n \in \mathbb{N}$, finite subsets $U \subseteq P_n$ and $S \subseteq P_{n+1}$, there exists $V \subseteq P_n$ such that $S$ moves $U$ to $V$ if and only if $S^\pm \subseteq U$ and $U \cap S^+ = \emptyset$.

*Proof.* If $S$ moves $U$ to $V$, then, by definition, $S^\pm \subseteq (V \cup S^-) \setminus S^+ = U$ and

$$U \cap S^+ = ((V \cup S^-) \setminus S^+) \cap S^+ = \emptyset.$$

Conversely, if $S^\pm \subseteq U$ and $U \cap S^+ = \emptyset$, let $V = (U \cup S^+) \setminus S^-$. Then

$$ \begin{aligned} (V \cup S^-) \setminus S^+ &= (U \cup S^+ \cup S^-) \setminus S^+ \\ &= (U \setminus S^+) \cup (S^- \setminus S^+) \\ &= U \cup S^\pm \qquad (\text{since } U \cap S^+ = \emptyset) \\ &= U \qquad (\text{since } S^\pm \subseteq U) \end{aligned} $$

and $S$ moves $U$ to $V$.

<figure><img src="image_20.png" /></figure>

The next property states that it is possible to modify a movement by adding or removing “independent” elements.

**Lemma A.1.2 ([23, Proposition 2.2])**. Let $n \in \mathbb{N}$, $U, V \subseteq P_n$ and $S \subseteq P_{n+1}$ be finite subsets such that $S$ moves $U$ to $V$. Then, given $X, Y \subseteq P_n$ such that $X \subseteq U$, $X \cap S^\pm = \emptyset$ and $Y \cap (S^- \cup S^+) = \emptyset$, we have that $S$ moves $(U \cup Y) \setminus X$ to $(V \cup Y) \setminus X$.

*Proof.* By Lemma A.1.1, we have $S^\pm \subseteq U$ and $U \cap S^+ = \emptyset$. Using the hypothesis, we can refine both equalities to $S^\pm \subseteq (U \cup Y) \setminus X$ and $((U \cup Y) \setminus X) \cap S^+ = \emptyset$. Using Lemma A.1.1 again, $S$ moves $(U \cup Y) \setminus X$ to $W$ where

$$ \begin{aligned} W &= (((U \cup Y) \setminus X) \cup S^+) \setminus S^- \\ &= ((U \cup S^+ \cup Y) \setminus X) \setminus S^- \qquad (\text{since } X \cap S^+ \subseteq U \cap S^+ = \emptyset) \\ &= (((U \cup S^+) \setminus S^-) \cup Y) \setminus X \qquad (\text{since } Y \cap S^- = \emptyset) \\ &= (V \cup Y) \setminus X. \end{aligned} $$

<figure><img src="image_21.png" /></figure>

The following property gives sufficient conditions for composing movements.

**Lemma A.1.3 ([23, Proposition 2.3])**. Let $n \in \mathbb{N}$, and $U, V, W \subseteq P_n$, $S, T \subseteq P_{n+1}$ be finite subsets such that $S$ moves $U$ to $V$ and $T$ moves $V$ to $W$, if $S^- \cap T^+ = \emptyset$ then $S \cup T$ moves $U$ to $W$.
Proof. We compute $(U \cup (S \cup T)^+) \setminus (S \cup T)^-$:

$$ (U \cup S^+ \cup T^+) \setminus (S^- \cup T^-) = (((U \cup S^+) \setminus S^-) \cup T^+) \setminus T^- = (V \cup T^+) \setminus T^- = W. $$

Similarly, $(W \cup (S \cup T)^-) \setminus (S \cup T)^+ = U$ and $S \cup T$ moves $U$ to $W$. $\square$

<figure><img src="image_16.png" /></figure>

Conversely, the next property enables decomposing movements, under a condition of orthogonality: given $n \in \mathbb{N}$ and finite sets $S, T \subseteq P_n$, we say that $S$ and $T$ are orthogonal, written $S \perp T$, when $(S^- \cap T^-) \cup (S^+ \cap T^+) = \emptyset$. We then have:

**Lemma A.1.4 ([23, Proposition 2.4]).** Given $n \in \mathbb{N}$, finite subsets $U, W \subseteq P_n$ and $S, T \subseteq P_{n+1}$ such that $S \cup T$ moves $U$ to $W$ and $S^\pm \subseteq U$, if $S \perp T$ then there exists $V$ such that $S$ moves $U$ to $V$ and $T$ moves $V$ to $W$.

*Proof.* Let $R = S \cup T$. By Lemma A.1.1, $R^\pm \subseteq U$ and $U \cap S^+ \subseteq U \cap R^+ = \emptyset$. By Lemma A.1.1 again, $S$ moves $U$ to $V = (U \cup S^+) \setminus S^-$. Moreover,

$$ \begin{align*} S^- \cap T^+ &= S^\pm \cap T^+ && (\text{since } S^+ \cap T^+ = \emptyset, \text{ by } S \perp T) \\ &\subseteq U \cap T^+ && (\text{since } S^\pm \subseteq U, \text{ by hypothesis}) \\ &\subseteq U \cap (S \cup T)^+ \\ &= \emptyset && (\text{by Lemma A.1.1}). \end{align*} $$

so that

$$ \begin{align*} R^\pm \subseteq U & \\ \Leftrightarrow & ((S^- \cup T^-) \setminus T^+) \setminus S^+ \subseteq U \\ \Leftrightarrow & ((T^- \cup T^+) \cup S^-) \setminus S^+ \subseteq U && (\text{since } S^- \cap T^+ = \emptyset) \\ \Leftrightarrow & T^\pm \cup S^- \subseteq U \cup S^+ \\ \Leftrightarrow & T^\pm \subseteq (U \cup S^+) \setminus S^- && (\text{since } T^\pm \cap S^- = \emptyset, \text{ by } S \perp T). \end{align*} $$

Hence, $T^\pm \subseteq (U \cup S^+) \setminus S^- = V$ and

$$ V \cap T^+ \subseteq (U \cup S^+) \cap T^+ \subseteq (U \cap R^+) \cup (S^+ \cap T^+) = \emptyset. $$

By Lemma A.1.1, $T$ moves $V$ to $(V \cup T^+) \setminus T^-$. Moreover,

$$ \begin{align*} S^- \cap T^+ &= S^\pm \cap T^+ && (\text{since } S \perp T) \\ &\subseteq U \cap R^+ && (\text{since } S^\pm \subseteq U \text{ by hypothesis}) \\ &= \emptyset. \end{align*} $$

Therefore,

$$ \begin{align*} (V \cup T^+) \setminus T^- &= (((U \cup S^+) \setminus S^-) \cup T^+) \setminus T^- \\ &= (U \cup S^+ \cup T^+) \setminus (S^- \cup T^-) && (\text{since } S^- \cap T^+ = \emptyset) \\ &= W. \end{align*} $$

Hence, $T$ moves $V$ to $W$. $\square$

<figure><img src="image_17.png" /></figure>

The next three properties (not in [23]) describe which elements are touched or left untouched by movement.
Proof. We compute $(U \cup (S \cup T)^+) \setminus (S \cup T)^-$:

$$(U \cup S^+ \cup T^+) \setminus (S^- \cup T^-) = (((U \cup S^+) \setminus S^-) \cup T^+) \setminus T^- = (V \cup T^+) \setminus T^- = W.$$

Similarly, $(W \cup (S \cup T)^-) \setminus (S \cup T)^+ = U$ and $S \cup T$ moves $U$ to $W$. □

Conversely, the next property enables decomposing movements, under a condition of orthogonality: given $n \in \mathbb{N}$ and finite sets $S, T \subseteq P_n$, we say that $S$ and $T$ are orthogonal, written $S \perp T$, when $(S^- \cap T^-) \cup (S^+ \cap T^+) = \emptyset$. We then have:

**Lemma A.1.4 ([23, Proposition 2.4]).** Given $n \in \mathbb{N}$, finite subsets $U, W \subseteq P_n$ and $S, T \subseteq P_{n+1}$ such that $S \cup T$ moves $U$ to $W$ and $S^\pm \subseteq U$, if $S \perp T$ then there exists $V$ such that $S$ moves $U$ to $V$ and $T$ moves $V$ to $W$.

*Proof.* Let $R = S \cup T$. By Lemma A.1.1, $R^\pm \subseteq U$ and $U \cap S^+ \subseteq U \cap R^+ = \emptyset$. By Lemma A.1.1 again, $S$ moves $U$ to $V = (U \cup S^+) \setminus S^-$. Moreover,

$$
\begin{align*}
S^- \cap T^+ &= S^\pm \cap T^+ && (\text{since } S^+ \cap T^+ = \emptyset, \text{ by } S \perp T) \\
&\subseteq U \cap T^+ && (\text{since } S^\pm \subseteq U, \text{ by hypothesis}) \\
&\subseteq U \cap (S \cup T)^+ \\
&= \emptyset && (\text{by Lemma A.1.1}).
\end{align*}
$$

so that

$$
\begin{align*}
& R^{\pm} \subseteq U \\
\Leftrightarrow \quad & ((S^{-} \cup T^{-}) \setminus T^{+}) \setminus S^{+} \subseteq U \\
\Leftrightarrow \quad & ((T^{-} \setminus T^{+}) \cup S^{-}) \setminus S^{+} \subseteq U && (\text{since } S^{-} \cap T^{+} = \emptyset) \\
\Leftrightarrow \quad & T^{\pm} \cup S^{-} \subseteq U \cup S^{+} \\
\Leftrightarrow \quad & T^{\pm} \subseteq (U \cup S^{+}) \setminus S^{-} && (\text{since } T^{\pm} \cap S^{-} = \emptyset, \text{ by } S \perp T).
\end{align*}
$$

Hence, $T^\pm \subseteq (U \cup S^+) \setminus S^- = V$ and

$$V \cap T^+ \subseteq (U \cup S^+) \cap T^+ \subseteq (U \cap R^+) \cup (S^+ \cap T^+) = \emptyset.$$

By Lemma A.1.1, $T$ moves $V$ to $(V \cup T^+) \setminus T^-$. Moreover,

$$
\begin{align*}
S^- \cap T^+ &= S^\pm \cap T^+ && (\text{since } S \perp T) \\
&\subseteq U \cap R^+ && (\text{since } S^\pm \subseteq U \text{ by hypothesis}) \\
&= \emptyset. &&
\end{align*}
$$

Therefore,

$$
\begin{align*}
(V \cup T^+) \setminus T^- &= (((U \cup S^+) \setminus S^-) \cup T^+) \setminus T^- \\
&= (U \cup S^+ \cup T^+) \setminus (S^- \cup T^-) && (\text{since } S^- \cap T^+ = \emptyset) \\
&= W. &&
\end{align*}
$$

Hence, $T$ moves $V$ to $W$. □

The next three properties (not in [23]) describe which elements are touched or left untouched by movement.
**Lemma A.1.5.** Given $n \in \mathbb{N}$, finite subsets $U, V \subseteq P_n$ and $S \subseteq P_{n+1}$, if $S$ moves $U$ to $V$, then

$$S^{\mp} = U \setminus V \quad \text{and} \quad S^{\pm} = V \setminus U.$$

In particular, if $T$ moves $U$ to $V$, then $S^{\mp} = T^{\mp}$ and $S^{\pm} = T^{\pm}$.

*Proof.* By the definition of movement, we have $V = (U \cup S^+) \setminus S^-$ and $U = (V \cup S^-) \setminus S^+$. Therefore, since $U \cap S^+ = \emptyset$, $U \cap V = U \cap ((U \setminus S^-) \cup S^+) = U \setminus S^{\mp}$. Similarly, $U \cap V = V \setminus S^{\pm}$. Hence, $S^{\mp} = U \setminus V$ and $S^{\pm} = V \setminus U$. $\square$

**Lemma A.1.6.** Given $n \in \mathbb{N}$, finite subsets $U, V \subseteq P_n$ and $S \subseteq P_{n+1}$, if $S$ moves $U$ to $V$, then

$$U \setminus S^{-} = U \setminus S^{\mp} = U \cap V = V \setminus S^{\pm} = V \setminus S^{+}.$$

*Proof.* We compute that

$$U \setminus S^{-} = U \setminus S^{\mp} \quad \text{(since } U \cap S^{+} = \emptyset, \text{ by definition of movement})$$

$$= U \cap V \quad \text{(by Lemma A.1.5)}$$

$$= V \setminus S^{\pm}$$

$$= V \setminus S^{+} \quad \text{(since } V \cap S^{-} = \emptyset, \text{ by definition of movement)} \quad \square$$

**Lemma A.1.7.** For $n \in \mathbb{N}$, finite subsets $U, V \subseteq P_n$ and $S \subseteq P_{n+1}$, if $S$ moves $U$ to $V$, then

$$U = (U \cap V) \cup S^{\mp} \quad \text{and} \quad V = (U \cap V) \cup S^{\pm}.$$

*Proof.* By Lemma A.1.6, we have

$$U = (V \cup S^{-}) \setminus S^{+} = (V \setminus S^{+}) \cup (S^{-} \setminus S^{+}) = (U \cap V) \cup S^{\mp}.$$

Moreover,

$$(U \cap V) \cap S^{\mp} \subseteq V \cap S^{-} = ((U \cup S^{-}) \setminus S^{-}) \cap S^{-} = \emptyset.$$

$$\text{Hence, } U = (U \cap V) \cup S^{\mp}. \text{ Similarly } V = (U \cap V) \cup S^{\pm}. \quad \square$$

For $n \in \mathbb{N}$, $V \subseteq U \subseteq P_n$, we say that $V$ is initial (resp. terminal) in $U$ for $\langle U \rangle$ when, for all $u \in U$, whenever there exists $v \in V$ such that $u \langle U \rangle v$ (resp. $v \langle U \rangle u$), we have $u \in V$. The last lemma enables decomposing a moving set starting from a subset which is a segment:

**Lemma A.1.8.** For $n \in \mathbb{N}$, finite subsets $U, V \subseteq P_n$, $S \subseteq P_{n+1}$ and $T \subseteq S$ such that $S$ is fork-free and moves $U$ to $V$, and $T$ is a segment in $S$ for $\langle S \rangle$, there exist $L, R \subseteq S$ and $A, B \subseteq P_n$ such that

- $L, T, R$ is a partition of $S$,

- $L$ is initial in $S$ for $\langle S \rangle$ and $R$ is final in $S$ for $\langle S \rangle$,

- $L$ moves $U$ to $A$, $T$ moves $A$ to $B$ and $R$ moves $B$ to $V$.

*Proof.* Let $L = \{x \in S \mid x \langle_S T \rangle\}$ and $R = S \setminus (L \cup T)$. $L, T, R$ is a partition of $S$, and since $S$ is fork-free, we have $L \perp T$, $L \perp R$ and $T \perp R$. Since $T$ is a segment for $\langle S \rangle$, we have that $L^{-} \cap T^{+} = \emptyset$, and, by definition of $L$ and $R$, $L^{-} \cap R^{+} = \emptyset$ so that $L$ is initial in $S$. In particular, $L^{\mp} \subseteq U$. Thus, by Lemma A.1.3, writing $A$ for the set $(U \cup L^{+}) \setminus L^{-}$, we have that $L$ moves $U$ to $A$. Furthermore, since $L \cap R = \emptyset$, we have $T^{-} \cap R^{+} = \emptyset$ so that $R$ is terminal in $S$. In particular, $R^{\pm} \subseteq V$. Thus, by the dual of Lemma A.1.3, writing $B$ for $(V \cup R^{-}) \setminus R^{+}$, we have that $R$ moves $B$ to $V$. $\square$
**Lemma A.1.5.** Given $n \in \mathbb{N}$, finite subsets $U, V \subseteq P_n$ and $S \subseteq P_{n+1}$, if $S$ moves $U$ to $V$, then

$$S^{\mp} = U \setminus V \quad \text{and} \quad S^{\pm} = V \setminus U.$$

In particular, if $T$ moves $U$ to $V$, then $S^{\mp} = T^{\mp}$ and $S^{\pm} = T^{\pm}$.

*Proof.* By the definition of movement, we have $V = (U \cup S^+) \setminus S^-$ and $U = (V \cup S^-) \setminus S^+$. Therefore, since $U \cap S^+ = \emptyset$, $U \cap V = U \cap ((U \setminus S^-) \cup S^+) = U \setminus S^{\mp}$. Similarly, $U \cap V = V \setminus S^{\pm}$. Hence, $S^{\mp} = U \setminus V$ and $S^{\pm} = V \setminus U$. $\square$

**Lemma A.1.6.** Given $n \in \mathbb{N}$, finite subsets $U, V \subseteq P_n$ and $S \subseteq P_{n+1}$, if $S$ moves $U$ to $V$, then

$$U \setminus S^{-} = U \setminus S^{\mp} = U \cap V = V \setminus S^{\pm} = V \setminus S^{+}.$$

*Proof.* We compute that

$$U \setminus S^{-} = U \setminus S^{\mp} \quad (\text{since } U \cap S^{+} = \emptyset, \text{ by definition of movement})$$

$$= U \cap V \quad (\text{by Lemma A.1.5})$$

$$= V \setminus S^{\pm}$$

$$= V \setminus S^{+} \quad (\text{since } V \cap S^{-} = \emptyset, \text{ by definition of movement}) \quad \square$$

**Lemma A.1.7.** For $n \in \mathbb{N}$, finite subsets $U, V \subseteq P_n$ and $S \subseteq P_{n+1}$, if $S$ moves $U$ to $V$, then

$$U = (U \cap V) \cup S^{\mp} \quad \text{and} \quad V = (U \cap V) \cup S^{\pm}.$$

*Proof.* By Lemma A.1.6, we have

$$U = (V \cup S^{-}) \setminus S^{+} = (V \setminus S^{+}) \cup (S^{-} \setminus S^{+}) = (U \cap V) \cup S^{\mp}.$$

Moreover,

$$(U \cap V) \cap S^{\mp} \subseteq V \cap S^{-} = ((U \cup S^{-}) \setminus S^{-}) \cap S^{-} = \emptyset.$$

$$\text{Hence, } U = (U \cap V) \cup S^{\mp}. \quad \text{Similarly } V = (U \cap V) \cup S^{\pm}. \quad \square$$

For $n \in \mathbb{N}$, $V \subseteq U \subseteq P_n$, we say that $V$ is initial (resp. terminal) in $U$ for $\langle U \rangle$ when, for all $u \in U$, whenever there exists $v \in V$ such that $u \langle U \rangle v$ (resp. $v \langle U \rangle u$), we have $u \in V$. The last lemma enables decomposing a moving set starting from a subset which is a segment:

**Lemma A.1.8.** For $n \in \mathbb{N}$, finite subsets $U, V \subseteq P_n$, $S \subseteq P_{n+1}$ and $T \subseteq S$ such that $S$ is fork-free and moves $U$ to $V$, and $T$ is a segment in $S$ for $\langle S \rangle$, there exist $L, R \subseteq S$ and $A, B \subseteq P_n$ such that

- $L, T, R$ is a partition of $S$,

- $L$ is initial in $S$ for $\langle S \rangle$ and $R$ is final in $S$ for $\langle S \rangle$,

- $L$ moves $U$ to $A$, $T$ moves $A$ to $B$ and $R$ moves $B$ to $V$.

*Proof.* Let $L = \{x \in S \mid x \langle_S T \rangle\}$ and $R = S \setminus (L \cup T)$. $L, T, R$ is a partition of $S$, and since $S$ is fork-free, we have $L \perp T$, $L \perp R$ and $T \perp R$. Since $T$ is a segment for $\langle S \rangle$, we have that $L^{-} \cap T^{+} = \emptyset$, and, by definition of $L$ and $R$, $L^{-} \cap R^{+} = \emptyset$ so that $L$ is initial in $S$. In particular, $L^{\mp} \subseteq U$. Thus, by Lemma A.1.3, writing $A$ for the set $(U \cup L^{+}) \setminus L^{-}$, we have that $L$ moves $U$ to $A$. Furthermore, since $L \cap R = \emptyset$, we have $T^{-} \cap R^{+} = \emptyset$ so that $R$ is terminal in $S$. In particular, $R^{\pm} \subseteq V$. Thus, by the dual of Lemma A.1.3, writing $B$ for $(V \cup R^{-}) \setminus R^{+}$, we have that $R$ moves $B$ to $V$. $\square$
**A.2 Proof of the gluing theorem** We have now enough material to prove the gluing theorem:

**Theorem 2.1.1.** Let $P$ be an $\omega$-hypergraph which satisfies Axioms (T0), (T1), (T2) and (T3). Given $n \in \mathbb{N}$, an $n$-cell $X$ of $P$ and a finite fork-free set $G \subseteq P_{n+1}$ such that $G$ is glueable on $X$, we have that

(a) $\mathrm{Act}(X, G)$ is a cell and $G^+ \cap X_n = \emptyset$,

(b) $\mathrm{Glue}(X, G)$ is a cell,

(c) given a finite fork-free subset $G' \subseteq P_{n+1}$ which is dually glueable on $X$, $G' \cap G^+ = \emptyset$. and dual properties hold when $G$ is dually glueable on $X$.

*Proof.* See Figure 3 for a representation of the cells in the statement of the theorem. In the following, write $S$ for

$$S = \mathrm{Act}(X, G)_n = (X_n \cup G^+) \setminus G^-.$$

We prove the different sub-properties (and their duals) of the theorem by induction on $n$.

*Proof of (a):* We prove (a) in two steps: first, in the case where $|G| = 1$, then, in the general case.

**Step 1:** (a) holds when $|G| = 1$. Let $x \in P_{n+1}$ be such that $\{x\} = G$. If $n = 0$, then there exists $y \in P_0$ such that $X_0 = \{y\}$. By Axioms (T1) and (T2), there exists $z \in P_0$ with $y \neq z$ such that $x^- = \{y\}$ and $x^+ = \{z\}$. So $\mathrm{Act}(X, G) = \{z\}$ is a cell. So suppose that $n > 0$. Then, we have $S = (X_n \cup x^+) \setminus x^-$ and, in order to prove that $\mathrm{Act}(X, G)$ is a cell, we are required to show that

- $S$ moves $X_{n-1,-}$ to $X_{n-1,+}$;

- $S$ is fork-free.

Using Axiom (T3), we get that $x^-$ is a segment in $X_n$ for $\langle X_n$. By Lemma A.1.8, we can decompose the set $X_n$ as a partition

$$X_n = U \cup x^- \cup V$$

with $U$ initial and $V$ final in $X_n$ and, writing $A, B \subseteq P_{n-1}$ for

$$A = (X_{n-1,-} \cup U^+) \setminus U^- \quad \text{and} \quad B = (X_{n-1,+} \cup V^-) \setminus V^+$$

we have that

$$U \text{ moves } X_{n-1,-} \text{ to } A, \quad x^- \text{ moves } A \text{ to } B, \quad V \text{ moves } B \text{ to } X_{n-1,+}$$

as pictured on Figure 5. In the following, for $Z \subseteq P_{n-1}$, we write $D(Z)$ for the $(n-1)$-pre-cell of $P$ defined by

$$D(Z)_{n-1} = Z \quad \text{and} \quad D(Z)_{i,\epsilon} = X_{i,\epsilon} \quad \text{for } i \in \mathbb{N}_{n-2} \text{ and } \epsilon \in \{-,+\}.$$

Since $D(A) = \mathrm{Act}(D(X_{n-1,-}), U)$, $D(B) = \mathrm{Act}(D(A), x^-)$, and $D(X_{n-1,-}) = \partial_{n-1}^-(X)$ is an $(n-1)$-cell and both $U$ and $x^-$ are fork-free, by using two times the induction hypothesis of Theorem 2.1.1, first on $D(X_{n-1,-})$, then on $D(A)$, we get that

$$D(A) \text{ and } D(B) \text{ are cells.} \qquad (17)$$

By Axiom (T2), we have that

$$x^+ \text{ is fork-free.} \qquad (18)$$
**A.2 Proof of the gluing theorem** We have now enough material to prove the gluing theorem:

**Theorem 2.1.1.** Let $P$ be an $\omega$-hypergraph which satisfies Axioms (T0), (T1), (T2) and (T3). Given $n \in \mathbb{N}$, an $n$-cell $X$ of $P$ and a finite fork-free set $G \subseteq P_{n+1}$ such that $G$ is glueable on $X$, we have that

(a) $\mathrm{Act}(X, G)$ is a cell and $G^+ \cap X_n = \emptyset$,

(b) $\mathrm{Glue}(X, G)$ is a cell,

(c) given a finite fork-free subset $G' \subseteq P_{n+1}$ which is dually glueable on $X$, $G'^- \cap G^+ = \emptyset$. and dual properties hold when $G$ is dually glueable on $X$.

*Proof.* See Figure 3 for a representation of the cells in the statement of the theorem. In the following, write $S$ for

$$S = \mathrm{Act}(X, G)_n = (X_n \cup G^+) \setminus G^-.$$

We prove the different sub-properties (and their duals) of the theorem by induction on $n$.

*Proof of (a):* We prove (a) in two steps: first, in the case where $|G| = 1$, then, in the general case.

**Step 1:** (a) holds when $|G| = 1$. Let $x \in P_{n+1}$ be such that $\{x\} = G$. If $n = 0$, then there exists $y \in P_0$ such that $X_0 = \{y\}$. By Axioms (T1) and (T2), there exists $z \in P_0$ with $y \neq z$ such that $x^- = \{y\}$ and $x^+ = \{z\}$. So $\mathrm{Act}(X, G) = \{z\}$ is a cell. So suppose that $n > 0$. Then, we have $S = (X_n \cup x^+) \setminus x^-$ and, in order to prove that $\mathrm{Act}(X, G)$ is a cell, we are required to show that

- $S$ moves $X_{n-1,-}$ to $X_{n-1,+}$;

- $S$ is fork-free.

Using Axiom (T3), we get that $x^-$ is a segment in $X_n$ for $\langle X_n$. By Lemma A.1.8, we can decompose the set $X_n$ as a partition

$$X_n = U \cup x^- \cup V$$

with $U$ initial and $V$ final in $X_n$ and, writing $A, B \subseteq P_{n-1}$ for

$$A = (X_{n-1,-} \cup U^+) \setminus U^- \quad \text{and} \quad B = (X_{n-1,+} \cup V^-) \setminus V^+$$

we have that

$$U \text{ moves } X_{n-1,-} \text{ to } A, \quad x^- \text{ moves } A \text{ to } B, \quad V \text{ moves } B \text{ to } X_{n-1,+}$$

as pictured on Figure 5. In the following, for $Z \subseteq P_{n-1}$, we write $D(Z)$ for the $(n-1)$-pre-cell of $P$ defined by

$$D(Z)_{n-1} = Z \quad \text{and} \quad D(Z)_{i,\epsilon} = X_{i,\epsilon} \quad \text{for } i \in \mathbb{N}_{n-2} \text{ and } \epsilon \in \{-,+\}.$$

Since $D(A) = \mathrm{Act}(D(X_{n-1,-}), U)$, $D(B) = \mathrm{Act}(D(A), x^-)$, and $D(X_{n-1,-}) = \partial_{n-1}^-(X)$ is an $(n-1)$-cell and both $U$ and $x^-$ are fork-free, by using two times the induction hypothesis of Theorem 2.1.1, first on $D(X_{n-1,-})$, then on $D(A)$, we get that

$$D(A) \text{ and } D(B) \text{ are cells.} \qquad (17)$$

By Axiom (T2), we have that

$$x^+ \text{ is fork-free.} \qquad (18)$$
<figure><img src="image_18.png" /><figcaption>Figure 5: The decomposition of $X_n$</figcaption></figure>

Since $x^-$ moves $A$ to $B$, by Lemma A.1.1, we get

$$A \cap x^{-+} = \emptyset. \tag{19}$$

By Axiom (T2), it holds that $x^{++} = x^{-\mp} \subseteq A$. By (17) and (18), using the induction hypothesis of Theorem 2.1.1 on $D(A)$, we get

$$A \cap x^{++} = \emptyset. \tag{20}$$

By Lemma A.1.1, there exists $B'$ such that $x^+$ moves $A$ to $B'$, and

$$
\begin{align*}
B' &= (A \cup x^{++}) \setminus x^{+-} \\
  &= (A \setminus x^{+-}) \cup (x^{++} \setminus x^{+-}) \\
  &= (A \setminus x^{+\mp}) \cup x^{\pm} \tag{by (20)} \\
  &= (A \setminus x^{-\mp}) \cup x^{-\pm} \tag{since $x^{+\mp} = x^{-\mp}$, by Axiom (T2)} \\
  &= (A \setminus x^{--}) \cup (x^{-+} \setminus x^{--}) \tag{by (19)} \\
  &= (A \cup x^{-+}) \setminus x^{--} \\
  &= B \tag{since $x^-$ moves $A$ to $B$).
\end{align*}
$$

Hence,

$$x^+ \text{ moves } A \text{ to } B. \tag{21}$$

Since $x^{+\mp} \subseteq D(A)_{n-1}$ and $U^{\pm} \subseteq D(A)_{n-1}$, using the induction hypothesis of Theorem 2.1.1, by (c) we get

$$U^{-} \cap x^{++} = \emptyset. \tag{22}$$

Similarly, with $D(B)$, we get

$$x^{+-} \cap V^{+} = \emptyset. \tag{23}$$

By definition, $U$ moves $X_{n-1,-}$ to $A$, and $x^+$ moves $A$ to $B$ by (21). Moreover, by (22), we have that $U^{-} \cap x^{++} = \emptyset$. Using Lemma A.1.3, we deduce that

$$U \cup x^+ \text{ moves } X_{n-1,-} \text{ to } B. \tag{24}$$

Since $U$ and $V$ are disjoint and respectively initial and terminal in $X_n$, $U^{-} \cap V^{+} = \emptyset$. Also, by (23), we have $(x^{+-} \cap V^{+}) = \emptyset$, therefore

$$(U \cup x^+)^- \cap V^+ \subseteq (U^- \cap V^+) \cup (x^{+-} \cap V^+) = \emptyset.$$
<figure><img src="image_22.png" /><figcaption>Figure 5: The decomposition of $X_n$</figcaption></figure>

Since $x^-$ moves $A$ to $B$, by Lemma A.1.1, we get

$$A \cap x^{-+} = \emptyset. \tag{19}$$

By Axiom (T2), it holds that $x^{++} = x^{-\mp} \subseteq A$. By (17) and (18), using the induction hypothesis of Theorem 2.1.1 on $D(A)$, we get

$$A \cap x^{++} = \emptyset. \tag{20}$$

By Lemma A.1.1, there exists $B'$ such that $x^+$ moves $A$ to $B'$, and

$$
\begin{align*}
B' &= (A \cup x^{++}) \setminus x^{+-} \\
  &= (A \setminus x^{+-}) \cup (x^{++} \setminus x^{+-}) \\
  &= (A \setminus x^{+\mp}) \cup x^{\pm} \tag{by (20)} \\
  &= (A \setminus x^{-\mp}) \cup x^{-\pm} \tag{since $x^{+\mp} = x^{-\mp}$, by Axiom (T2)} \\
  &= (A \setminus x^{--}) \cup (x^{-+} \setminus x^{--}) \tag{by (19)} \\
  &= (A \cup x^{-+}) \setminus x^{--} \\
  &= B \tag{since $x^-$ moves $A$ to $B$).
\end{align*}
$$

Hence,

$$x^+ \text{ moves } A \text{ to } B. \tag{21}$$

Since $x^{+\mp} \subseteq D(A)_{n-1}$ and $U^{\pm} \subseteq D(A)_{n-1}$, using the induction hypothesis of Theorem 2.1.1, by (c) we get

$$U^{-} \cap x^{++} = \emptyset. \tag{22}$$

Similarly, with $D(B)$, we get

$$x^{+-} \cap V^{+} = \emptyset. \tag{23}$$

By definition, $U$ moves $X_{n-1,-}$ to $A$, and $x^+$ moves $A$ to $B$ by (21). Moreover, by (22), we have that $U^{-} \cap x^{++} = \emptyset$. Using Lemma A.1.3, we deduce that

$$U \cup x^+ \text{ moves } X_{n-1,-} \text{ to } B. \tag{24}$$

Since $U$ and $V$ are disjoint and respectively initial and terminal in $X_n$, $U^{-} \cap V^{+} = \emptyset$. Also, by (23), we have $(x^{+-} \cap V^{+}) = \emptyset$, therefore

$$(U \cup x^+)^- \cap V^+ \subseteq (U^- \cap V^+) \cup (x^{+-} \cap V^+) = \emptyset.$$
Using (24) and Lemma A.1.3, knowing that $S = U \cup x^+ \cup V$, we deduce that

$$S \text{ moves } X_{n-1,-} \text{ to } X_{n-1,+}. \qquad (25)$$

The set $U \cup V$ is fork-free as a subset of the fork-free $X_n$, and $x^+$ is fork-free since $x$ is relevant by Axiom (T2). Moreover,

$$U^- \cap x^{+-} = U^- \cap x^{+\mp} \qquad (\text{by (22)})$$

$$\subseteq U^- \cap A \qquad (\text{by (21) and Lemma A.1.1})$$

$$= \emptyset \qquad (\text{since } U \text{ moves } X_{n-1,-} \text{ to } A),$$

$$U^+ \cap x^{++} = U^\pm \cap x^{++} \qquad (\text{by (22)})$$

$$\subseteq A \cap x^{++} \qquad (\text{by Lemma A.1.1 since } U \text{ moves } X_{n-1,-} \text{ to } A)$$

$$= \emptyset \qquad (\text{by (21) and Lemma A.1.1}).$$

So $U \perp x^+$. Similarly, $x^+ \perp V$. Hence, since $S = U \cup x^+ \cup V$,

$$S \text{ is fork-free.} \qquad (26)$$

Then, by (25) and (26), $\text{Act}(X, G)$ is a cell.

We now prove the second part of (a). By Axiom (T1), $x^- \cap x^+ = \emptyset$. Since $U \perp x^+$ and $x^+ \perp V$ (by (26)), using Axiom (T0), we deduce that $U \cap x^+ = x^+ \cap V = \emptyset$ so that

$$X_n \cap x^+ = (U \cup x^- \cup V) \cap x^+ = \emptyset$$

which concludes the proof of the Step 1.

*Step 2: (a) holds.* We prove this by induction on $|G|$. If $|G| = 0$, then the result is trivial. Moreover, the case $|G| = 1$ was proved in Step 1. So suppose that $|G| \ge 2$. Since the relation $\triangleleft$ is acyclic by Axiom (T1), we can consider a minimal $x \in G$ for $\triangleleft_G$. Let

$$\tilde{G} = G \setminus \{x\}, \quad U = (X_n \cup x^+) \setminus x^-, \quad V = (U \cup \tilde{G}^+) \setminus \tilde{G}^-$$

and recall that we defined $S$ as $(X_n \cup G^+) \setminus G^-$. In order to show that $\text{Act}(X, G)$ is a cell, we are required to prove that $S$ moves $X_{n-1,-}$ to $X_{n-1,+}$, and that $S$ is fork-free. For this purpose, we will first move $X_n$ with $\{x\}$ to $U$ and use Step 1, then move $U$ by $\tilde{G}$ to $V$ and use the induction of Step 2. Finally, we will prove that $V = S$. So, using Step 1 with $X$ and $\{x\}$, we get that

- $\text{Act}(X, \{x\})$ is a cell;

- in particular, $U$ is fork-free and, when $n > 0$, $U$ moves $X_{n-1,-}$ to $X_{n-1,+}$;

- $X_n \cap x^+ = \emptyset$.

By Lemma A.1.1, we deduce that $\{x\}$ moves $X_n$ to $U$. Moreover,

$$\begin{align*}
\tilde{G}^{\mp} &= \tilde{G}^{-} \setminus \tilde{G}^{+} \\
&= (G^{-} \setminus x^{-}) \setminus (G^{+} \setminus x^{+}) \qquad (\text{since fork-freeness implies that } G^{\epsilon} = \sqcup_{u \in G} u^{\epsilon}) \\
&\subseteq ((G^{-} \setminus x^{-}) \setminus G^{+}) \cup x^{+} \\
&= ((G^{-} \setminus G^{+}) \setminus x^{-}) \cup x^{+} \\
&\subseteq (X_n \setminus x^{-}) \cup x^{+} \qquad (\text{since } G^{\mp} \subseteq X_n \text{ by Lemma A.1.1}) \\
&\subseteq (X_n \cup x^{+}) \setminus x^{-} \qquad (\text{since } x^{-} \cap x^{+} = \emptyset \text{ by Axiom (T1)}) \\
&= U.
\end{align*}$$

Also, $\tilde{G}$ is fork-free as a subset of the fork-free set $G$. Using the induction hypothesis of Step 2 for $\tilde{G}$, we get that
Using (24) and Lemma A.1.3, knowing that $S = U \cup x^+ \cup V$, we deduce that

$$S \text{ moves } X_{n-1,-} \text{ to } X_{n-1,+}. \qquad (25)$$

The set $U \cup V$ is fork-free as a subset of the fork-free $X_n$, and $x^+$ is fork-free since $x$ is relevant by Axiom (T2). Moreover,

$$U^- \cap x^{+-} = U^- \cap x^{+\mp} \qquad (\text{by (22)})$$

$$\subseteq U^- \cap A \qquad (\text{by (21) and Lemma A.1.1})$$

$$= \emptyset \qquad (\text{since } U \text{ moves } X_{n-1,-} \text{ to } A),$$

$$U^+ \cap x^{++} = U^\pm \cap x^{++} \qquad (\text{by (22)})$$

$$\subseteq A \cap x^{++} \qquad (\text{by Lemma A.1.1 since } U \text{ moves } X_{n-1,-} \text{ to } A)$$

$$= \emptyset \qquad (\text{by (21) and Lemma A.1.1}).$$

So $U \perp x^+$. Similarly, $x^+ \perp V$. Hence, since $S = U \cup x^+ \cup V$,

$$S \text{ is fork-free.} \qquad (26)$$

Then, by (25) and (26), $\text{Act}(X, G)$ is a cell.

We now prove the second part of (a). By Axiom (T1), $x^- \cap x^+ = \emptyset$. Since $U \perp x^+$ and $x^+ \perp V$ (by (26)), using Axiom (T0), we deduce that $U \cap x^+ = x^+ \cap V = \emptyset$ so that

$$X_n \cap x^+ = (U \cup x^- \cup V) \cap x^+ = \emptyset$$

which concludes the proof of the Step 1.

**Step 2: (a) holds.** We prove this by induction on $|G|$. If $|G| = 0$, then the result is trivial. Moreover, the case $|G| = 1$ was proved in Step 1. So suppose that $|G| \ge 2$. Since the relation $\triangleleft$ is acyclic by Axiom (T1), we can consider a minimal $x \in G$ for $\triangleleft_G$. Let

$$\tilde{G} = G \setminus \{x\}, \quad U = (X_n \cup x^+) \setminus x^-, \quad V = (U \cup \tilde{G}^+) \setminus \tilde{G}^-$$

and recall that we defined $S$ as $(X_n \cup G^+) \setminus G^-$. In order to show that $\text{Act}(X, G)$ is a cell, we are required to prove that $S$ moves $X_{n-1,-}$ to $X_{n-1,+}$, and that $S$ is fork-free. For this purpose, we will first move $X_n$ with $\{x\}$ to $U$ and use Step 1, then move $U$ by $\tilde{G}$ to $V$ and use the induction of Step 2. Finally, we will prove that $V = S$. So, using Step 1 with $X$ and $\{x\}$, we get that

- $\text{Act}(X, \{x\})$ is a cell;

- in particular, $U$ is fork-free and, when $n > 0$, $U$ moves $X_{n-1,-}$ to $X_{n-1,+}$;

- $X_n \cap x^+ = \emptyset$.

By Lemma A.1.1, we deduce that $\{x\}$ moves $X_n$ to $U$. Moreover,

$$\begin{align*}
\tilde{G}^{\mp} &= \tilde{G}^{-} \setminus \tilde{G}^{+} \\
&= (G^{-} \setminus x^{-}) \setminus (G^{+} \setminus x^{+}) \quad (\text{since fork-freeness implies that } G^{\epsilon} = \sqcup_{u \in G} u^{\epsilon}) \\
&\subseteq ((G^{-} \setminus x^{-}) \setminus G^{+}) \cup x^{+} \\
&= ((G^{-} \setminus G^{+}) \setminus x^{-}) \cup x^{+} \\
&\subseteq (X_n \setminus x^{-}) \cup x^{+} \quad (\text{since } G^{\mp} \subseteq X_n \text{ by Lemma A.1.1}) \\
&\subseteq (X_n \cup x^{+}) \setminus x^{-} \quad (\text{since } x^{-} \cap x^{+} = \emptyset \text{ by Axiom (T1)}) \\
&= U.
\end{align*}$$

Also, $\tilde{G}$ is fork-free as a subset of the fork-free set $G$. Using the induction hypothesis of Step 2 for $\tilde{G}$, we get that
- Act($\mathrm{Act}(X, \{x\})$, $\tilde{G}$) is a cell;

- In particular, $V = (U \cup \tilde{G}^+) \setminus \tilde{G}^-$ is fork-free, and, when $n > 0$, $V$ moves $X_{n-1,-}$ to $X_{n-1,+}$;

- $U \cap \tilde{G}^+ = \emptyset$.

By Lemma A.1.1, we deduce that $\tilde{G}$ moves $U$ to $V$. Also, note that $x^- \cap \tilde{G}^+ = \emptyset$ since $x$ was taken minimal in $G$. Using Lemma A.1.3, we deduce that $G = \{x\} \cup \tilde{G}$ moves $X_n$ to $V$. But $S = (X_n \cup G^+) \setminus G^-$ so that $S = V$.

The second part of (a) is left to show, that is, $X_n \cap G^+ = \emptyset$. We compute that

$$
\begin{align*}
X_n \cap G^+ &= (U \cup x^- \setminus x^+) \cap G^+ && \text{(by Lemma A.1.1, since } \{x\} \text{ moves } X_n \text{ to } U) \\
&= ((U \cup x^-) \cap G^+) \setminus x^+ \\
&= (U \cap G^+) \setminus x^+ && \text{(since } x^- \cap G = x^- \cap (x^+ \cup \tilde{G}) = \emptyset) \\
&= (U \cap \tilde{G}^+) = \emptyset
\end{align*}
$$

which concludes the proofs of Step 2 and (a).

*Proof of (b):* By (a), $\mathrm{Act}(X, G)$ is a cell. To conclude, we need to show that $G$ moves $X_n$ to $S$. By definition of $S$, we have that $S = (X_n \cup G^+) \setminus G^-$. Moreover,

$$
\begin{align*}
(S \cup G^-) \setminus G^+ &= (((X_n \cup G^+) \setminus G^-) \cup G^-) \setminus G^+ \\
&= (X_n \cup G^+ \cup G^-) \setminus G^+ \\
&= (X_n \setminus G^+) \cup G^\mp \\
&= X_n \cup G^\mp && \text{(since } X_n \cap G^+ = \emptyset \text{ by (a)}) \\
&= X_n && \text{(since } G \text{ is glueable on } X).
\end{align*}
$$

Hence, $\mathrm{Glue}(X, G)$ is a cell.

*Proof of (c):* By contradiction, suppose that $G'^- \cap G^+ \neq \emptyset$. Then, there are $x \in G'$, $y \in G$ and $z \in x^- \cap y^+$. Consider $U = \{x' \in G' | x \prec_G x'\} \cup \{x\}$, and $V = \{y' \in G | y' \prec_G y\} \cup \{y\}$. By the acyclicity Axiom (T1), we have $U^+ \cap V^- = \emptyset$. Since $U$ is a terminal set for $\prec_G$, we have in particular $U^+ \cap G'^- \subseteq U^-$. So,

$$U^+ = (U^+ \setminus G'^-) \cup (U^+ \cap G'^-) \subseteq G'^\pm \cup U^-.$$

Hence, $U^\pm \subseteq G'^\pm \subseteq X_n$ (since $G'$ is dually glueable on $X$). Similarly, $V^\mp \subseteq X_n$. Using the dual version of (a), the $n$-pre-cell $Y = \overline{\mathrm{Act}}(X, U)$ is an $n$-cell where $Y_n = (X_n \cup U^-) \setminus U^+$ (see Figure 6) and we have

$$
\begin{align*}
V^{\mp} &= V^{\mp} \setminus U^{+} && \text{(since } V^{-} \cap U^{+} = \emptyset) \\
&\subseteq X_n \setminus U^{+} && \text{(since } V^{\mp} \subseteq X_n) \\
&\subseteq (X_n \cup U^{-}) \setminus U^{+} = Y_n.
\end{align*}
$$

Using Theorem 2.1.1(a) with $Y$ and $V$, we get $Y_n \cap V^+ = \emptyset$. But, since $z \in U^\mp \subseteq Y_n$ (by Axiom (T1)) and $U^\mp \subseteq Y_n$, $z \in Y_n \cap V^+$, which is a contradiction. Hence, $G'^- \cap G^+ = \emptyset$ which ends the proof of (c). $\square$
- Act($\mathrm{Act}(X, \{x\})$, $\tilde{G}$) is a cell;

- In particular, $V = (U \cup \tilde{G}^+) \setminus \tilde{G}^-$ is fork-free, and, when $n > 0$, $V$ moves $X_{n-1,-}$ to $X_{n-1,+}$;

- $U \cap \tilde{G}^+ = \emptyset$.

By Lemma A.1.1, we deduce that $\tilde{G}$ moves $U$ to $V$. Also, note that $x^- \cap \tilde{G}^+ = \emptyset$ since $x$ was taken minimal in $G$. Using Lemma A.1.3, we deduce that $G = \{x\} \cup \tilde{G}$ moves $X_n$ to $V$. But $S = (X_n \cup G^+) \setminus G^-$ so that $S = V$.

The second part of (a) is left to show, that is, $X_n \cap G^+ = \emptyset$. We compute that

$$
\begin{align*}
X_n \cap G^+ &= (U \cup x^- \setminus x^+) \cap G^+ && \text{(by Lemma A.1.1, since $\{x\}$ moves $X_n$ to $U$)} \\
&= ((U \cup x^-) \cap G^+) \setminus x^+ \\
&= (U \cap G^+) \setminus x^+ && \text{(since $x^- \cap G = x^- \cap (x^+ \cup \tilde{G}) = \emptyset$)} \\
&= (U \cap \tilde{G}^+) = \emptyset
\end{align*}
$$

which concludes the proofs of Step 2 and (a).

*Proof of (b):* By (a), $\mathrm{Act}(X, G)$ is a cell. To conclude, we need to show that $G$ moves $X_n$ to $S$. By definition of $S$, we have that $S = (X_n \cup G^+) \setminus G^-$. Moreover,

$$
\begin{align*}
(S \cup G^-) \setminus G^+ &= (((X_n \cup G^+) \setminus G^-) \cup G^-) \setminus G^+ \\
&= (X_n \cup G^+ \cup G^-) \setminus G^+ \\
&= (X_n \setminus G^+) \cup G^\mp \\
&= X_n \cup G^\mp && \text{(since } X_n \cap G^+ = \emptyset \text{ by (a)}) \\
&= X_n && \text{(since } G \text{ is glueable on } X).
\end{align*}
$$

Hence, $\mathrm{Glue}(X, G)$ is a cell.

*Proof of (c):* By contradiction, suppose that $G'^- \cap G^+ \neq \emptyset$. Then, there are $x \in G'$, $y \in G$ and $z \in x^- \cap y^+$. Consider $U = \{x' \in G' | x \prec_G x'\} \cup \{x\}$, and $V = \{y' \in G | y' \prec_G y\} \cup \{y\}$. By the acyclicity Axiom (T1), we have $U^+ \cap V^- = \emptyset$. Since $U$ is a terminal set for $\prec_G$, we have in particular $U^+ \cap G'^- \subseteq U^-$. So,

$$U^+ = (U^+ \setminus G'^-) \cup (U^+ \cap G'^-) \subseteq G'^\pm \cup U^-.$$

Hence, $U^\pm \subseteq G'^\pm \subseteq X_n$ (since $G'$ is dually glueable on $X$). Similarly, $V^\mp \subseteq X_n$. Using the dual version of (a), the $n$-pre-cell $Y = \overline{\mathrm{Act}}(X, U)$ is an $n$-cell where $Y_n = (X_n \cup U^-) \setminus U^+$ (see Figure 6) and we have

$$
\begin{align*}
V^{\mp} &= V^{\mp} \setminus U^{+} && \text{(since } V^{-} \cap U^{+} = \emptyset) \\
&\subseteq X_n \setminus U^{+} && \text{(since } V^{\mp} \subseteq X_n) \\
&\subseteq (X_n \cup U^{-}) \setminus U^{+} = Y_n.
\end{align*}
$$

Using Theorem 2.1.1(a) with $Y$ and $V$, we get $Y_n \cap V^+ = \emptyset$. But, since $z \in U^\mp \subseteq Y_n$ (by Axiom (T1)) and $U^\mp \subseteq Y_n$, $z \in Y_n \cap V^+$, which is a contradiction. Hence, $G'^- \cap G^+ = \emptyset$ which ends the proof of (c). $\square$
<figure><img src="image_19.png" /><figcaption>Figure 6: $V$, $U$ and $Y_n$</figcaption></figure>

**Appendix B: Details on the freeness property**

This section is devoted to prove Theorem 2.4.1, stating that the $\omega$-category of cells of a torsion-free complex is a tower of cellular extensions. For this purpose, we first need a concrete description of cells of free extensions. Such a description was already introduced by Makkai when he gave a solution to the word problem on free strict categories [16]. It is based on an alternate definition of strict categories using another set of operations. Indeed, the standard set of operations, i.e., identity operations $id$ and operations $*$ which allow composing pairs of cells of homogeneous dimensions, is inconvenient for finding such a description, since this homogeneous constraint requires an intensive use of identity operations to lift dimensions of cells, in order to compose them with other cells of higher dimensions. Instead, strict categories can be seen as instances of another kind of higher categories called *precategories* which satisfy an additional condition. In precategories, the composition operation $\bullet$ of precategories allows composing cells of heterogeneous dimensions, which is more convenient for formal handling. Based on this new definition, syntactical devices called *contexts* and *context classes* can be developed. These formally represent whiskering operations on generators as composites with one hole. Using them, cells of free extensions can be described as adequately quotiented sequences of context classes formally applied to generators. From this description, the freeness of the $\omega$-category of cells of torsion-free complexes can be proved.

In Appendix B.1, we first introduce the definition of precategories and show that strict categories can be interpreted as precategories satisfying an additional exchange condition. Then, in Appendix B.2, we introduce contexts and context classes for strict categories, together with several natural operations on them. In Appendix B.3, we give a description, in the form of Theorem B.3.3, of the functor $-[\phantom{-}]$ from cellular extensions to strict categories using the intermediate notion of categorical actions, the latter describing the structure that context classes have with regard to the underlying strict category. Concretely, the cells of free extensions will be classes of sequences of formally applied context classes. In Appendix B.4, we prove that the cells of torsion-free complexes admit a decomposition of this form (Theorem B.4.6). In Appendix B.5, we finally prove Theorem 2.4.1 by showing that this decomposition is essentially unique, which precisely characterizes that a strict category is a free extension. If required, more detailed proofs can be found in [7].

**B.1 Another definition of strict categories** In this section, we introduce an alternate definition of strict categories as particular instances of precategories.
<figure><img src="image_23.png" /><figcaption>Figure 6: $V$, $U$ and $Y_n$</figcaption></figure>

## Appendix B: Details on the freeness property

This section is devoted to prove Theorem 2.4.1, stating that the $\omega$-category of cells of a torsion-free complex is a tower of cellular extensions. For this purpose, we first need a concrete description of cells of free extensions. Such a description was already introduced by Makkai when he gave a solution to the word problem on free strict categories [16]. It is based on an alternate definition of strict categories using another set of operations. Indeed, the standard set of operations, i.e., identity operations $id$ and operations $*$ which allow composing pairs of cells of homogeneous dimensions, is inconvenient for finding such a description, since this homogeneous constraint requires an intensive use of identity operations to lift dimensions of cells, in order to compose them with other cells of higher dimensions. Instead, strict categories can be seen as instances of another kind of higher categories called *precategories* which satisfy an additional condition. In precategories, the composition operation $\bullet$ of precategories allows composing cells of heterogeneous dimensions, which is more convenient for formal handling. Based on this new definition, syntactical devices called *contexts* and *context classes* can be developed. These formally represent whiskering operations on generators as composites with one hole. Using them, cells of free extensions can be described as adequately quotiented sequences of context classes formally applied to generators. From this description, the freeness of the $\omega$-category of cells of torsion-free complexes can be proved.

In Appendix B.1, we first introduce the definition of precategories and show that strict categories can be interpreted as precategories satisfying an additional exchange condition. Then, in Appendix B.2, we introduce contexts and context classes for strict categories, together with several natural operations on them. In Appendix B.3, we give a description, in the form of Theorem B.3.3, of the functor $-[\phantom{-}]$ from cellular extensions to strict categories using the intermediate notion of categorical actions, the latter describing the structure that context classes have with regard to the underlying strict category. Concretely, the cells of free extensions will be classes of sequences of formally applied context classes. In Appendix B.4, we prove that the cells of torsion-free complexes admit a decomposition of this form (Theorem B.4.6). In Appendix B.5, we finally prove Theorem 2.4.1 by showing that this decomposition is essentially unique, which precisely characterizes that a strict category is a free extension. If required, more detailed proofs can be found in [7].

**B.1 Another definition of strict categories** In this section, we introduce an alternate definition of strict categories as particular instances of precategories.
**Precategories** Precategories can be described, in a sense that will be made precise in the next paragraph, as “strict categories without exchange law” and generalize in higher dimensions the 2-dimensional theory of sesquicategories defined by Street in [25]. They were already introduced by Makkai in order to describe the cells of free strict categories [16].

Given $n \in \mathbb{N} \cup \{\omega\}$, an $n$-precategory $C$ is an $n$-globular set together with, for $k \in \mathbb{N}_{n-1}$, identity operations

$$ \mathrm{id}^{k+1}: C_k \to C_{k+1} $$

for which we use the same notation conventions than the identity operations on strict categories, and, for $k, l \in \mathbb{N}_n^*$, composition operations

$$ \bullet_{k,l}: C_k \times_{\min(k,l)-1} C_l \to C_{\max(k,l)} $$

which satisfy the axioms below. Given $i, k, l \in \mathbb{N}_n$ with $i = \min(k, l)$, since the dimensions of the cells determine the functions to be used, we often write $\bullet_i$ for $\bullet_{k,l}$. This way, we still display the most important information which is the dimension $i$ of composition. The axioms of $n$-precategories are the following:

(P-i) for $k \in \mathbb{N}_{n-1}$ and $u \in C_k$,

$$ \partial_k^{-}(\mathrm{id}_u^{k+1}) = u = \partial_k^{+}(\mathrm{id}_u^{k+1}), $$

(P-ii) for $i, k, l \in \mathbb{N}_n$ such that $i = \min(k, l) - 1$, $(u, v) \in C_k \times_i C_l$, and $\epsilon \in \{-, +\}$,

$$ \partial^{\epsilon}(u \bullet_i v) = \begin{cases} u \bullet_i \partial^{\epsilon}(v) & \text{if } k < l, \\ \partial^{-}(u) & \text{if } k = l \text{ and } \epsilon = -, \\ \partial^{+}(v) & \text{if } k = l \text{ and } \epsilon = +, \\ \partial^{\epsilon}(u) \bullet_i v & \text{if } k > l, \end{cases} $$

(P-iii) for $i, k, l \in \mathbb{N}_n$ with $i = \min(k, l) - 1$, given $(u, v) \in C_{k-1} \times_i C_l$,

$$ \mathrm{id}_u \bullet_i v = \begin{cases} v & \text{if } k \le l, \\ \mathrm{id}_{u \bullet_i v} & \text{if } k > l, \end{cases} $$

and, given $(u, v) \in C_k \times_i C_{l-1}$,

$$ u \bullet_i \mathrm{id}_v = \begin{cases} u & \text{if } l \le k, \\ \mathrm{id}_{u \bullet_i v} & \text{if } l > k, \end{cases} $$

(P-iv) for $i, k, l, m \in \mathbb{N}_n$ with $i = \min(k, l) - 1 = \min(l, m) - 1$, and $u \in C_k, v \in C_l$ and $w \in C_w$ such that $u, v, w$ are $i$-composable,

$$ (u \bullet_i v) \bullet_i w = u \bullet_i (v \bullet_i w), $$

(P-v) for every $i, j, k \in \mathbb{N}_n$ satisfying $i < j < k$, and cells $u_1, u_2 \in C_{i+1}, v_1, v_2 \in C_{j+1}$ and $w \in C_k$ such that $u_1, w, u_2$ are $i$-composable and $v_1, w, v_2$ are $j$-composable, we have

$$ u_1 \bullet_i (v_1 \bullet_j w \bullet_j v_2) \bullet_i u_2 = (u_1 \bullet_i v_1 \bullet_i u_2) \bullet_j (u_1 \bullet_i w \bullet_i u_2) \bullet_j (u_1 \bullet_i v_2 \bullet_i u_2). $$

Given two $n$-precategories $C$ and $D$, a *morphism of $n$-precategories* between $C$ and $D$ (also called *$n$-prefunctor*), is a morphism of $n$-globular sets $F: C \to D$ which moreover commutes with the operations $\mathrm{id}$ and $\bullet$. We write $\mathbf{PCat}_n$ for the category of $n$-precategories.
**Precategories** Precategories can be described, in a sense that will be made precise in the next paragraph, as “strict categories without exchange law” and generalize in higher dimensions the 2-dimensional theory of sesquicategories defined by Street in [25]. They were already introduced by Makkai in order to describe the cells of free strict categories [16].

Given $n \in \mathbb{N} \cup \{\omega\}$, an $n$-precategory $C$ is an $n$-globular set together with, for $k \in \mathbb{N}_{n-1}$, identity operations

$$ \mathrm{id}^{k+1}: C_k \to C_{k+1} $$

for which we use the same notation conventions than the identity operations on strict categories, and, for $k, l \in \mathbb{N}_n^*$, composition operations

$$ \bullet_{k,l}: C_k \times_{\min(k,l)-1} C_l \to C_{\max(k,l)} $$

which satisfy the axioms below. Given $i, k, l \in \mathbb{N}_n$ with $i = \min(k, l)$, since the dimensions of the cells determine the functions to be used, we often write $\bullet_i$ for $\bullet_{k,l}$. This way, we still display the most important information which is the dimension $i$ of composition. The axioms of $n$-precategories are the following:

(P-i) for $k \in \mathbb{N}_{n-1}$ and $u \in C_k$,

$$ \partial_k^{-}(\mathrm{id}_u^{k+1}) = u = \partial_k^{+}(\mathrm{id}_u^{k+1}), $$

(P-ii) for $i, k, l \in \mathbb{N}_n$ such that $i = \min(k, l) - 1$, $(u, v) \in C_k \times_i C_l$, and $\epsilon \in \{-, +\}$,

$$ \partial^{\epsilon}(u \bullet_i v) = \begin{cases} u \bullet_i \partial^{\epsilon}(v) & \text{if } k < l, \\ \partial^{-}(u) & \text{if } k = l \text{ and } \epsilon = -, \\ \partial^{+}(v) & \text{if } k = l \text{ and } \epsilon = +, \\ \partial^{\epsilon}(u) \bullet_i v & \text{if } k > l, \end{cases} $$

(P-iii) for $i, k, l \in \mathbb{N}_n$ with $i = \min(k, l) - 1$, given $(u, v) \in C_{k-1} \times_i C_l$,

$$ \mathrm{id}_u \bullet_i v = \begin{cases} v & \text{if } k \le l, \\ \mathrm{id}_{u \bullet_i v} & \text{if } k > l, \end{cases} $$

and, given $(u, v) \in C_k \times_i C_{l-1}$,

$$ u \bullet_i \mathrm{id}_v = \begin{cases} u & \text{if } l \le k, \\ \mathrm{id}_{u \bullet_i v} & \text{if } l > k, \end{cases} $$

(P-iv) for $i, k, l, m \in \mathbb{N}_n$ with $i = \min(k, l) - 1 = \min(l, m) - 1$, and $u \in C_k, v \in C_l$ and $w \in C_w$ such that $u, v, w$ are $i$-composable,

$$ (u \bullet_i v) \bullet_i w = u \bullet_i (v \bullet_i w), $$

(P-v) for every $i, j, k \in \mathbb{N}_n$ satisfying $i < j < k$, and cells $u_1, u_2 \in C_{i+1}, v_1, v_2 \in C_{j+1}$ and $w \in C_k$ such that $u_1, w, u_2$ are $i$-composable and $v_1, w, v_2$ are $j$-composable, we have

$$ u_1 \bullet_i (v_1 \bullet_j w \bullet_j v_2) \bullet_i u_2 = (u_1 \bullet_i v_1 \bullet_i u_2) \bullet_j (u_1 \bullet_i w \bullet_i u_2) \bullet_j (u_1 \bullet_i v_2 \bullet_i u_2). $$

Given two $n$-precategories $C$ and $D$, a *morphism of $n$-precategories* between $C$ and $D$ (also called *$n$-prefunctor*), is a morphism of $n$-globular sets $F: C \to D$ which moreover commutes with the operations $\mathrm{id}$ and $\bullet$. We write $\mathbf{PCat}_n$ for the category of $n$-precategories.
**Strict categories as precategories** In this section, we recall from [16] how strict categories can be expressed as precategories satisfying a condition analogous to the exchange law.

For $n \in \mathbb{N} \cup \{\omega\}$ and $C \in \mathbf{PCat}_n$, we write (E) for the following property on $C$:

(E) for $i, k, l \in \mathbb{N}_n$ with $1 \le i = \min(k, l) - 1$, $u \in C_k$ and $v \in C_l$, if $u, v$ are $(i-1)$-composable, then

$$ (u \bullet_{i-1} \partial_i^-(v)) \bullet_i (\partial_i^+(u) \bullet_{i-1} v) = (\partial_i^-(u) \bullet_{i-1} v) \bullet_i (u \bullet_{i-1} \partial_i^+(v)). $$

Let $\mathbf{PCat}_n^{(E)}$ be the full subcategory of $\mathbf{PCat}_n$ of those $n$-precategories $C$ that satisfy (E). The condition (E) can be thought as an equivalent for precategories of the exchange law (S-vi) of strict categories.

Given $n \in \mathbb{N} \cup \{\omega\}$ and $C \in \mathbf{Cat}_n$, we define a structure of $n$-precategory on the underlying $n$-globular set of $C$. We keep the identities given by the strict $n$-category structure and define the composition operations $\bullet_{(-)}$ on $C$ based on the composition operations $*_{(-)}$. Given $i, k, l \in \mathbb{N}_n$ with $i = \min(k, l) - 1$, $u \in C_k$ and $v \in C_l$ such that $u, v$ are $i$-composable, we put

$$ u \bullet_i v = \mathrm{id}_u^m *_i \mathrm{id}_v^m $$

where $m = \max(k, l)$. We can then prove that we indeed obtain an $n$-precategory which satisfies (E) and that the construction is functorial.

Conversely, given $C \in \mathbf{PCat}_n^{(E)}$, we define a structure of strict $n$-category on the underlying $n$-globular set of $C$. We keep the identities given by the structure of $n$-precategory of $C$ as before and define the multiple composition operations $*_{(-)}$ based on the precategorical composition operations $\bullet_{(-)}$. For $i, k \in \mathbb{N}_n$ with $i < k$, we define $u *_i v$ for $i$-composable $u, v \in C_k$ by induction on $k - i$. If $i = k - 1$, we put $u *_i v = u \bullet_i v$. Otherwise, if $i < k - 1$, we define $u *_i v$ inductively by

$$ u *_i v = (u \bullet_i \partial_{i+1}^-(v)) *_{i+1} (\partial_{i+1}^+(u) \bullet_i v). $$

We can then prove that we obtain a strict $n$-category and that the construction is functorial.

**Theorem B.1.1.** *The two constructions define an isomorphism between $\mathbf{Cat}_n$ and $\mathbf{PCat}_n^{(E)}$.*

Thus, a strict $n$-category $C$ is canonically an $n$-precategory satisfying (E) (and vice versa). For our purposes, we will often prefer the definition of strict categories and use the precategorical compositions $\bullet_{(-)}$ on a strict category without invoking Theorem B.1.1.

**B.2 Contexts and contexts classes** Here, we introduce *contexts* and *context classes*, that represent formal cells of strict categories with “holes” in them. Our definitions are similar to the one of context given by Métayer in [17], but with a stronger syntactical perspective.

**Definition** Let $n \in \mathbb{N} \cup \{\omega\}$ and $G$ be an $n$-globular set. Given $k \in \mathbb{N}_n$ and $u, v \in X_k$, $u$ and $v$ are said *parallel* when $k = 0$ or $\partial_{k-1}^\epsilon(u) = \partial_{k-1}^\epsilon(v)$ for $\epsilon \in \{-, +\}$. Given $m \in \mathbb{N}_n$, an $m$-type is a pair $(u, u')$ of parallel $(m-1)$-globes of $G$ (we use the convention that there is a unique $(-1)$-globe $*$ which is parallel with itself). Given $k \in \mathbb{N}_n$ with $k \ge m$ and $v \in G_k$, the $m$-type of $v$ is the $m$-type $(\partial_{m-1}^-(v), \partial_{m-1}^+(v))$ so that every $k$-cell can be implicitly considered as an $m$-type.

Let $C \in \mathbf{Cat}_n$. For every $m \in \mathbb{N}_n$ and $m$-type $(u, u')$, we define, by induction on $m$,

- the notion of *m-context of type* $(u, u')$ of $C$,
**Strict categories as precategories** In this section, we recall from [16] how strict categories can be expressed as precategories satisfying a condition analogous to the exchange law.

For $n \in \mathbb{N} \cup \{\omega\}$ and $C \in \mathbf{PCat}_n$, we write (E) for the following property on $C$:

(E) for $i, k, l \in \mathbb{N}_n$ with $1 \le i = \min(k, l) - 1$, $u \in C_k$ and $v \in C_l$, if $u, v$ are $(i-1)$-composable, then

$$ (u \bullet_{i-1} \partial_i^-(v)) \bullet_i (\partial_i^+(u) \bullet_{i-1} v) = (\partial_i^-(u) \bullet_{i-1} v) \bullet_i (u \bullet_{i-1} \partial_i^+(v)). $$

Let $\mathbf{PCat}_n^{(E)}$ be the full subcategory of $\mathbf{PCat}_n$ of those $n$-precategories $C$ that satisfy (E). The condition (E) can be thought as an equivalent for precategories of the exchange law (S-vi) of strict categories.

Given $n \in \mathbb{N} \cup \{\omega\}$ and $C \in \mathbf{Cat}_n$, we define a structure of $n$-precategory on the underlying $n$-globular set of $C$. We keep the identities given by the strict $n$-category structure and define the composition operations $\bullet_{(-)}$ on $C$ based on the composition operations $*_{(-)}$. Given $i, k, l \in \mathbb{N}_n$ with $i = \min(k, l) - 1$, $u \in C_k$ and $v \in C_l$ such that $u, v$ are $i$-composable, we put

$$ u \bullet_i v = \mathrm{id}_u^m *_i \mathrm{id}_v^m $$

where $m = \max(k, l)$. We can then prove that we indeed obtain an $n$-precategory which satisfies (E) and that the construction is functorial.

Conversely, given $C \in \mathbf{PCat}_n^{(E)}$, we define a structure of strict $n$-category on the underlying $n$-globular set of $C$. We keep the identities given by the structure of $n$-precategory of $C$ as before and define the multiple composition operations $*_{(-)}$ based on the precategorical composition operations $\bullet_{(-)}$. For $i, k \in \mathbb{N}_n$ with $i < k$, we define $u *_i v$ for $i$-composable $u, v \in C_k$ by induction on $k - i$. If $i = k - 1$, we put $u *_i v = u \bullet_i v$. Otherwise, if $i < k - 1$, we define $u *_i v$ inductively by

$$ u *_i v = (u \bullet_i \partial_{i+1}^-(v)) *_{i+1} (\partial_{i+1}^+(u) \bullet_i v). $$

We can then prove that we obtain a strict $n$-category and that the construction is functorial.

**Theorem B.1.1.** *The two constructions define an isomorphism between $\mathbf{Cat}_n$ and $\mathbf{PCat}_n^{(E)}$.*

Thus, a strict $n$-category $C$ is canonically an $n$-precategory satisfying (E) (and vice versa). For our purposes, we will often prefer the definition of strict categories and use the precategorical compositions $\bullet_{(-)}$ on a strict category without invoking Theorem B.1.1.

**B.2 Contexts and contexts classes** Here, we introduce *contexts* and *context classes*, that represent formal cells of strict categories with “holes” in them. Our definitions are similar to the one of context given by Métayer in [17], but with a stronger syntactical perspective.

**Definition** Let $n \in \mathbb{N} \cup \{\omega\}$ and $G$ be an $n$-globular set. Given $k \in \mathbb{N}_n$ and $u, v \in X_k$, $u$ and $v$ are said *parallel* when $k = 0$ or $\partial_{k-1}^\epsilon(u) = \partial_{k-1}^\epsilon(v)$ for $\epsilon \in \{-, +\}$. Given $m \in \mathbb{N}_n$, an *m-type* is a pair $(u, u')$ of parallel $(m-1)$-globes of $G$ (we use the convention that there is a unique $(-1)$-globe $*$ which is parallel with itself). Given $k \in \mathbb{N}_n$ with $k \ge m$ and $v \in G_k$, the *m-type* of $v$ is the *m-type* $(\partial_{m-1}^-(v), \partial_{m-1}^+(v))$ so that every $k$-cell can be implicitly considered as an *m-type*.

Let $C \in \mathbf{Cat}_n$. For every $m \in \mathbb{N}_n$ and $m$-type $(u, u')$, we define, by induction on $m$,

- the notion of *m-context of type* $(u, u')$ of $C$,
- the notion of *m-context class of type* $(u, u')$ of *C*,

- for $k \in \mathbb{N}_n$ with $k \ge m$, the evaluation of an *m-context* $E$ (resp. *m-context class* $F$) of type $(u, u')$ at a cell $w \in C_k$ of type $(u, u')$ which is a *k-cell* denoted $E[w]$ (resp. $F[w]$).

For $m \in \mathbb{N}_n$, an *m-context class of type* $(u, u')$ of *C* will be an equivalence class of *m-contexts* of type $(u, u')$ under a relation denoted $\approx_m$, so that we write $[[E]]$ for the associated *m-context* class of an *m-context* $E$. This relation witnesses that two contexts are equivalent up to the equalities (E) considered in dimension *m*.

There is a unique 0-context, denoted $[-]$, and the relation $\approx_0$ is the identity relation, so that a 0-context class is exactly a 0-context. Given $k \in \mathbb{N}_n$ and $k$-cell $v \in C_k$, the evaluation of the unique 0-context (class) $[-]$ at $v$ is $v$.

Given $m \in \mathbb{N}_{n-1}$ and an $(m+1)$-type $(u, u')$, an $(m+1)$-context of type $(u, u')$ is defined as a triple $E = (l, F, r)$ where

- $F$ is an *m-context class of type* $(\partial_{m-1}^{-}(u), \partial_{m-1}^{+}(u'))$,

- $l$ and $r$ are $(m+1)$-cells of $C$ such that $\partial_{m}^{+}(l) = F[u]$ and $\partial_{m}^{-}(r) = F[u]$.

Moreover, given $k \in \mathbb{N}_n$ with $k \ge m + 1$ and $w \in C_k$ of type $(u, u')$, the evaluation $E[w]$ of $E$ at $w$ is the $k$-cell

$$E[w] = l \bullet_m F[w] \bullet_m r.$$

We define the relation $\approx_{m+1}$ on $(m+1)$-contexts of type $(u, u')$. When $m = 0$, for all 1-contexts $E_1$ and $E_2$ of type $(u, u')$, we put $E_1 \approx_1 E_2$ if and only if $E_1 = E_2$. When $m > 0$, we define $\approx_{m+1}$ to be the reflexive symmetrical transitive closure of $\approx_{m+1}^1$, where $\approx_{m+1}^1$ is the relation such that, for all $(m+1)$-contexts

$$E_1 = (l_1, F_1, r_1) \quad \text{and} \quad E_2 = (l_2, F_2, r_2)$$

of type $(u, u')$, we have $E_1 \approx_{m+1}^1 E_2$ if there exist *m*-contexts

$$E'_1 = (l'_1, F'_1, r'_1) \quad \text{and} \quad E'_2 = (l'_2, F'_2, r'_2)$$

of type $(\partial_{m-1}^{-}(u), \partial_{m-1}^{+}(u'))$ with $F_i = [[E_i']]$ for $i \in \{1, 2\}$, and $l, r, w \in C_{m+1}$ such that at least one of the two sets of conditions ($\approx$-L) and ($\approx$-R) is satisfied, where the set of conditions ($\approx$-L) is

$$ \begin{array}{l@{\qquad}l} \text{($\approx$-L)} & \begin{array}{l} l_1 = l \bullet_m (w \bullet_{m-1} F'_1[u] \bullet_{m-1} r'_1) \quad r_1 = r \\ l_2 = l \quad r_2 = (w \bullet_{m-1} F'_2[u'] \bullet_{m-1} r'_2) \bullet_m r \end{array} \\ \text{($\approx$-R)} & \begin{array}{l} l'_1 = \partial_m^+(w) \quad r'_1 = r'_2 \\ l'_2 = \partial_m^-(w) \quad F'_1 = F'_2 \end{array} \end{array} $$

and the set of conditions ($\approx$-R) is

$$ \begin{array}{l@{\qquad}l} \text{($\approx$-R)} & \begin{array}{l} l_1 = l \bullet_m (l'_1 \bullet_{m-1} F'_1[u] \bullet_{m-1} w) \quad r_1 = r \\ l_2 = l \quad r_2 = (l'_2 \bullet_{m-1} F'_2[u'] \bullet_{m-1} w) \bullet_m r \end{array} \\ \text{($\approx$-L)} & \begin{array}{l} l'_1 = l'_2 \quad r'_1 = \partial_m^+(w) \\ r'_2 = \partial_m^-(w) \quad F'_1 = F'_2 \end{array} \end{array} $$

We give a graphical representation of ($\approx$-L) and ($\approx$-R) in the case $m = 1$ in Figure 7. An $(m+1)$-context class of type $(u, u')$ is an equivalence class of $(m+1)$-contexts of type $(u, u')$ under $\approx_{m+1}$. Note that if $E_1 \approx_{m+1} E_2$ and $w$ is a $k$-cell of type $(u, u')$, then $E_1[w] = E_2[w]$, so that we can define the evaluation $F[w]$ of an $(m+1)$-context class $F$ by a $k$-cell $w$, both of type $(u, u')$, as $E[w]$, where $E$ is an $(m+1)$-context of type $(u, u')$ such that $F = [[E]]$.
- the notion of *m-context class of type* $(u, u')$ of *C*,

- for $k \in \mathbb{N}_n$ with $k \ge m$, the evaluation of an *m-context* $E$ (resp. *m-context class* $F$) of type $(u, u')$ at a cell $w \in C_k$ of type $(u, u')$ which is a *k-cell* denoted $E[w]$ (resp. $F[w]$).

For $m \in \mathbb{N}_n$, an *m-context class of type* $(u, u')$ of *C* will be an equivalence class of *m-contexts* of type $(u, u')$ under a relation denoted $\approx_m$, so that we write $[[E]]$ for the associated *m-context* class of an *m-context* $E$. This relation witnesses that two contexts are equivalent up to the equalities (E) considered in dimension *m*.

There is a unique 0-context, denoted $[-]$, and the relation $\approx_0$ is the identity relation, so that a 0-context class is exactly a 0-context. Given $k \in \mathbb{N}_n$ and $k$-cell $v \in C_k$, the evaluation of the unique 0-context (class) $[-]$ at $v$ is $v$.

Given $m \in \mathbb{N}_{n-1}$ and an $(m+1)$-type $(u, u')$, an $(m+1)$-context of type $(u, u')$ is defined as a triple $E = (l, F, r)$ where

- $F$ is an *m-context class of type* $(\partial_{m-1}^{-}(u), \partial_{m-1}^{+}(u'))$,

- $l$ and $r$ are $(m+1)$-cells of $C$ such that $\partial_{m}^{+}(l) = F[u]$ and $\partial_{m}^{-}(r) = F[u]$.

Moreover, given $k \in \mathbb{N}_n$ with $k \ge m + 1$ and $w \in C_k$ of type $(u, u')$, the evaluation $E[w]$ of $E$ at $w$ is the $k$-cell

$$E[w] = l \cdot_m F[w] \cdot_m r.$$

We define the relation $\approx_{m+1}$ on $(m+1)$-contexts of type $(u, u')$. When $m = 0$, for all 1-contexts $E_1$ and $E_2$ of type $(u, u')$, we put $E_1 \approx_1 E_2$ if and only if $E_1 = E_2$. When $m > 0$, we define $\approx_{m+1}$ to be the reflexive symmetrical transitive closure of $\approx_{m+1}^1$, where $\approx_{m+1}^1$ is the relation such that, for all $(m+1)$-contexts

$$E_1 = (l_1, F_1, r_1) \quad \text{and} \quad E_2 = (l_2, F_2, r_2)$$

of type $(u, u')$, we have $E_1 \approx_{m+1}^1 E_2$ if there exist *m-contexts*

$$E'_1 = (l'_1, F'_1, r'_1) \quad \text{and} \quad E'_2 = (l'_2, F'_2, r'_2)$$

of type $(\partial_{m-1}^{-}(u), \partial_{m-1}^{+}(u'))$ with $F_i = [[E'_i]]$ for $i \in \{1, 2\}$, and $l, r, w \in C_{m+1}$ such that at least one of the two sets of conditions ($\approx$-L) and ($\approx$-R) is satisfied, where the set of conditions ($\approx$-L) is

$$ \begin{array}{ll} l_1 = l \cdot_m (w \cdot_{m-1} F'_1[u] \cdot_{m-1} r'_1) & r_1 = r \\ l_2 = l & r_2 = (w \cdot_{m-1} F'_2[u'] \cdot_{m-1} r'_2) \cdot_m r \\ l'_1 = \partial_m^+(w) & r'_1 = r'_2 \\ l'_2 = \partial_m^-(w) & F'_1 = F'_2 \end{array} \qquad (\approx-L) $$

and the set of conditions ($\approx$-R) is

$$ \begin{array}{ll} l_1 = l \cdot_m (l'_1 \cdot_{m-1} F'_1[u] \cdot_{m-1} w) & r_1 = r \\ l_2 = l & r_2 = (l'_2 \cdot_{m-1} F'_2[u'] \cdot_{m-1} w) \cdot_m r \\ l'_1 = l'_2 & r'_1 = \partial_m^+(w) \\ r'_2 = \partial_m^-(w) & F'_1 = F'_2 \end{array} \qquad (\approx-R) $$

We give a graphical representation of ($\approx$-L) and ($\approx$-R) in the case $m = 1$ in Figure 7. An $(m+1)$-context class of type $(u, u')$ is an equivalence class of $(m+1)$-contexts of type $(u, u')$ under $\approx_{m+1}$. Note that if $E_1 \approx_{m+1} E_2$ and $w$ is a $k$-cell of type $(u, u')$, then $E_1[w] = E_2[w]$, so that we can define the evaluation $F[w]$ of an $(m+1)$-context class $F$ by a $k$-cell $w$, both of type $(u, u')$, as $E[w]$, where $E$ is an $(m+1)$-context of type $(u, u')$ such that $F = [[E]]$.