# Unifying notions of pasting diagrams

Simon Forest

► To cite this version:

Simon Forest. Unifying notions of pasting diagrams. Higher Structures, 2022, 6 (1), pp.1-79.
10.21136/HS.2022.01. hal-02051506v2

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https://hal.science/hal-02051506v2

Submitted on 19 Feb 2024

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<figure><img src="image_1.png" /></figure>
# Unifying notions of pasting diagrams

Simon Forest

February 19, 2024

## Abstract

In this work, we relate the three main formalisms for the notion of pasting diagram in strict $\omega$-categories: Street's *parity complexes*, Johnson's *pasting schemes* and Steiner's *augmented directed complexes*. In the process, we show that the axioms of parity complexes and pasting schemes are not strong enough for them to correctly represent pasting diagrams, and we do so by providing a counter-example. Then, we introduce a new formalism, called *torsion-free complexes*, which aims at encompassing the three other ones. We prove its correctness by providing a detailed proof that an instance induces a free $\omega$-category. Next, we prove that the three other formalisms can be embedded in some sense in the new one. Finally, we show that there are no other embedding between these four formalisms.

## Contents

<table><tbody><tr><td><b>1 Formalisms of pasting diagrams</b></td><td><b>8</b></td></tr><tr><td>1.1 Globular sets</td><td>8</td></tr><tr><td>1.2 Strict categories</td><td>9</td></tr><tr><td>1.3 Hypergraphs</td><td>10</td></tr><tr><td>1.4 Parity complexes</td><td>11</td></tr><tr><td>1.5 Pasting schemes</td><td>15</td></tr><tr><td>1.6 Augmented directed complexes</td><td>18</td></tr><tr><td>1.7 Torsion-free complexes</td><td>21</td></tr><tr><td><b>2 The free &omega;-category of cells</b></td><td><b>24</b></td></tr><tr><td>2.1 Gluing sets on cells</td><td>25</td></tr><tr><td>2.2 Structure of &omega;-category</td><td>26</td></tr><tr><td>2.3 The notion of freeness</td><td>28</td></tr><tr><td>2.4 Freeness of the &omega;-category of cells</td><td>29</td></tr><tr><td><b>3 Relating formalisms</b></td><td><b>31</b></td></tr><tr><td>3.1 Closed and maximal cells</td><td>31</td></tr><tr><td>3.2 Embedding parity complexes</td><td>40</td></tr><tr><td>3.3 Embedding pasting schemes</td><td>42</td></tr><tr><td>3.4 Embedding augmented directed complexes</td><td>45</td></tr><tr><td>3.5 Absence of other embeddings</td><td>51</td></tr><tr><td><b>A Details on the gluing theorem</b></td><td><b>54</b></td></tr><tr><td>A.1 Movement properties</td><td>54</td></tr><tr><td>A.2 Proof of the gluing theorem</td><td>57</td></tr></tbody></table>
Unifying notions of pasting diagrams

Simon Forest

February 19, 2024

Abstract

In this work, we relate the three main formalisms for the notion of pasting diagram in strict $\omega$-categories: Street's *parity complexes*, Johnson's *pasting schemes* and Steiner's *augmented directed complexes*. In the process, we show that the axioms of parity complexes and pasting schemes are not strong enough for them to correctly represent pasting diagrams, and we do so by providing a counter-example. Then, we introduce a new formalism, called *torsion-free complexes*, which aims at encompassing the three other ones. We prove its correctness by providing a detailed proof that an instance induces a free $\omega$-category. Next, we prove that the three other formalisms can be embedded in some sense in the new one. Finally, we show that there are no other embedding between these four formalisms.

Contents

<table><tbody><tr><td><b>1 Formalisms of pasting diagrams</b></td><td><b>8</b></td></tr><tr><td>1.1 Globular sets</td><td>8</td></tr><tr><td>1.2 Strict categories</td><td>9</td></tr><tr><td>1.3 Hypergraphs</td><td>10</td></tr><tr><td>1.4 Parity complexes</td><td>11</td></tr><tr><td>1.5 Pasting schemes</td><td>15</td></tr><tr><td>1.6 Augmented directed complexes</td><td>18</td></tr><tr><td>1.7 Torsion-free complexes</td><td>21</td></tr><tr><td><b>2 The free &omega;-category of cells</b></td><td><b>24</b></td></tr><tr><td>2.1 Gluing sets on cells</td><td>25</td></tr><tr><td>2.2 Structure of &omega;-category</td><td>26</td></tr><tr><td>2.3 The notion of freeness</td><td>28</td></tr><tr><td>2.4 Freeness of the &omega;-category of cells</td><td>29</td></tr><tr><td><b>3 Relating formalisms</b></td><td><b>31</b></td></tr><tr><td>3.1 Closed and maximal cells</td><td>31</td></tr><tr><td>3.2 Embedding parity complexes</td><td>40</td></tr><tr><td>3.3 Embedding pasting schemes</td><td>42</td></tr><tr><td>3.4 Embedding augmented directed complexes</td><td>45</td></tr><tr><td>3.5 Absence of other embeddings</td><td>51</td></tr><tr><td><b>A Details on the gluing theorem</b></td><td><b>54</b></td></tr><tr><td>A.1 Movement properties</td><td>54</td></tr><tr><td>A.2 Proof of the gluing theorem</td><td>57</td></tr></tbody></table>
**Pasting diagrams in 1-categories** The simplest instances of pasting diagrams are the ones of dimension 1: in this case, they are of the form

$$x_0 \xrightarrow{a_1} x_1 \xrightarrow{a_2} x_2 \xrightarrow{a_3} \dots \xrightarrow{a_n} x_n \qquad (2)$$

and admit $a_n \circ \dots \circ a_1$ as composite. On the contrary, diagrams such as

$$y \xleftarrow{a} x \xrightarrow{b} z \quad \text{or} \quad x \xrightarrow{\circlearrowleft{a}} \qquad (3)$$

are not expected to be pasting diagrams: in the first one, the two arrows are not even composable, and the second one is ambiguous in the sense that it might denote $a$, or $a \circ a$, etc. Note that the diagram (2) can be freely obtained as the composite of generating diagrams of the form

$$x_i \xrightarrow{a_i} x_{i+1}$$

(composition amounts here to identify the target object of a diagram with the source of the second), whereas this is not the case for the diagrams of (3). Pasting diagrams of the form (2) can also be characterized as finite graphs which are connected, acyclic and non-branching, in the sense that no two arrows have the same source or the same target.

**Pasting diagrams in higher dimensions** In order to extend the definition of pasting diagrams to higher dimensions, we first need to extend the one of graph: an *ω-hypergraph* is the data of sets of hyperedges, or *generators*, for each dimension, where each generator of dimension $i+1$ has specified source and target sets of edges of dimension $i$. Then, the definition of higher-dimensional pasting diagrams can be sketched as follows: an $(n+1)$-pasting diagram is an *ω-hypergraph* with edges up to dimension $n+1$ whose source and target are valid $n$-pasting diagrams, and whose $(n+1)$-generator can be composed unambiguously in an $(n+1)$-category. The conditions for which the composition is unambiguous cannot be formulated as easily as in dimension 1. Indeed, on the one hand, the complexity of the definition of higher strict categories makes it difficult to check whether a set of generators can be composed in at least one way, and if two composites are formally equivalent. On the other hand, the sources for non-composability or ambiguity are much more varied. For example, the order in which we are supposed to compose the elements of (1) is ambiguous. Considering only the 2-generators, the orders of composition $\alpha, \beta, \gamma, \delta$ and $\alpha, \gamma, \delta, \beta$ are both possible. However, it can be proved that all possible orders of composition are equivalent by the axioms of strict *ω*-categories, so this ambiguity is not important. On the contrary, given the 2-cells $\alpha$ and $\beta$ described by the diagrams

<figure><img src="image_1.png" /></figure>

$\alpha$ and $\beta$ can be composed together in two possible orders: $\alpha$ then $\beta$ or $\beta$ then $\alpha$, which can be represented as

<figure><img src="image_2.png" /></figure>
**Pasting diagrams in 1-categories** The simplest instances of pasting diagrams are the ones of dimension 1: in this case, they are of the form

$$x_0 \xrightarrow{a_1} x_1 \xrightarrow{a_2} x_2 \xrightarrow{a_3} \dots \xrightarrow{a_n} x_n \qquad (2)$$

and admit $a_n \circ \dots \circ a_1$ as composite. On the contrary, diagrams such as

$$y \xleftarrow{a} x \xrightarrow{b} z \quad \text{or} \quad x \xrightarrow{\circ a} \qquad (3)$$

are not expected to be pasting diagrams: in the first one, the two arrows are not even composable, and the second one is ambiguous in the sense that it might denote $a$, or $a \circ a$, etc. Note that the diagram (2) can be freely obtained as the composite of generating diagrams of the form

$$x_i \xrightarrow{a_i} x_{i+1}$$

(composition amounts here to identify the target object of a diagram with the source of the second), whereas this is not the case for the diagrams of (3). Pasting diagrams of the form (2) can also be characterized as finite graphs which are connected, acyclic and non-branching, in the sense that no two arrows have the same source or the same target.

**Pasting diagrams in higher dimensions** In order to extend the definition of pasting diagrams to higher dimensions, we first need to extend the one of graph: an *ω-hypergraph* is the data of sets of hyperedges, or *generators*, for each dimension, where each generator of dimension $i+1$ has specified source and target sets of edges of dimension $i$. Then, the definition of higher-dimensional pasting diagrams can be sketched as follows: an $(n+1)$-pasting diagram is an *ω-hypergraph* with edges up to dimension $n+1$ whose source and target are valid $n$-pasting diagrams, and whose $(n+1)$-generator can be composed unambiguously in an $(n+1)$-category. The conditions for which the composition is unambiguous cannot be formulated as easily as in dimension 1. Indeed, on the one hand, the complexity of the definition of higher strict categories makes it difficult to check whether a set of generators can be composed in at least one way, and if two composites are formally equivalent. On the other hand, the sources for non-composability or ambiguity are much more varied. For example, the order in which we are supposed to compose the elements of (1) is ambiguous. Considering only the 2-generators, the orders of composition $\alpha, \beta, \gamma, \delta$ and $\alpha, \gamma, \delta, \beta$ are both possible. However, it can be proved that all possible orders of composition are equivalent by the axioms of strict $\omega$-categories, so this ambiguity is not important. On the contrary, given the 2-cells $\alpha$ and $\beta$ described by the diagrams

<figure><img src="image_2.png" /></figure>

$\alpha$ and $\beta$ can be composed together in two possible orders: $\alpha$ then $\beta$ or $\beta$ then $\alpha$, which can be represented as

<figure><img src="image_3.png" /></figure>
But here, these two composites are different. Even more subtle problems arise starting from dimension 3, justifying the use of sophisticated formalisms for recognizing pasting diagrams.

**Pasting diagram formalisms** Until now, different proposals of pasting diagram formalisms, each one giving a set of conditions to recognize pasting diagrams, have been made. The main ones are Johnson's *pasting schemes* [13], Street's *parity complexes* [23, 24] and Steiner's *augmented directed complexes* [21]. Even though the ideas underlying the definitions of these formalisms are quite similar, they differ on many points and comparing them precisely is uneasy, and actually, to the best of our knowledge, no formal account of the differences was ever made.

One of the main difference between these formalisms is the notion of sub-pasting diagram, or *cell*, which is used. The formalism of pasting schemes has the most evident notion of cell: it is simply a set which gathers all the generators which appear in the pasting diagram it represents, regardless of their dimension or source/target status. For example, the diagram (1) will be represented by the set

$$X = \{u, v, w, x, y, a, b, c, d, e, f, g, h, \alpha, \beta, \gamma, \delta\}.$$

Parity complexes use cells defined as tuples of sets of generators that are kept organized by dimension and by source/target status. For instance, the pasting diagram (1) is represented by five sets

$$X_2 = \{\alpha, \beta, \gamma, \delta\}, \\ X_{1,-} = \{a, b, e, h\}, \quad X_{1,+} = \{a, d, g, h\}, \\ X_{0,-} = \{u\}, \quad X_{0,+} = \{y\}$$

where $X_{i,-}$ represents the *i*-source, $X_{i,+}$ the *i*-target, and $X_2$ the 2-dimensional part of the pasting diagram. This notion of cell seems less natural at first than the one of pasting schemes, since the translations from a pasting diagram to a cell, and *vice versa*, are not evident. The notion of cell used by augmented directed complexes can be obtained by considering the abelian groups induced by an $\omega$-hypergraph. As a variant of the ones of parity complexes, cells are now given by sums of generators for each dimension and source/target status. For example, the pasting diagram (1) will be represented by the five elements

$$X_2 = \alpha + \beta + \gamma + \delta, \\ X_{1,-} = a + b + e + h, \quad X_{1,+} = a + d + g + h, \\ X_{0,-} = u, \quad X_{0,+} = y.$$

Thus, one can use tools from group theory and commutative algebra when manipulating augmented directed complexes, which make them an interesting alternative to the two other set-based formalisms.

Another important point of divergence between the different formalisms is the conditions, or *axioms*, they require on diagrams in order for them to be pasting diagrams. This naturally raises the question of the difference of expressivity, *i.e.*, ability to recognize more or fewer diagrams, between these formalisms. Since the *axioms* are quite sophisticated and rely on different definitions used by each formalism and, in particular, the different notions of cells, these comparisons cannot be done so easily.
But here, these two composites are different. Even more subtle problems arise starting from dimension 3, justifying the use of sophisticated formalisms for recognizing pasting diagrams.

**Pasting diagram formalisms** Until now, different proposals of pasting diagram formalisms, each one giving a set of conditions to recognize pasting diagrams, have been made. The main ones are Johnson's *pasting schemes* [13], Street's *parity complexes* [23, 24] and Steiner's *augmented directed complexes* [21]. Even though the ideas underlying the definitions of these formalisms are quite similar, they differ on many points and comparing them precisely is uneasy, and actually, to the best of our knowledge, no formal account of the differences was ever made.

One of the main difference between these formalisms is the notion of sub-pasting diagram, or *cell*, which is used. The formalism of pasting schemes has the most evident notion of cell: it is simply a set which gathers all the generators which appear in the pasting diagram it represents, regardless of their dimension or source/target status. For example, the diagram (1) will be represented by the set

$$X = \{u, v, w, x, y, a, b, c, d, e, f, g, h, \alpha, \beta, \gamma, \delta\}.$$

Parity complexes use cells defined as tuples of sets of generators that are kept organized by dimension and by source/target status. For instance, the pasting diagram (1) is represented by five sets

$$X_2 = \{\alpha, \beta, \gamma, \delta\}, \\ X_{1,-} = \{a, b, e, h\}, \quad X_{1,+} = \{a, d, g, h\}, \\ X_{0,-} = \{u\}, \quad X_{0,+} = \{y\}$$

where $X_{i,-}$ represents the *i*-source, $X_{i,+}$ the *i*-target, and $X_2$ the 2-dimensional part of the pasting diagram. This notion of cell seems less natural at first than the one of pasting schemes, since the translations from a pasting diagram to a cell, and *vice versa*, are not evident. The notion of cell used by augmented directed complexes can be obtained by considering the abelian groups induced by an $\omega$-hypergraph. As a variant of the ones of parity complexes, cells are now given by sums of generators for each dimension and source/target status. For example, the pasting diagram (1) will be represented by the five elements

$$X_2 = \alpha + \beta + \gamma + \delta, \\ X_{1,-} = a + b + e + h, \quad X_{1,+} = a + d + g + h, \\ X_{0,-} = u, \quad X_{0,+} = y.$$

Thus, one can use tools from group theory and commutative algebra when manipulating augmented directed complexes, which make them an interesting alternative to the two other set-based formalisms.

Another important point of divergence between the different formalisms is the conditions, or *axioms*, they require on diagrams in order for them to be pasting diagrams. This naturally raises the question of the difference of expressivity, *i.e.*, ability to recognize more or fewer diagrams, between these formalisms. Since the *axioms* are quite sophisticated and rely on different definitions used by each formalism and, in particular, the different notions of cells, these comparisons cannot be done so easily.
**Outline and results** In Section 1, we recall the definitions of the main structures involved in this article. We first introduce the definitions of globular sets (Section 1.1) and strict categories (Section 1.2), and then recall the definitions of the three existing pasting diagram formalisms that we consider: parity complexes (Section 1.4), pasting schemes (Section 1.5) and augmented directed complexes (Section 1.6). We relate each definition to the unifying notion of $\omega$-hypergraph (Section 1.3): a formalism is then a class of $\omega$-hypergraphs (defined by axioms) together with a notion of cell and operations on these cells. In Section 1.4, we discuss a counter-example to the freeness property claimed in the respective articles of parity complexes and pasting schemes, i.e., that the diagrams they accept are pasting diagrams. It involves the diagram made of

<figure><img src="image_3.png" /></figure>

together with two 3-generators

<figure><img src="image_4.png" /></figure>

and

<figure><img src="image_5.png" /></figure>

This shortcoming motivated the introduction of a new formalism, called *torsion-free complexes*, whose axioms aim at correcting and generalizing the ones of parity complexes and pasting schemes (Section 1.7).

In Section 2, we show the correctness of torsion-free complexes as a pasting diagram formalism, i.e., that the set of cells associated with a torsion-free complex has a canonical structure of a free $\omega$-category. For this purpose, we state in Section 2.1 the correctness of a “gluing” operation (Theorem 2.1.1), as an adapted version of an existing result for parity complexes [23, Lemma 3.2]. This operation allows constructing new cells by gluing higher-dimensional generators on existing cells. In Section 2.2, we prove that the cells of a torsion-free complex admit a structure of an $\omega$-category (Theorem 2.2.3). Then, in Section 2.3, in order to show that the $\omega$-category is free, we first introduce the notion of freeness that we use by recalling the definition of polygraphs [22, 4], which describe sets of generators of different dimensions from which one can generate a free $\omega$-category. Finally, in Section 2.3, we state the freeness properties of the $\omega$-category of cells of a torsion-free complex (Theorem 2.4.1 and corollary 2.4.2).

In Section 3, we relate the different pasting diagram formalisms that were introduced. We first make the link between torsion-free complexes and the three other ones. For this purpose, in Section 3.1, we define other notions of cells for torsion-free complexes, namely *maximal-well-formed* and *closed-well-formed* sets. Closed-well-formed sets should be understood as the
**Outline and results** In Section 1, we recall the definitions of the main structures involved in this article. We first introduce the definitions of globular sets (Section 1.1) and strict categories (Section 1.2), and then recall the definitions of the three existing pasting diagram formalisms that we consider: parity complexes (Section 1.4), pasting schemes (Section 1.5) and augmented directed complexes (Section 1.6). We relate each definition to the unifying notion of $\omega$-hypergraph (Section 1.3): a formalism is then a class of $\omega$-hypergraphs (defined by axioms) together with a notion of cell and operations on these cells. In Section 1.4, we discuss a counter-example to the freeness property claimed in the respective articles of parity complexes and pasting schemes, i.e., that the diagrams they accept are pasting diagrams. It involves the diagram made of

<figure><img src="image_4.png" /></figure>

together with two 3-generators

<figure><img src="image_5.png" /></figure>

and

<figure><img src="image_6.png" /></figure>

This shortcoming motivated the introduction of a new formalism, called *torsion-free complexes*, whose axioms aim at correcting and generalizing the ones of parity complexes and pasting schemes (Section 1.7).

In Section 2, we show the correctness of torsion-free complexes as a pasting diagram formalism, i.e., that the set of cells associated with a torsion-free complex has a canonical structure of a free $\omega$-category. For this purpose, we state in Section 2.1 the correctness of a “gluing” operation (Theorem 2.1.1), as an adapted version of an existing result for parity complexes [23, Lemma 3.2]. This operation allows constructing new cells by gluing higher-dimensional generators on existing cells. In Section 2.2, we prove that the cells of a torsion-free complex admit a structure of an $\omega$-category (Theorem 2.2.3). Then, in Section 2.3, in order to show that the $\omega$-category is free, we first introduce the notion of freeness that we use by recalling the definition of polygraphs [22, 4], which describe sets of generators of different dimensions from which one can generate a free $\omega$-category. Finally, in Section 2.3, we state the freeness properties of the $\omega$-category of cells of a torsion-free complex (Theorem 2.4.1 and corollary 2.4.2).

In Section 3, we relate the different pasting diagram formalisms that were introduced. We first make the link between torsion-free complexes and the three other ones. For this purpose, in Section 3.1, we define other notions of cells for torsion-free complexes, namely *maximal-well-formed* and *closed-well-formed* sets. Closed-well-formed sets should be understood as the
equivalent of the notion of cell for pasting schemes in torsion-free complexes. Maximal-well-
formed sets are then a convenient intermediate for proofs between the original notion of cell
for parity complexes and closed-well-formed sets. We show that the both new notions induce
ω-categories of cells isomorphic to the original one (Theorems 3.1.18 and 3.1.21). We then prove
the embedding results into torsion-free complexes for the three other formalisms: in Section 3.2,
we show that parity complexes are torsion-free complexes (Theorem 3.2.3); in Section 3.3, we
show that loop-free pasting schemes are torsion-free complexes (Theorem 3.3.9) and that both
formalisms induce isomorphic ω-categories (Theorem 3.3.10); in Section 3.4, we show that loop-
free unital augmented directed complexes are torsion-free complexes (Theorem 3.4.17) and that
both formalisms induce isomorphic ω-categories (Theorem 3.4.18). Finally, in Section 3.5, we
give counter-examples to the other embeddings between the formalisms.

**Applications and related works** Pasting diagram formalisms are an effective description of cells of free ω-categories. In particular, they give a precise definition to the notion of commutative diagram and can represent generic compositions. Moreover, they make it possible to study higher categories by probing them through pasting diagrams. For example, augmented directed complexes were used to give an effective description of the Gray tensor product in [21]. In a related manner, Kapranov and Voevodsky studied topological properties of pasting schemes in [15] and used them in an attempt to give a description of ω-groupoids in [14], but their results were shown paradoxical [20].

Several other works studied pasting diagrams. In [3], Buckley gives a mechanized Coq proof of the results of [23] but stops at the excision theorem [23, Theorem 4.1]. In particular, the proof of the freeness claim [23, Theorem 4.2] was not formally verified, and could not be, since this claim does not hold in general, as is shown in the present paper. In [5], Campbell isolates a common structure behind parity complexes and pasting schemes, called *parity structure*, and gives stronger axioms than the ones of parity complexes and pasting schemes, taking an opposite path from this work which seeks a more general formalism. In [18], Nguyen studies *pre-polytopes with labeled structures* and shows that they give a parity structure that satisfies a variant of Campbell's axioms that are enough to obtain another correct notion of pasting diagrams. In [10], Henry defines a theoretical notion of pasting diagrams, called polyplexes, to show that certain classes of polygraphs are presheaf categories, and uses them to prove a variant of the Simpson's conjecture in [11]. However, his pasting diagrams can involve some looping behaviors, and are then out of the scope of the formalisms studied in the present work. Using similar ideas, Hadzihasanovic [9] defines a class of pasting diagrams, called *regular polygraphs*, that is “big enough” to study semi-strict categories and which is well-behaved for several constructions (notably, their realizations as topological spaces are CW complexes).

**Acknowledgements** I would like to deeply thank Samuel Mimram and Yves Guiraud for their supervision, help and useful feedback during this work. I would also like to thank Simon Henry, Ross Street and Léonard Guetta for the interesting exchanges on the subject. Finally, I would like to thank École Normale Supérieure de Paris for funding my PhD thesis, during which most of this work was carried out.

**Notations** We write $\mathbb{N}$ for the set of natural numbers, $\mathbb{N}^*$ for $\mathbb{N} \setminus \{0\}$ and $\omega$ for the first infinite ordinal. Given $n \in \mathbb{N}$, we write $\mathbb{N}_n$ for the set $\{0, \dots, n\}$ and $\mathbb{N}_n^*$ for $\mathbb{N}_n \setminus \{0\}$. We use the convention that $\mathbb{N}_\omega$ denotes $\mathbb{N}$.
equivalent of the notion of cell for pasting schemes in torsion-free complexes. Maximal-well-
formed sets are then a convenient intermediate for proofs between the original notion of cell
for parity complexes and closed-well-formed sets. We show that the both new notions induce
ω-categories of cells isomorphic to the original one (Theorems 3.1.18 and 3.1.21). We then prove
the embedding results into torsion-free complexes for the three other formalisms: in Section 3.2,
we show that parity complexes are torsion-free complexes (Theorem 3.2.3); in Section 3.3, we
show that loop-free pasting schemes are torsion-free complexes (Theorem 3.3.9) and that both
formalisms induce isomorphic ω-categories (Theorem 3.3.10); in Section 3.4, we show that loop-
free unital augmented directed complexes are torsion-free complexes (Theorem 3.4.17) and that
both formalisms induce isomorphic ω-categories (Theorem 3.4.18). Finally, in Section 3.5, we
give counter-examples to the other embeddings between the formalisms.

**Applications and related works** Pasting diagram formalisms are an effective description of cells of free ω-categories. In particular, they give a precise definition to the notion of commutative diagram and can represent generic compositions. Moreover, they make it possible to study higher categories by probing them through pasting diagrams. For example, augmented directed complexes were used to give an effective description of the Gray tensor product in [21]. In a related manner, Kapranov and Voevodsky studied topological properties of pasting schemes in [15] and used them in an attempt to give a description of ω-groupoids in [14], but their results were shown paradoxical [20].

Several other works studied pasting diagrams. In [3], Buckley gives a mechanized Coq proof of the results of [23] but stops at the excision theorem [23, Theorem 4.1]. In particular, the proof of the freeness claim [23, Theorem 4.2] was not formally verified, and could not be, since this claim does not hold in general, as is shown in the present paper. In [5], Campbell isolates a common structure behind parity complexes and pasting schemes, called *parity structure*, and gives stronger axioms than the ones of parity complexes and pasting schemes, taking an opposite path from this work which seeks a more general formalism. In [18], Nguyen studies *pre-polytopes with labeled structures* and shows that they give a parity structure that satisfies a variant of Campbell’s axioms that are enough to obtain another correct notion of pasting diagrams. In [10], Henry defines a theoretical notion of pasting diagrams, called polyplexes, to show that certain classes of polygraphs are presheaf categories, and uses them to prove a variant of the Simpson’s conjecture in [11]. However, his pasting diagrams can involve some looping behaviors, and are then out of the scope of the formalisms studied in the present work. Using similar ideas, Hadzihasanovic [9] defines a class of pasting diagrams, called *regular polygraphs*, that is “big enough” to study semi-strict categories and which is well-behaved for several constructions (notably, their realizations as topological spaces are CW complexes).

**Acknowledgements** I would like to deeply thank Samuel Mimram and Yves Guiraud for their supervision, help and useful feedback during this work. I would also like to thank Simon Henry, Ross Street and Léonard Guetta for the interesting exchanges on the subject. Finally, I would like to thank École Normale Supérieure de Paris for funding my PhD thesis, during which most of this work was carried out.

**Notations** We write $\mathbb{N}$ for the set of natural numbers, $\mathbb{N}^*$ for $\mathbb{N} \setminus \{0\}$ and $\omega$ for the first infinite ordinal. Given $n \in \mathbb{N}$, we write $\mathbb{N}_n$ for the set $\{0, \dots, n\}$ and $\mathbb{N}_n^*$ for $\mathbb{N}_n \setminus \{0\}$. We use the convention that $\mathbb{N}_\omega$ denotes $\mathbb{N}$.
# 1. Formalisms of pasting diagrams

In this section, we recall some basic definitions about strict categories and introduce the definitions of the formalisms of pasting diagrams that we will consider in this article. We present them through the common perspective of $\omega$-hypergraphs, that are structures which encode the information in diagrams of generators like (1). Then, the definition of each formalism roughly follows the same pattern. First, a definition for cells that represent pasting diagrams is introduced, together with an identity and composition operations that aim at equipping those cells with a structure of $\omega$-category. Then, a class of $\omega$-hypergraphs that are correctly handled by the considered formalism is defined by the mean of axioms or conditions.

We first recall the definition of globular sets (Section 1.1) and of strict categories (Section 1.2) as globular sets with additional operations. We then introduce $\omega$-hypergraphs (Section 1.3) and recall the definitions of the three main existing formalisms for pasting diagrams: parity complexes (Section 1.4), pasting schemes (Section 1.5) and augmented directed complexes (Section 1.6). Then, we introduce the new formalism of torsion-free complexes that share the definitions of parity complexes but have different axioms on $\omega$-hypergraphs (Section 1.7).

## 1.1 Globular sets
Given $n \in \mathbb{N} \cup \{\omega\}$, an $n$-globular set $(X, \partial^{-}, \partial^{+})$ is the data of sets $X_k$ for $k \in \mathbb{N}_n$, the elements of $X_k$ being called $k$-cells, together with, for $i \in \mathbb{N}_{n-1}$, functions

$$ \partial_i^{-}, \partial_i^{+} : X_{i+1} \to X_i $$

as in

$$ X_0 \underset{\partial_0^+}{\overset{\partial_0^-}{\rightleftarrows}} X_1 \underset{\partial_1^+}{\overset{\partial_1^-}{\rightleftarrows}} X_2 \underset{\partial_2^+}{\overset{\partial_2^-}{\rightleftarrows}} \cdots \underset{\partial_{n-2}^+}{\overset{\partial_{n-2}^-}{\rightleftarrows}} X_{n-1} \underset{\partial_{n-1}^+}{\overset{\partial_{n-1}^-}{\rightleftarrows}} X_n $$

satisfying the following globular identities for every $i \in \mathbb{N}_{n-2}$:

$$ \partial_i^{-} \circ \partial_{i+1}^{-} = \partial_i^{-} \circ \partial_{i+1}^{+} \quad \text{and} \quad \partial_i^{+} \circ \partial_{i+1}^{-} = \partial_i^{+} \circ \partial_{i+1}^{+}. $$

Given $k \in \mathbb{N}_n^*$, the elements of $X_k$ are called the $k$-globes of $X$. Given $i, j \in \mathbb{N}$ with $i \le j$, by abusing notation, we write $\partial_i^{-} : X_j \to X_i$ for the function

$$ \partial_i^{-} = \partial_i^{-} \circ \partial_{i+1}^{-} \circ \cdots \circ \partial_{j-1}^{-} $$

and similarly for $\partial^+$. Given $i, k \in \mathbb{N}_n$ with $i \le k$, for $u \in X_k$, $\partial_i^{-}(u)$ and $\partial_i^{+}(u)$ are respectively the $i$-source and $i$-target of $u$. We write $X_k \times_i X_k$ for the pullback

<figure><img src="image_6.png" /></figure>

Given $u, v \in X_k$, we say that $u$ and $v$ are $i$-composable when $\partial_i^+(u) = \partial_i^-(v)$. More generally, given $p \ge 0$ and $u_1, \dots, u_p \in X_k$, we say that $u_1, \dots, u_p$ are $i$-composable when, for $j \in \mathbb{N}_{p-1}^*$, $u_j$ and $u_{j+1}$ are $i$-composable.

Given two $n$-globular sets $X$ and $Y$, a morphism $F: X \to Y$ between $X$ and $Y$ is the data of functions $F_k: X_k \to Y_k$ for $k \in \mathbb{N}_n$ such that $F_i \circ \partial_i^\epsilon = \partial_i^\epsilon \circ F_{i+1}$ for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$.
# 1. Formalisms of pasting diagrams

In this section, we recall some basic definitions about strict categories and introduce the definitions of the formalisms of pasting diagrams that we will consider in this article. We present them through the common perspective of $\omega$-hypergraphs, that are structures which encode the information in diagrams of generators like (1). Then, the definition of each formalism roughly follows the same pattern. First, a definition for cells that represent pasting diagrams is introduced, together with an identity and composition operations that aim at equipping those cells with a structure of $\omega$-category. Then, a class of $\omega$-hypergraphs that are correctly handled by the considered formalism is defined by the mean of axioms or conditions.

We first recall the definition of globular sets (Section 1.1) and of strict categories (Section 1.2) as globular sets with additional operations. We then introduce $\omega$-hypergraphs (Section 1.3) and recall the definitions of the three main existing formalisms for pasting diagrams: parity complexes (Section 1.4), pasting schemes (Section 1.5) and augmented directed complexes (Section 1.6). Then, we introduce the new formalism of torsion-free complexes that share the definitions of parity complexes but have different axioms on $\omega$-hypergraphs (Section 1.7).

## 1.1 Globular sets
Given $n \in \mathbb{N} \cup \{\omega\}$, an $n$-globular set $(X, \partial^{-}, \partial^{+})$ is the data of sets $X_k$ for $k \in \mathbb{N}_n$, the elements of $X_k$ being called $k$-cells, together with, for $i \in \mathbb{N}_{n-1}$, functions

$$ \partial_i^{-}, \partial_i^{+} : X_{i+1} \to X_i $$

as in

$$ X_0 \underset{\partial_0^+}{\overset{\partial_0^-}{\rightleftarrows}} X_1 \underset{\partial_1^+}{\overset{\partial_1^-}{\rightleftarrows}} X_2 \underset{\partial_2^+}{\overset{\partial_2^-}{\rightleftarrows}} \cdots \underset{\partial_{n-2}^+}{\overset{\partial_{n-2}^-}{\rightleftarrows}} X_{n-1} \underset{\partial_{n-1}^+}{\overset{\partial_{n-1}^-}{\rightleftarrows}} X_n $$

satisfying the following globular identities for every $i \in \mathbb{N}_{n-2}$:

$$ \partial_i^{-} \circ \partial_{i+1}^{-} = \partial_i^{-} \circ \partial_{i+1}^{+} \quad \text{and} \quad \partial_i^{+} \circ \partial_{i+1}^{-} = \partial_i^{+} \circ \partial_{i+1}^{+}. $$

Given $k \in \mathbb{N}_n^*$, the elements of $X_k$ are called the $k$-globes of $X$. Given $i, j \in \mathbb{N}$ with $i \le j$, by abusing notation, we write $\partial_i^{-} : X_j \to X_i$ for the function

$$ \partial_i^{-} = \partial_i^{-} \circ \partial_{i+1}^{-} \circ \cdots \circ \partial_{j-1}^{-} $$

and similarly for $\partial^+$. Given $i, k \in \mathbb{N}_n$ with $i \le k$, for $u \in X_k$, $\partial_i^{-}(u)$ and $\partial_i^{+}(u)$ are respectively the $i$-source and $i$-target of $u$. We write $X_k \times_i X_k$ for the pullback

<figure><img src="image_7.png" /></figure>

Given $u, v \in X_k$, we say that $u$ and $v$ are $i$-composable when $\partial_i^+(u) = \partial_i^-(v)$. More generally, given $p \ge 0$ and $u_1, \dots, u_p \in X_k$, we say that $u_1, \dots, u_p$ are $i$-composable when, for $j \in \mathbb{N}_{p-1}^*$, $u_j$ and $u_{j+1}$ are $i$-composable.

Given two $n$-globular sets $X$ and $Y$, a morphism $F: X \to Y$ between $X$ and $Y$ is the data of functions $F_k: X_k \to Y_k$ for $k \in \mathbb{N}_n$ such that $F_i \circ \partial_i^\epsilon = \partial_i^\epsilon \circ F_{i+1}$ for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$.
**1.2 Strict categories** Given $n \in \mathbb{N} \cup \{\omega\}$, a strict $n$-category $(C, \partial^{-}, \partial^{+}, \mathrm{id}, *)$ (often simply denoted $C$) is an $n$-globular set $(C, \partial^{-}, \partial^{+})$ together with, for $k \in \mathbb{N}$ with $k < n$, identity operations

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

often written $\mathrm{id}$ when there is no ambiguity on $k$, and, for $i, k \in \mathbb{N}_n$ with $i < k$, composition operations

$$ *_{i,k}: C_k \times_i C_k \to C_k $$

often denoted $*_{i}$ when there is no ambiguity on $k$, which satisfy the axioms (S-i) to (S-vi) below. Given $k, l \in \mathbb{N}_n$ such that $k \le l$ and $u \in C_k$, we extend the notations for identity operations and write $\mathrm{id}^l(u)$ for

$$ \mathrm{id}^l(u) = \mathrm{id}^l \circ \cdots \circ \mathrm{id}^{k+1}(u) $$

and, for the sake of conciseness, we often write $\mathrm{id}_u^l$ for $\mathrm{id}^l(u)$, or even $\mathrm{id}_u$ when $l = k + 1$. The axioms are the following:

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

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

(S-ii) for $i, k \in \mathbb{N}_n$ with $i < k$, $(u, v) \in C_k \times_i C_k$ and $\epsilon \in \{-, +\}$,

$$ \partial_{k-1}^{\epsilon}(u *_{i} v) = \begin{cases} \partial_{k-1}^{\epsilon}(u) *_{i} \partial_{k-1}^{\epsilon}(v) & \text{if } i < k-1, \\ \partial_{k-1}^{-}(u) & \text{if } i = k-1 \text{ and } \epsilon = -, \\ \partial_{k-1}^{+}(v) & \text{if } i = k-1 \text{ and } \epsilon = +, \end{cases} $$

(S-iii) for $i, k \in \mathbb{N}_n$ such that $i < k$, and $u \in C_k$,

$$ \mathrm{id}^k(\partial_i^{-}(u)) *_{i} u = u = u *_{i} \mathrm{id}^k(\partial_i^{+}(u)), $$

(S-iv) for $i, k \in \mathbb{N}_n$ such that $i < k$, and $i$-composable $u, v, w \in C_k$,

$$ (u *_{i} v) *_{i} w = u *_{i} (v *_{i} w), $$

(S-v) for $i, k \in \mathbb{N}_{n-1}$ such that $i < k$, and $(u, v) \in C_k \times_i C_k$,

$$ \mathrm{id}^{k+1}(u *_{i} v) = \mathrm{id}_{u}^{k+1} *_{i} \mathrm{id}_{v}^{k+1}, $$

(S-vi) ("exchange law") for $i, j, k \in \mathbb{N}_n$ such that $i < j < k$, and $u, u', v, v' \in C_k$ such that $u, v$ are $i$-composable, and $u, u'$ are $j$-composable, and $v, v'$ are $j$-composable,

$$ (u *_{i} v) *_{j} (u' *_{i} v') = (u *_{j} u') *_{i} (v *_{j} v'). $$

Given two strict $n$-categories $C$ and $D$, a morphism $F$ between $C$ and $D$ is the data of an $n$-globular morphism $F: C \to D$ which moreover satisfies that

$$ - F(\mathrm{id}_u^{k+1}) = \mathrm{id}_{F(u)}^{k+1} \text{ for every } k \in \mathbb{N}_{n-1} \text{ and } u \in C_k, $$

$$ - F(u *_{i} v) = F(u) *_{i} F(v) \text{ for every } i, k \in \mathbb{N}_{n} \text{ with } i < k \text{ and } i\text{-composable } u, v \in C_{k}. $$

We often call such morphisms $n$-functors. We write $\mathbf{Cat}_n$ for the category of strict $n$-categories. Given $k, l \in \mathbb{N} \cup \{\omega\}$ with $k < l$, there is an evident truncation functor

$$ (-)_{\le k}: \mathbf{Cat}_l \to \mathbf{Cat}_k $$

which forgets the cells of dimension $> k$ from an $l$-category.

**Remark 1.2.1.** Using the truncation functors, the category $\mathbf{Cat}_{\omega}$ could equivalently be defined as the strict limit in **CAT** on the diagram

$$ \mathbf{Cat}_0 \xleftarrow{(-)_{\le 0}} \mathbf{Cat}_1 \xleftarrow{(-)_{\le 1}} \mathbf{Cat}_2 \xleftarrow{(-)_{\le 2}} \dots \xleftarrow{(-)_{\le k-1}} \mathbf{Cat}_k \xleftarrow{(-)_{\le k}} \mathbf{Cat}_{k+1} \xleftarrow{(-)_{\le k+1}} \dots $$
**1.2 Strict categories** Given $n \in \mathbb{N} \cup \{\omega\}$, a strict $n$-category $(C, \partial^{-}, \partial^{+}, \mathrm{id}, *)$ (often simply denoted $C$) is an $n$-globular set $(C, \partial^{-}, \partial^{+})$ together with, for $k \in \mathbb{N}$ with $k < n$, identity operations

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

often written $\mathrm{id}$ when there is no ambiguity on $k$, and, for $i, k \in \mathbb{N}_n$ with $i < k$, composition operations

$$ *_{i,k}: C_k \times_i C_k \to C_k $$

often denoted $*_i$ when there is no ambiguity on $k$, which satisfy the axioms (S-i) to (S-vi) below. Given $k, l \in \mathbb{N}_n$ such that $k \le l$ and $u \in C_k$, we extend the notations for identity operations and write $\mathrm{id}^l(u)$ for

$$ \mathrm{id}^l(u) = \mathrm{id}^l \circ \cdots \circ \mathrm{id}^{k+1}(u) $$

and, for the sake of conciseness, we often write $\mathrm{id}_u^l$ for $\mathrm{id}^l(u)$, or even $\mathrm{id}_u$ when $l = k + 1$. The axioms are the following:

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

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

(S-ii) for $i, k \in \mathbb{N}_n$ with $i < k$, $(u, v) \in C_k \times_i C_k$ and $\epsilon \in \{-, +\}$,

$$ \partial_{k-1}^{\epsilon}(u *_{i} v) = \begin{cases} \partial_{k-1}^{\epsilon}(u) *_{i} \partial_{k-1}^{\epsilon}(v) & \text{if } i < k-1, \\ \partial_{k-1}^{-}(u) & \text{if } i = k-1 \text{ and } \epsilon = -, \\ \partial_{k-1}^{+}(v) & \text{if } i = k-1 \text{ and } \epsilon = +, \end{cases} $$

(S-iii) for $i, k \in \mathbb{N}_n$ such that $i < k$, and $u \in C_k$,

$$ \mathrm{id}^k(\partial_i^{-}(u)) *_{i} u = u = u *_{i} \mathrm{id}^k(\partial_i^{+}(u)), $$

(S-iv) for $i, k \in \mathbb{N}_n$ such that $i < k$, and $i$-composable $u, v, w \in C_k$,

$$ (u *_{i} v) *_{i} w = u *_{i} (v *_{i} w), $$

(S-v) for $i, k \in \mathbb{N}_{n-1}$ such that $i < k$, and $(u, v) \in C_k \times_i C_k$,

$$ \mathrm{id}^{k+1}(u *_{i} v) = \mathrm{id}_{u}^{k+1} *_{i} \mathrm{id}_{v}^{k+1}, $$

(S-vi) ("exchange law") for $i, j, k \in \mathbb{N}_n$ such that $i < j < k$, and $u, u', v, v' \in C_k$ such that $u, v$ are $i$-composable, and $u, u'$ are $j$-composable, and $v, v'$ are $j$-composable,

$$ (u *_{i} v) *_{j} (v' *_{i} v') = (u *_{j} u') *_{i} (v *_{j} v'). $$

Given two strict $n$-categories $C$ and $D$, a morphism $F$ between $C$ and $D$ is the data of an $n$-globular morphism $F: C \to D$ which moreover satisfies that

$$ - F(\mathrm{id}_u^{k+1}) = \mathrm{id}_{F(u)}^{k+1} \text{ for every } k \in \mathbb{N}_{n-1} \text{ and } u \in C_k, $$

$$ - F(u *_{i} v) = F(u) *_{i} F(v) \text{ for every } i, k \in \mathbb{N}_{n} \text{ with } i < k \text{ and } i\text{-composable } u, v \in C_{k}. $$

We often call such morphisms $n$-functors. We write $\mathbf{Cat}_n$ for the category of strict $n$-categories. Given $k, l \in \mathbb{N} \cup \{\omega\}$ with $k < l$, there is an evident truncation functor

$$ (-)_{\le k}: \mathbf{Cat}_l \to \mathbf{Cat}_k $$

which forgets the cells of dimension $> k$ from an $l$-category.

**Remark 1.2.1.** Using the truncation functors, the category $\mathbf{Cat}_\omega$ could equivalently be defined as the strict limit in **CAT** on the diagram

$$ \mathbf{Cat}_0 \xleftarrow{(-)_{\le 0}} \mathbf{Cat}_1 \xleftarrow{(-)_{\le 1}} \mathbf{Cat}_2 \xleftarrow{(-)_{\le 2}} \dots \xleftarrow{(-)_{\le k-1}} \mathbf{Cat}_k \xleftarrow{(-)_{\le k}} \mathbf{Cat}_{k+1} \xleftarrow{(-)_{\le k+1}} \dots $$
**Other source and target operations** Given an $\omega$-hypergraph $P$, for $n \ge 2$, $u \in P_n$ and $\epsilon, \eta \in \{-, +\}$, we write $u^{\epsilon\eta}$ for $(u^\epsilon)^\eta$. We extend the notation to subsets $U \subseteq P_n$ and write $U^{\epsilon\eta}$ for $(U^\epsilon)^\eta$. Moreover, we write $u^\mp$ and $u^\pm$ for

$$u^{\mp} = u^{-} \setminus u^{+} \quad \text{and} \quad u^{\pm} = u^{+} \setminus u^{-}.$$

We also extend the notation to subsets $U \subseteq P_n$ and write $U^{\mp}$ and $U^{\pm}$ for

$$U^{\mp} = U^{-} \setminus U^{+} \quad \text{and} \quad U^{\pm} = U^{+} \setminus U^{-}.$$

*Example 1.3.4.* Consider the $\omega$-hypergraph represented by the diagram

$$t \xrightarrow{a} u \xrightarrow{b} \xrightarrow{\downarrow \alpha} w' \xrightarrow{c''} w \xrightarrow{\downarrow \beta} x \xrightarrow{c} w'' \xrightarrow{d} d' \xrightarrow{e} y \xrightarrow{f} z . \quad (6)$$

For this $\omega$-hypergraph, we have

$$\alpha^{--} = \{u, v\}, \quad \alpha^{+-} = \{u, v'\}, \quad \alpha^{-\mp} = \{u\}, \quad \alpha^{\pm\pm} = \{w'\}$$

and, writing $U$ for the set $\{a, b, c, d, e, f\}$,

$$\begin{align*} U^{-} &= \{t, u, v, w, x, y\}, & U^{+} &= \{u, v, w, x, y, z\}, \\ U^{\mp} &= \{t\}, & U^{\pm} &= \{z\} \end{align*}$$

and, writing $V$ for the set $\{\alpha, \beta, \gamma, \delta\}$,

$$\begin{align*} V^{-} &= \{b, c, c'', c''', d, d'', d''', e\}, & V^{+} &= \{b', c', c'', c''', d', d'', d''', e'\}, \\ V^{\mp} &= \{b, c, d, e\}, & V^{\pm} &= \{b', c', d', e'\}. \end{align*}$$

From the above examples, one can intuitively describe the operations $(-)^{-}$ and $(-)^{+}$ as computing the “inner” sources and targets of a set of generators, whereas the operations $(-)^{\mp}$ and $(-)^{\pm}$ compute the source and target “borders” of a set of generators.

**1.4 Parity complexes** In this section, we recall the formalism of parity complexes developed by Street in [23]. Most of the content will be reused when defining torsion-free complexes. The idea behind the formalism is to represent an $(n+1)$-cell as a pair of source and target $n$-cells together with a subset of $P_{n+1}$ which “moves” the source $n$-cell to the target $n$-cell.

**Pre-cells** Let $P$ be an $\omega$-hypergraph. For $n \in \mathbb{N}$, an $n$-pre-cell of $P$ is a tuple

$$X = (X_{0,-}, X_{0,+}, \dots, X_{n-1,-}, X_{n-1,+}, X_n)$$

of finite subsets of $P$, such that $X_{i,\epsilon} \subseteq P_i$ for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$, and $X_n \subseteq P_n$. By convention, we often denote $X_n$ by $X_{n,-}$ or $X_{n,+}$. We write $PCell(P)$ for the graded set of pre-cells of $P$. Given $n \in \mathbb{N}$, $\epsilon \in \{-, +\}$ and an $(n+1)$-pre-cell $X$ of $P$, we define the $n$-pre-cell $\partial_n^\epsilon(X)$ as

$$\partial_n^\epsilon(X) = (X_{0,-}, X_{0,+}, \dots, X_{n-1,-}, X_{n-1,+}, X_{n,\epsilon}).$$

The functions $\partial^{-}, \partial^{+}$ then equip $PCell(P)$ with a structure of an $\omega$-globular set.
<figure><img src="image_7.png" /><figcaption>Figure 1: Movements in a cell</figcaption></figure>

**Movement and orthogonality** Let $P$ be an $\omega$-hypergraph. Given $n \in \mathbb{N}$ and finite sets $M \subseteq P_{n+1}$, $U \subseteq P_n$ and $V \subseteq P_n$, we say that $M$ moves $U$ to $V$ when

$$U = (V \cup M^-) \setminus M^+ \quad \text{and} \quad V = (U \cup M^+) \setminus M^-.$$

Intuitively, the first equation means that $U$ is the subset obtained from $V$ by replacing the target of $M$ by its source, and the second equation has a dual meaning.

*Example 1.4.1.* In the $\omega$-hypergraph (6), the set $\{\alpha, \beta, \gamma, \delta\}$ moves the set $\{a, b, c, d, e, f\}$ to the set $\{a, b', c', d', e', f\}$.

**Cells** Let $P$ be an $\omega$-hypergraph. Given $n \in \mathbb{N}$, an $n$-cell of $P$ is an $n$-pre-cell of $P$, such that

(i) $X_{i+1,\epsilon}$ moves $X_{i,-}$ to $X_{i,+}$ for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$,

(ii) $X_{i,\epsilon}$ is fork-free for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$.

We denote by $\text{Cell}(P)$ the graded set of cells of $P$, which inherits the structure of globular set from $\text{PCell}(P)$. An $n$-cell $X$ can be represented as on Figure 1 where each arrow

$$U \xrightarrow{M} V$$

means that $M$ moves $U$ to $V$.

*Example 1.4.2.* The $\omega$-hypergraph represented by (6) has, among others,

- a 0-cell ($\{t\}$),

- a 1-cell ($\{t\}$, $\{w'\}$, $\{a, b, c''\}$, $\{a, b, c'''\}$, $\{\alpha\}$),

- a 2-cell ($\{t\}$, $\{z\}$, $\{a, b, c, d, e, f\}$, $\{a, b', c', d', e', f\}$, $\{\alpha, \beta, \gamma, \delta\}$), etc.

**Identity and composition of operations** Let $P$ be an $\omega$-hypergraph. Given $n \in \mathbb{N}$ and an $n$-cell $X$, the identity of $X$ is the $(n+1)$-cell

$$\text{id}^{n+1}(X) = (X_{0,-}, X_{0,+}, \dots, X_{n-1,-}, X_{n-1,+}, X_n, X_n, \emptyset).$$

Given $i, n \in \mathbb{N}$ with $i < n$, and $i$-composable $n$-cells $X, Y \in \text{Cell}(P)_n$, the $i$-composition $X *_i Y$ of $X$ and $Y$ is defined as the $n$-pre-cell $Z$ such that, for $j \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$,

$$Z_{j,\epsilon} = \begin{cases} X_{j,\epsilon} & \text{if } j < i, \\ X_{i,-} & \text{if } j = i \text{ and } \epsilon = -, \\ Y_{i,+} & \text{if } j = i \text{ and } \epsilon = +, \\ X_{j,\epsilon} \cup Y_{j,\epsilon} & \text{if } j > i. \end{cases}$$

It will be shown in Section 2 that, under suitable assumptions, the composite of two $n$-cells is actually an $n$-cell.
<figure><img src="image_8.png" /><figcaption>Figure 1: Movements in a cell</figcaption></figure>

**Movement and orthogonality** Let $P$ be an $\omega$-hypergraph. Given $n \in \mathbb{N}$ and finite sets $M \subseteq P_{n+1}$, $U \subseteq P_n$ and $V \subseteq P_n$, we say that $M$ moves $U$ to $V$ when

$$U = (V \cup M^-) \setminus M^+ \quad \text{and} \quad V = (U \cup M^+) \setminus M^-.$$

Intuitively, the first equation means that $U$ is the subset obtained from $V$ by replacing the target of $M$ by its source, and the second equation has a dual meaning.

*Example 1.4.1.* In the $\omega$-hypergraph (6), the set $\{\alpha, \beta, \gamma, \delta\}$ moves the set $\{a, b, c, d, e, f\}$ to the set $\{a, b', c', d', e', f\}$.

**Cells** Let $P$ be an $\omega$-hypergraph. Given $n \in \mathbb{N}$, an $n$-cell of $P$ is an $n$-pre-cell of $P$, such that

(i) $X_{i+1,\epsilon}$ moves $X_{i,-}$ to $X_{i,+}$ for $i \in \mathbb{N}_{n-1}$ and $\epsilon \in \{-, +\}$,

(ii) $X_{i,\epsilon}$ is fork-free for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$.

We denote by $\text{Cell}(P)$ the graded set of cells of $P$, which inherits the structure of globular set from $\text{PCell}(P)$. An $n$-cell $X$ can be represented as on Figure 1 where each arrow

$$U \stackrel{M}{\rightleftharpoons} V$$

means that $M$ moves $U$ to $V$.

*Example 1.4.2.* The $\omega$-hypergraph represented by (6) has, among others,

- a 0-cell ($\{t\}$),

- a 1-cell ($\{t\}$, $\{w'\}$, $\{a, b, c''\}$, $\{a, b, c'''\}$, $\{\alpha\}$),

- a 2-cell ($\{t\}$, $\{z\}$, $\{a, b, c, d, e, f\}$, $\{a, b', c', d', e', f\}$, $\{\alpha, \beta, \gamma, \delta\}$), etc.

**Identity and composition of operations** Let $P$ be an $\omega$-hypergraph. Given $n \in \mathbb{N}$ and an $n$-cell $X$, the identity of $X$ is the $(n+1)$-cell

$$\text{id}^{n+1}(X) = (X_{0,-}, X_{0,+}, \dots, X_{n-1,-}, X_{n-1,+}, X_n, X_n, \emptyset).$$

Given $i, n \in \mathbb{N}$ with $i < n$, and $i$-composable $n$-cells $X, Y \in \text{Cell}(P)_n$, the $i$-composition $X *_i Y$ of $X$ and $Y$ is defined as the $n$-pre-cell $Z$ such that, for $j \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$,

$$Z_{j,\epsilon} = \begin{cases} X_{j,\epsilon} & \text{if } j < i, \\ X_{i,-} & \text{if } j = i \text{ and } \epsilon = -, \\ Y_{i,+} & \text{if } j = i \text{ and } \epsilon = +, \\ X_{j,\epsilon} \cup Y_{j,\epsilon} & \text{if } j > i. \end{cases}$$

It will be shown in Section 2 that, under suitable assumptions, the composite of two $n$-cells is actually an $n$-cell.
**Atoms and relevance** Let $P$ be an $\omega$-hypergraph. Given $n \in \mathbb{N}$ and $u \in P_n$, we define sets $\langle u \rangle_{i,\epsilon} \subseteq P_i$ for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$ with a downward induction by

$$ \langle u \rangle_{n,-} = \langle u \rangle_{n,+} = \{u\} $$

and

$$ \langle u \rangle_{j,-} = \langle u \rangle_{j+1,-}^{\mp} \quad \langle u \rangle_{j,+} = \langle u \rangle_{j+1,+}^{\pm} $$

for $j \in \mathbb{N}_{n-1}$. We often write $\langle u \rangle_n$ for both $\langle u \rangle_{n,-}$ and $\langle u \rangle_{n,+}$. The atom associated to $u$ is then the $n$-pre-cell of $P$

$$ \langle u \rangle = (\langle u \rangle_{0,-}, \langle u \rangle_{0,+}, \dots, \langle u \rangle_{n-1,-}, \langle u \rangle_{n-1,+}, \langle u \rangle_n). $$

A generator $u$ is said relevant when the atom $\langle u \rangle$ is a cell. When $P$ is a parity complex, the relevant generators of $P$ will have the role of generating cells in the $\omega$-category $\mathrm{Cell}(P)$.

**Example 1.4.3.** The atom associated to $\alpha$ in (4) is $\langle \alpha \rangle$ with

$$ \begin{align*} \langle \alpha \rangle_{0,-} &= \{u\}, & \langle \alpha \rangle_{1,-} &= \{a, c\}, & \langle \alpha \rangle_{2} &= \{\alpha\}, \\ \langle \alpha \rangle_{0,+} &= \{z\}, & \langle \alpha \rangle_{1,+} &= \{b, d\}, \end{align*} $$

and, since it is a cell, $\alpha$ is relevant.

**Tightness** Some defects were found in the first definition of parity complexes given in [23], so that Street fixed his definition in [24]. His correction involves the notion of tightness defined as follows. Given $n \in \mathbb{N}$, a subset $U \subseteq P_n$ is said to be tight when, for all $u, v \in P_n$ such that $u \triangleleft v$ and $v \in U$, we have $u^{-} \cap U^{\pm} = \emptyset$.

**Example 1.4.4.** In (6), $U = \{\beta, \gamma\}$ is not tight since $\alpha \triangleleft \gamma$ and $c'' \in \alpha^{-} \cap U^{\pm}$. However, the set $U' = \{\alpha, \beta, \gamma, \delta\}$ is tight.

**Parity complexes** A parity complex [23, 24] is an $\omega$-hypergraph $P$ satisfying the axioms (C0) to (C5) below:

(C0) for $n \in \mathbb{N}^*$ and $u \in P_n$, $u^{-} \neq \emptyset$ and $u^{+} \neq \emptyset$;

(C1) for $n \in \mathbb{N}$ with $n \ge 2$ and $u \in P_n$, $u^{-} \cup u^{++} = u^{-} \cup u^{+-}$;

(C2) for $n \in \mathbb{N}^*$ and $u \in P_n$, $u^{-}$ and $u^{+}$ are fork-free;

(C3) $P$ is acyclic;

(C4) for $n \in \mathbb{N}^*$, $u, v \in P_n$, $w \in P_{n+1}$, if $u \triangleleft v$, $u \in w^{\epsilon}$ and $v \in w^{\eta}$ for some $\epsilon, \eta \in \{-, +\}$, then $\epsilon = \eta$;

(C5) for $i, n \in \mathbb{N}$ with $i < n$ and $u \in P_n$, $\langle u \rangle_{i,-}$ is tight.

**Axiom (C0)** ensures that each generator has defined source and target. **Axiom (C1)** enforces basic globular properties on generators. For example, it forbids the $\omega$-hypergraph

$$ w \underset{b}{\overset{a}{\rightleftharpoons}} x \underset{b}{\overset{\downarrow}{\rightleftharpoons}} y \rightarrow z \qquad (7) $$

since $\alpha^{-} \cup \alpha^{++} = \{w, y\}$ and $\alpha^{+-} \cup \alpha^{-+} = \{x, z\}$. **Axiom (C2)** forbids generators with parallel elements in their sources or targets. For example, it forbids the $\omega$-hypergraph

$$ \begin{array}{ccc} x & \xrightarrow{a} & z \\ y & \longrightarrow & \end{array} \qquad (8) $$
**Atoms and relevance** Let $P$ be an $\omega$-hypergraph. Given $n \in \mathbb{N}$ and $u \in P_n$, we define sets $\langle u \rangle_{i,\epsilon} \subseteq P_i$ for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$ with a downward induction by

$$ \langle u \rangle_{n,-} = \langle u \rangle_{n,+} = \{u\} $$

and

$$ \langle u \rangle_{j,-} = \langle u \rangle_{j+1,-}^{\mp} \quad \langle u \rangle_{j,+} = \langle u \rangle_{j+1,+}^{\pm} $$

for $j \in \mathbb{N}_{n-1}$. We often write $\langle u \rangle_n$ for both $\langle u \rangle_{n,-}$ and $\langle u \rangle_{n,+}$. The atom associated to $u$ is then the $n$-pre-cell of $P$

$$ \langle u \rangle = (\langle u \rangle_{0,-}, \langle u \rangle_{0,+}, \dots, \langle u \rangle_{n-1,-}, \langle u \rangle_{n-1,+}, \langle u \rangle_n). $$

A generator $u$ is said relevant when the atom $\langle u \rangle$ is a cell. When $P$ is a parity complex, the relevant generators of $P$ will have the role of generating cells in the $\omega$-category $\mathrm{Cell}(P)$.

**Example 1.4.3.** The atom associated to $\alpha$ in (4) is $\langle \alpha \rangle$ with

$$ \begin{align*} \langle \alpha \rangle_{0,-} &= \{u\}, & \langle \alpha \rangle_{1,-} &= \{a, c\}, & \langle \alpha \rangle_{2} &= \{\alpha\}, \\ \langle \alpha \rangle_{0,+} &= \{z\}, & \langle \alpha \rangle_{1,+} &= \{b, d\}, \end{align*} $$

and, since it is a cell, $\alpha$ is relevant.

**Tightness** Some defects were found in the first definition of parity complexes given in [23], so that Street fixed his definition in [24]. His correction involves the notion of tightness defined as follows. Given $n \in \mathbb{N}$, a subset $U \subseteq P_n$ is said to be tight when, for all $u, v \in P_n$ such that $u \triangleleft v$ and $v \in U$, we have $u^{-} \cap U^{\pm} = \emptyset$.

**Example 1.4.4.** In (6), $U = \{\beta, \gamma\}$ is not tight since $\alpha \triangleleft \gamma$ and $c'' \in \alpha^{-} \cap U^{\pm}$. However, the set $U' = \{\alpha, \beta, \gamma, \delta\}$ is tight.

**Parity complexes** A parity complex [23, 24] is an $\omega$-hypergraph $P$ satisfying the axioms (C0) to (C5) below:

(C0) for $n \in \mathbb{N}^*$ and $u \in P_n$, $u^{-} \neq \emptyset$ and $u^{+} \neq \emptyset$;

(C1) for $n \in \mathbb{N}$ with $n \ge 2$ and $u \in P_n$, $u^{-} \cup u^{++} = u^{-} \cup u^{+-}$;

(C2) for $n \in \mathbb{N}^*$ and $u \in P_n$, $u^{-}$ and $u^{+}$ are fork-free;

(C3) $P$ is acyclic;

(C4) for $n \in \mathbb{N}^*$, $u, v \in P_n$, $w \in P_{n+1}$, if $u \triangleleft v$, $u \in w^{\epsilon}$ and $v \in w^{\eta}$ for some $\epsilon, \eta \in \{-, +\}$, then $\epsilon = \eta$;

(C5) for $i, n \in \mathbb{N}$ with $i < n$ and $u \in P_n$, $\langle u \rangle_{i,-}$ is tight.

**Axiom (C0)** ensures that each generator has defined source and target. **Axiom (C1)** enforces basic globular properties on generators. For example, it forbids the $\omega$-hypergraph

$$ w \underset{b}{\overset{a}{\rightleftharpoons}} x \underset{b}{\overset{\downarrow}{\rightleftharpoons}} y \rightarrow z \qquad (7) $$

since $\alpha^{-} \cup \alpha^{++} = \{w, y\}$ and $\alpha^{+-} \cup \alpha^{-+} = \{x, z\}$. **Axiom (C2)** forbids generators with parallel elements in their sources or targets. For example, it forbids the $\omega$-hypergraph

$$ x \xrightarrow{a} y \qquad (8) $$
since $a^{-} = \{x, y\}$ is not fork-free. Axiom (C3) forbids $\omega$-hypergraphs with some loops like (5). Axiom (C4) can be informally described as forbidding “bridges”. For instance, the $\omega$-hypergraph

$$ \begin{tikzcd}[column sep=2.8em, row sep=2.8em]
    & a \arrow[r, "a"] \arrow[d, "b"] & y \arrow[d, "z"] \\
x & \alpha \arrow[r, "\alpha"] \arrow[d, "a'"'] & \beta \arrow[r, "\beta"] \\
y' & b \arrow[u, "b'"'] \arrow[r, "b"] & c
\end{tikzcd} \quad (9) $$

does not satisfy Axiom (C4). Indeed, $a \triangleleft c \triangleleft b'$ and $a \in \alpha^{-}$ and $b' \in \alpha^{+}$. Axiom (C5) prevents more subtle problems, like the one exposed by the $\omega$-hypergraph (14) (even though the latter does not satisfy Axiom (C3) in the first place). It entails that the sources and targets of each generator are segments (as defined in Section 1.3), which is a condition that we will motivate in Section 1.7 when discussing Axiom (T3) of torsion-free complexes.

**A counter-example to the freeness property** Given a parity complex $P$, the main result claimed in [23] is that the globular set $\text{Cell}(P)$ together with the source, target, identity and composition operations, has the structure of an $\omega$-category, which is freely generated by the atoms $\langle u \rangle$ for $u \in P$ ([23, Theorem 4.2]). More precisely, using the terminology of Section 2.3, this result states that there is an $\omega$-polygraph $Q$ and an $\omega$-functor $F: Q^* \to \text{Cell}(P)$ such that

$$ - Q_k = P_k \text{ for } k \in \mathbb{N}, $$

$$ - F(u) = \langle u \rangle \text{ for } u \in Q, $$

$$ - F \text{ is an isomorphism.} $$

Since the cells of a free $\omega$-category on a polygraph are equivalent classes of formal composites of generators (c.f. [17, Lemma 4.1]), this property intuitively says that the cells of parity complexes adequately represent pasting diagrams, i.e., diagrams associated with a unique class of equivalent formal composites of generators. However, this property does not hold as we illustrate with a counter-example. Consider the $\omega$-hypergraph $P$ defined by the diagram given by

$$ \begin{tikzcd}[column sep=2.8em, row sep=2.8em]
    & \alpha \arrow[r, "\alpha"] \arrow[d, "\alpha'"'] & y \arrow[d, "z"] \\
x & b \arrow[r, "b"] \arrow[d, "\delta"'] & \beta \arrow[r, "\beta"] \\
y' & \gamma \arrow[r, "\gamma"] \arrow[d, "\delta'"'] & z
\end{tikzcd} \quad (10) $$

together with two 3-generators

$$ \begin{tikzcd}[column sep=2.8em, row sep=2.8em]
    & \alpha \arrow[r, "\alpha"] \arrow[d, "\alpha'"'] & y \arrow[d, "z"] \\
x & b \arrow[r, "b"] \arrow[d, "\delta"'] & \beta \arrow[r, "\beta"] \\
y' & \gamma \arrow[r, "\gamma"] \arrow[d, "\delta'"'] & z
\end{tikzcd} \quad \begin{array}{l} A \Rightarrow \alpha \arrow[r, "\alpha"] \arrow[d, "\alpha'"'] & y \arrow[d, "z"] \\ x & b \arrow[r, "b"] \arrow[d, "\delta"'] & z \\ y' & \gamma \arrow[r, "\gamma"] \arrow[d, "\delta'"'] & z \end{array} \\
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
    & \beta \arrow[r, "\beta"] \arrow[d, "\beta'"'] & z \\
x & b \arrow[r, "b"] \arrow[d, "\delta"'] & \gamma \arrow[r, "\gamma"] \\
y' & \beta \arrow[r, "\beta"] \arrow[d, "\beta'"'] & z
\end{tikzcd} \quad \begin{array}{l} B \Rightarrow \beta \arrow[r, "\beta"] \arrow[d, "\beta'"'] & z \\ x & b \arrow[r, "b"] \arrow[d, "\delta"'] & \gamma \arrow[r, "\gamma"] \\ y' & \beta \arrow[r, "\beta"] \arrow[d, "\beta'"'] & z \end{array} $$

By carefully checking Axioms (C0) to (C5), it can be shown that $P$ is a parity complex. The diagram (10) moreover defines a polygraph $Q$, whose induced $\omega$-category $Q^*$ is supposed to be
isomorphic to Cell($P$), as a consequence of [23, Theorem 4.2], but it is not the case here. Indeed, we can find two expressions that compose together the 3-generators $A$ and $B$ in $Q^*$, inducing two 3-cells $H_1$ and $H_2$ with

$$H_1 = ((a*_0 \gamma) *_1 A *_1 (\beta*_0 f)) *_2 ((\alpha'*_0 d) *_1 B *_1 (c*_0 \delta')) \\ H_2 = ((\alpha*_0 d) *_1 B *_1 (c*_0 \delta)) *_2 ((a*_0 \gamma') *_1 A *_1 (\beta'*_0 f))$$

where, for the sake of readability, we omitted the operations id required to lift the generators to dimension 3. The canonical morphism $F: Q^* \to \text{Cell}(P)$ maps $H_1$ and $H_2$ to the same 3-cell $X$ defined by:

$$X_3 = \{A, B\},$$

$$\begin{align*}
X_{2,-} &= \{\alpha, \beta, \gamma, \delta\}, & X_{2,+} &= \{\alpha', \beta', \gamma', \delta'\}, \\
X_{1,-} &= \{a, d\}, & X_{1,+} &= \{c, f\}, \\
X_{0,-} &= \{x\}, & X_{0,-} &= \{z\}.
\end{align*}$$

However, $H_1$ and $H_2$ are different cells in $Q^*$. This has first been proved using Agda as a proof assistant [8], but can be proved more quickly using a solver for the word problem on strict categories like **cateq** [6]. Hence, the distinct cells $H_1$ and $H_2$ of $Q^*$ are sent to the same cell of Cell($P$) by $F$, since the information that makes $H_1$ and $H_2$ different is the order in which $A$ and $B$ are composed, which can not be expressed by a cell of a parity complex. This refutes [23, Theorem 4.2] which asserts that $F$ is an isomorphism. Thus, parity complexes do not necessarily induce free $\omega$-categories in general.

**1.5 Pasting schemes** Johnson's loop-free pasting schemes [13] is another proposed formalism for pasting diagrams. Like parity complexes, they are based on $\omega$-hypergraphs, but the cells will now be represented as single subsets of generators instead of tuples like for parity complexes. This formalism relies on set relations, namely $B$ and $E$, to encode which generators to remove in order to obtain respectively the target and the source of a cell.

**Conventions for relations** A relation between two sets $X$ and $Y$ is a subset $L \subseteq X \times Y$. For $(x, y) \in X \times Y$, we write $xLy$ when $(x, y) \in L$. The identity relation on a set $X$ is the relation $L \subseteq X \times X$ such that $xLy$ if and only if $x = y$. Given a relation $L$ between $X$ and $Y$, and $x \in X$, we write $L(x)$ for the set $L(x) = \{y \in Y \mid xLy\}$. Given a subset $X' \subseteq X$, we denote by $L(X')$ the set $\{y \in Y \mid \exists x \in X', xLy\}$. The relation $L$ is said *finitary* when, for all $x \in X$, $L(x)$ is a finite set. If $L$ is a relation on a graded set $P = \cup_{n \in \mathbb{N}} P_n$, given $k, l \in \mathbb{N}$, we write $L_k^l$ for the relation between $P_l$ and $P_k$ defined as $L \cap (P_l \times P_k)$. Similarly, we write $L^l$ for the relation between $P_l$ and $P$ defined as $L \cap (P_l \times P)$. Given relations $L$ between $X$ and $Y$ and $L'$ between $Y$ and $Z$, we write $LL'$ for the relation between $X$ and $Z$ which is the composite relation defined as

$$LL' = \{(x, z) \in X \times Z \mid \exists y \in Y, xLy \text{ and } yL'z\}.$$

**Pre-pasting schemes** A pre-pasting scheme $(P, B, E)$ is given by a graded set $P$ and two relations $B, E$ (for “beginning” and “end”) on $P$ such that

(i) $B$ and $E$ are finitary,

(ii) for $k, l \in \mathbb{N}$ with $l < k$, $B_k^l = E_k^l = \emptyset$,

(iii) $B_k^k$ (resp. $E_k^k$) is the identity relation on $P_k$,
isomorphic to Cell($P$), as a consequence of [23, Theorem 4.2], but it is not the case here. Indeed, we can find two expressions that compose together the 3-generators $A$ and $B$ in $Q^*$, inducing two 3-cells $H_1$ and $H_2$ with

$$H_1 = ((a*_0 \gamma)*_1 A*_1 (\beta*_0 f))*_2 ((\alpha'*_0 d)*_1 B*_1 (c*_0 \delta'))$$ $$H_2 = ((\alpha*_0 d)*_1 B*_1 (c*_0 \delta))*_2 ((a*_0 \gamma')*_1 A*_1 (\beta'*_0 f))$$

where, for the sake of readability, we omitted the operations id required to lift the generators to dimension 3. The canonical morphism $F: Q^* \to \text{Cell}(P)$ maps $H_1$ and $H_2$ to the same 3-cell $X$ defined by:

$$X_3 = \{A, B\},$$

$$X_{2,-} = \{\alpha, \beta, \gamma, \delta\}, \quad X_{2,+} = \{\alpha', \beta', \gamma', \delta'\},$$ $$X_{1,-} = \{a, d\}, \quad X_{1,+} = \{c, f\},$$ $$X_{0,-} = \{x\}, \quad X_{0,+} = \{z\}.$$

However, $H_1$ and $H_2$ are different cells in $Q^*$. This has first been proved using Agda as a proof assistant [8], but can be proved more quickly using a solver for the word problem on strict categories like `cateq` [6]. Hence, the distinct cells $H_1$ and $H_2$ of $Q^*$ are sent to the same cell of Cell($P$) by $F$, since the information that makes $H_1$ and $H_2$ different is the order in which $A$ and $B$ are composed, which can not be expressed by a cell of a parity complex. This refutes [23, Theorem 4.2] which asserts that $F$ is an isomorphism. Thus, parity complexes do not necessarily induce free $\omega$-categories in general.

**1.5 Pasting schemes** Johnson's loop-free pasting schemes [13] is another proposed formalism for pasting diagrams. Like parity complexes, they are based on $\omega$-hypergraphs, but the cells will now be represented as single subsets of generators instead of tuples like for parity complexes. This formalism relies on set relations, namely $B$ and $E$, to encode which generators to remove in order to obtain respectively the target and the source of a cell.

**Conventions for relations** A relation between two sets $X$ and $Y$ is a subset $L \subseteq X \times Y$. For $(x, y) \in X \times Y$, we write $xL_y$ when $(x, y) \in L$. The identity relation on a set $X$ is the relation $L \subseteq X \times X$ such that $xL_y$ if and only if $x = y$. Given a relation $L$ between $X$ and $Y$, and $x \in X$, we write $L(x)$ for the set $L(x) = \{y \in Y \mid xLy\}$. Given a subset $X' \subseteq X$, we denote by $L(X')$ the set $\{y \in Y \mid \exists x \in X', xLy\}$. The relation $L$ is said *finitary* when, for all $x \in X$, $L(x)$ is a finite set. If $L$ is a relation on a graded set $P = \cup_{n \in \mathbb{N}} P_n$, given $k, l \in \mathbb{N}$, we write $L_k^l$ for the relation between $P_l$ and $P_k$ defined as $L \cap (P_l \times P_k)$. Similarly, we write $L^l$ for the relation between $P_l$ and $P$ defined as $L \cap (P_l \times P)$. Given relations $L$ between $X$ and $Y$ and $L'$ between $Y$ and $Z$, we write $LL'$ for the relation between $X$ and $Z$ which is the composite relation defined as

$$LL' = \{(x, z) \in X \times Z \mid \exists y \in Y, xLy \text{ and } yL'z\}.$$

**Pre-pasting schemes** A pre-pasting scheme $(P, B, E)$ is given by a graded set $P$ and two relations $B, E$ (for “beginning” and “end”) on $P$ such that

(i) $B$ and $E$ are finitary,

(ii) for $k, l \in \mathbb{N}$ with $l < k$, $B_k^l = E_k^l = \emptyset$,

(iii) $B_k^k$ (resp. $E_k^k$) is the identity relation on $P_k$,
(iv) for $k, l \in \mathbb{N}$ with $k < l$, $L \in \{\mathbf{B}, \mathbf{E}\}$, $u \in P_{l+1}$ and $v \in P_k$, $u L_k^{l+1} v$ if and only if
$$u L_l^{l+1} \mathbf{B}_k^l v \quad \text{and} \quad u L_l^{l+1} \mathbf{E}_k^l v.$$

**Example 1.5.1.** The diagram (4) can be encoded as a pre-pasting scheme

$$
\begin{align*}
\mathbf{B}_1^2(\alpha) &= \{a, c\}, & \mathbf{B}_0^2(\alpha) &= \{y\}, & \mathbf{B}_0^1(\alpha) &= \{x\}, \\
\mathbf{E}_1^2(\alpha) &= \{b, d\}, & \mathbf{E}_0^2(\alpha) &= \{y'\}, & \mathbf{E}_0^1(\alpha) &= \{y\} \dots
\end{align*}
$$

Note that the relations B and E of a pre-pasting scheme P are completely determined by the data of $\mathbf{B}_k^{k+1}(u)$ and $\mathbf{E}_k^{k+1}(u)$ for $k \in \mathbb{N}$ and $u \in P_k$. As a consequence, the data of a pre-pasting scheme structure on P is equivalent to the data of an $\omega$-hypergraph structure on P: the correspondence is given by $u^{-} = \mathbf{B}_k^{k+1}(u)$ and $u^{+} = \mathbf{E}_k^{k+1}(u)$ for $k \in \mathbb{N}^*$ and $u \in P_{k+1}$. In particular, the relation $\triangleleft$ on a pasting scheme is defined as the one on the associated $\omega$-hypergraph.

**Direct loops** Given an $\omega$-hypergraph $P$, $P$ has a direct loop when

(i) either there exist $n \in \mathbb{N}^*$ and $u, v \in P_n$ such that $u \triangleleft v$ and $E(v) \cap B(u) \neq \emptyset$,

(ii) or there exists $w \in P$ such that $E(w) \cap B(w) \neq \{w\}$.

**Example 1.5.2.** The $\omega$-hypergraph

$$ \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.4, every node/.style={scale=0.8}] 
  \node (x1) at (0,0) {$y$};
  \node (x2) at (1,0) {$a_2$};
  \node (y1) at (0,1) {$\alpha$};
  \node (y2) at (1,1) {$\beta$};
  \node (z) at (0.5,0.5) {$z$};
  \node (a1) at (0.2,0.2) {$a_1$};
  \node (b1) at (0.2,0.8) {$b$};
  \node (c1) at (0.2,1.2) {$c_1$};
  \node (a2) at (0.8,0.2) {$a_2$};
  \node (b2) at (0.8,0.8) {$b$};
  \node (c2) at (0.8,1.2) {$c_2$};
  \node (y) at (0.5,0.5) {$y$};
  \node (x) at (0.5,1.5) {$x$};
  \draw[->] (x1) -- (y1);
  \draw[->] (x2) -- (y2);
  \draw[->] (a1) -- (b1);
  \draw[->] (b1) -- (c1);
  \draw[->] (a2) -- (b2);
  \draw[->] (b2) -- (c2);
  \draw[->] (x1) -- (a1);
  \draw[->] (y1) -- (a2);
  \draw[->] (x2) -- (b2);
  \draw[->] (y2) -- (b1);
  \draw[->] (x) -- (y);
  \draw[->] (y) -- (x);
\end{tikzpicture} \quad (11) $$

has a direct loop by the first criterion, because $\alpha \triangleleft \beta$ and $y \in B(\alpha) \cap E(\beta)$. Examples of direct loops by the second condition are given by the $\omega$-hypergraphs

$$ P^1 = \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.4, every node/.style={scale=0.8}] 
  \node (v) at (0,0) {$v$};
  \node (u) at (0,1) {$w$};
  \node (a) at (0.5,0.5) {$\alpha$};
  \node (b) at (0.5,1.5) {$\beta$};
  \draw[->] (v) -- (a);
  \draw[->] (v) -- (b);
  \draw[->] (u) -- (a);
  \draw[->] (u) -- (b);
  \node (x) at (0.5,0.5) {$x$};
  \node (y) at (0.5,1.5) {$y$};
  \draw[->] (x) -- (y);
  \draw[->] (x) -- (b);
  \draw[->] (y) -- (a);
  \draw[->] (y) -- (b);
\end{tikzpicture} \quad \text{and} \quad P^2 = \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.4, every node/.style={scale=0.8}] 
  \node (x) at (0,0) {$x$};
  \node (y) at (0,1) {$z$};
  \node (b) at (0.5,0.5) {$\beta$};
  \node (c) at (0.5,1.5) {$\alpha$};
  \draw[->] (x) -- (b);
  \draw[->] (x) -- (c);
  \draw[->] (y) -- (b);
  \draw[->] (y) -- (c);
  \node (a) at (0.5,0.5) {$a$};
  \node (b1) at (0.5,1.5) {$b$};
  \node (c1) at (0.5,0.5) {$c$};
  \node (a1) at (0.5,1.5) {$a_1$};
  \node (b2) at (0.5,0.5) {$b_2$};
  \node (c2) at (0.5,1.5) {$c_2$};
  \node (y) at (0.5,0.5) {$y$};
  \node (x) at (0.5,1.5) {$x$};
  \draw[->] (a1) -- (b1);
  \draw[->] (b2) -- (c2);
  \draw[->] (a2) -- (b2);
  \draw[->] (b1) -- (c1);
  \draw[->] (c2) -- (a2);
  \draw[->] (c1) -- (a1);
\end{tikzpicture} \quad (12) $$

**Finite graded subsets** Let $P$ be a pre-pasting scheme. We define the relation $\mathbb{R} \subseteq P \times P$ as the smallest reflexive transitive relation on $P$ such that, for all $k \in \mathbb{N}$ and $x \in P_{k+1}$, we have

$$ \mathbf{B}(x) \cup \mathbf{E}(x) \subseteq \mathbb{R}(x). $$

**Example 1.5.3.** In the case of the $\omega$-hypergraph (11), we have

$$ \mathbf{R}(\alpha) = \{x, y, z, a_1, a_2, b, \alpha\} \quad \text{and} \quad \mathbf{R}(\beta) = \{x, y, z, b, c_1, c_2, \beta\}. $$

A finite graded subset of dimension $n$ of $P$ (abbreviated $n$-fgs) is an $(n+1)$-tuple

$$ X = (X_0, \dots, X_n) $$

such that $X_k \subseteq P_k$ and $X_k$ is finite for $k \in \mathbb{N}_n$. We often identify the $n$-fgs $X$ with the set $\cup_{k \in \mathbb{N}_n} X_k$, but one should keep in mind that the $n$-fgs $X$ and the $(n+1)$-fgs $(X_0, \dots, X_n, \emptyset)$ are two different objects. We say that $X$ is *closed* when $\mathbb{R}(X) = X$. Given $n \in \mathbb{N}$ and an $(n+1)$-fgs $X$ of $P$, we define the *source* and the *target* of $X$ as the $n$-fgs's $\partial_n^-(X)$ and $\partial_n^+(X)$ of $P$ such that

$$ \partial_n^-(X) = X \setminus \mathbf{E}^n(X) \quad \text{and} \quad \partial_n^+(X) = X \setminus \mathbf{B}^n(Y). $$
**Example 1.5.4.** Considering the $\omega$-hypergraph (11), we have

$$ \partial_n^-(R(\alpha)) = R(\alpha) \setminus \{b, \alpha\} = \{x, y, z, a_1, a_2\} \quad \text{and} \quad \partial_n^+(R(\alpha)) = R(\alpha) \setminus \{y, a_1, a_2\} = \{x, z, b\}. $$

**Remark 1.5.5.** The fgs's of the form $R(u)$ for $u \in P$ are the analogue of the atoms defined for parity complexes.

**Well-formed sets** Given a pre-pasting scheme $P$, we define by induction on $n$ the notion of well-formed $n$-fgs (abbreviated $n$-wfs): given $n \in \mathbb{N}$, an $n$-fgs $X$ of $P$ is well-formed when

(i) $X$ is closed,

(ii) $X_n$ is fork-free,

(iii) when $n > 0$, $\partial_n^-(X)$ and $\partial_n^+(X)$ are well-formed $(n-1)$-fgs.

We denote by $WF(P)$ the graded set of $n$-wfs's of $P$ for $n \in \mathbb{N}$. By [13, Theorem 3], for $n \in \mathbb{N}$, the operations $\partial_n^-$ and $\partial_n^+$ on $(n+1)$-fgs's restrict to functions

$$ \partial_n^-, WF(P)_{n+1} \to WF(P)_n \quad \text{and} \quad \partial_n^+, WF(P)_{n+1} \to WF(P)_n $$

and they equip $WF(P)$ with a structure of $\omega$-globular set. In the following, the wfs's will be the "cells" of the pasting diagram formalism of pasting schemes.

**Example 1.5.6.** The pre-pasting scheme

$$ x \xrightarrow[c_1]{a_1} \xrightarrow{\downarrow \alpha} y_1 \xrightarrow{b} c_2 \xrightarrow{\downarrow \beta} y_2 \qquad (13) $$

has, among others, the 0-wfs's $\{x\}$ and $\{z\}$, the 1-wfs's $\{x, y_1, z, a_1, a_2\}$ and $\{x, y_2, z, c_1, c_2\}$, and the 2-wfs $\{x, y_1, y_2, z, a_1, a_2, b, c_1, c_2, \alpha, \beta\}$.

**Identity and composition operations** Let $P$ be a pre-pasting scheme. Given $n \in \mathbb{N}$ and an $n$-wfs $X = (X_0, \dots, X_n)$ of $P$, the identity of $X$ is the $(n+1)$-wfs $\mathrm{id}^{n+1}(X)$ defined by

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

Given $i, n \in \mathbb{N}$ with $i < n$ and $X, Y$ two $n$-wfs such that $\partial_i^+(X) = \partial_i^-(Y)$, the $i$-composition of $X$ and $Y$ is the $n$-fgs $X *_i Y$ such that

$$ X *_i Y = X \cup Y. $$

Under the conditions of a pre-pasting scheme, it is not necessarily the case that the composite of two $n$-wfs's is an $n$-wfs, but it will under the axioms of a pasting scheme introduced below.

**Loop-free pasting schemes** A pasting scheme [13] is a pre-pasting scheme $P$ satisfying the following two axioms:

(S0) for $k \in \mathbb{N}$ and $u \in P_{k+1}$, $B_k^{k+1}(u) \neq \emptyset$ and $E_k^{k+1}(u) \neq \emptyset$;

(S1) for $k, l \in \mathbb{N}$ with $k \le l$, $L \in \{\mathrm{B}, \mathrm{E}\}$, $u \in P_{l+1}$ and $v \in P_k$,

- if $u E_l^{l+1} L_k^l v$ then $u E_k^{l+1} v$ or $u B_k^{l+1} L_k^l v$,

- if $u B_l^{l+1} L_k^l v$ then $u B_k^{l+1} v$ or $u E_l^{l+1} L_k^l v$.
**Example 1.5.4.** Considering the $\omega$-hypergraph (11), we have

$$ \partial_n^-(R(\alpha)) = R(\alpha) \setminus \{b, \alpha\} = \{x, y, z, a_1, a_2\} \quad \text{and} \quad \partial_n^+(R(\alpha)) = R(\alpha) \setminus \{y, a_1, a_2\} = \{x, z, b\}. $$

**Remark 1.5.5.** The fgs's of the form $R(u)$ for $u \in P$ are the analogue of the atoms defined for parity complexes.

**Well-formed sets** Given a pre-pasting scheme $P$, we define by induction on $n$ the notion of well-formed $n$-fgs (abbreviated $n$-wfs): given $n \in \mathbb{N}$, an $n$-fgs $X$ of $P$ is well-formed when

(i) $X$ is closed,

(ii) $X_n$ is fork-free,

(iii) when $n > 0$, $\partial_n^-(X)$ and $\partial_n^+(X)$ are well-formed $(n-1)$-fgs.

We denote by $WF(P)$ the graded set of $n$-wfs's of $P$ for $n \in \mathbb{N}$. By [13, Theorem 3], for $n \in \mathbb{N}$, the operations $\partial_n^-$ and $\partial_n^+$ on $(n+1)$-fgs's restrict to functions

$$ \partial_n^-: WF(P)_{n+1} \to WF(P)_n \quad \text{and} \quad \partial_n^+: WF(P)_{n+1} \to WF(P)_n $$

and they equip $WF(P)$ with a structure of $\omega$-globular set. In the following, the wfs's will be the "cells" of the pasting diagram formalism of pasting schemes.

**Example 1.5.6.** The pre-pasting scheme

$$ \xrightarrow{x} \xrightarrow{\substack{y_1 \\ \downarrow \alpha \\ c_1}} \xrightarrow{\substack{b \\ \downarrow \beta \\ y_2}} \xrightarrow{z} \quad (13) $$

has, among others, the 0-wfs's $\{x\}$ and $\{z\}$, the 1-wfs's $\{x, y_1, z, a_1, a_2\}$ and $\{x, y_2, z, c_1, c_2\}$, and the 2-wfs $\{x, y_1, y_2, z, a_1, a_2, b, c_1, c_2, \alpha, \beta\}$.

**Identity and composition operations** Let $P$ be a pre-pasting scheme. Given $n \in \mathbb{N}$ and an $n$-wfs $X = (X_0, \dots, X_n)$ of $P$, the identity of $X$ is the $(n+1)$-wfs $\mathrm{id}^{n+1}(X)$ defined by

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

Given $i, n \in \mathbb{N}$ with $i < n$ and $X, Y$ two $n$-wfs such that $\partial_i^+(X) = \partial_i^-(Y)$, the $i$-composition of $X$ and $Y$ is the $n$-fgs $X *_i Y$ such that

$$ X *_i Y = X \cup Y. $$

Under the conditions of a pre-pasting scheme, it is not necessarily the case that the composite of two $n$-wfs's is an $n$-wfs, but it will under the axioms of a pasting scheme introduced below.

**Loop-free pasting schemes** A pasting scheme [13] is a pre-pasting scheme $P$ satisfying the following two axioms:

(S0) for $k \in \mathbb{N}$ and $u \in P_{k+1}$, $B_k^{k+1}(u) \neq \emptyset$ and $E_k^{k+1}(u) \neq \emptyset$;

(S1) for $k, l \in \mathbb{N}$ with $k \le l$, $L \in \{\mathrm{B}, \mathrm{E}\}$, $u \in P_{l+1}$ and $v \in P_k$,

- if $u E_l^{l+1} L_k^l v$ then $u E_k^{l+1} v$ or $u B_k^{l+1} L_k^l v$,

- if $u B_l^{l+1} L_k^l v$ then $u B_k^{l+1} v$ or $u E_l^{l+1} L_k^l v$.
The pasting scheme $P$ is a loop-free pasting scheme when it moreover satisfies the following:

(S2) $P$ has no direct loops;

(S3) for $u \in P$, $R(u) \in WF(P)$;

(S4) for $k, n \in \mathbb{N}$ with $k < n$, $X \in WF(P)_k$ and $u \in P_n$,

- if $\partial_k^-(R(u)) \subseteq X$, then $\langle u \rangle_{k,-}$ is a segment for $\langle X \rangle_{k,}$,

- if $\partial_k^+(R(u)) \subseteq X$, then $\langle u \rangle_{k,+}$ is a segment for $\langle X \rangle_{k,}$;

(S5) for $n \in \mathbb{N}$, $X \in WF(P)_n$ and $u \in P_{n+1}$ with $\partial_n^-(R(u)) \subseteq X$, the following hold:

(a) $X \cap E(u) = \emptyset$,

(b) for $y \in X$, if $B(u) \cap R(y) \neq \emptyset$, then $y \in B(u)$.

**Axiom (S1)** enforces basic globular properties on generators and forbids, the $\omega$-hypergraph (7) for example. **Axiom (S2)** forbids $\omega$-hypergraphs with loops like (5), (11) and (12). **Axiom (S3)** enforces fork-freeness on the iterated sources and targets of a generator (for example, it forbids the $\omega$-hypergraph (8)). **Axiom (S4)** relates to **Axiom (C5)** of parity complexes and prevent situations in the spirit of the $\omega$-hypergraph (14) (even though the latter does not satisfy **Axiom (S2)** in the first place). We motivate this axiom in Section 1.7 when we discuss a similar axiom for torsion-free complexes. **Axiom (S5)** can be deduced from the other axioms (c.f. [12, Theorem 3.7]) but it simplifies the proofs of [13]. An example of a sensible pre-pasting scheme that satisfy **Axioms (S0)** to (S3), but neither **Axiom (S4)** nor **Axiom (S5)**, exists in dimension four (see [19, Example 3.11]).

**A counter-example to the freeness property** The main result claimed in [13] is similar to the one of [23]: given a loop-free pasting scheme $P$, the globular set $WF(P)$ together with the source, target, identity and composition operations has the structure of an $\omega$-category, which is freely generated by the wfs's $R(u)$ for $u \in P$ ([13, Theorem 13]). Using the terminology of Section 2.3, this amounts to say that there exist an $\omega$-polygraph $Q$ and an $\omega$-functor $F: Q^* \to WF(P)$ such that

- $Q_k = P_k$ for $k \in \mathbb{N}$,

- $F(u) = R(u)$ for $u \in Q$,

- $F$ is an isomorphism.

But the same flaw as for parity complexes is present here too, which makes the freeness result wrong. In fact, the counter-example to the freeness property of parity complexes, introduced in Section 1.4, is also a counter-example to the freeness property of pasting schemes: the $\omega$-hypergraph $P$ is a loop-free pasting scheme and the canonical morphism $F: Q^* \to WF(P)$ sends $H_1$ and $H_2$ to the same 3-wfs $X = \{x, y, z, \alpha, \beta, \gamma, \delta, \alpha', \beta', \gamma', \delta', A, B\}$ refuting the freeness property [13, Theorem 13].

**1.6 Augmented directed complexes** Augmented directed complexes, designed by Steiner in [21], are not directly based on $\omega$-hypergraphs, but on chain complexes. Under the conditions required by Steiner, it happens that the data of a chain complex is equivalent to the data of an $\omega$-hypergraph. The definition of cells for this formalism strongly resembles the one of parity complexes. The only difference is that the cells are tuples of group elements instead of subsets of an $\omega$-hypergraph.

**Augmented directed complex** A pre-augmented directed complexes $(K, \mathrm{d}, \mathrm{e})$ (abbreviated pre-adc) is the data of
The pasting scheme $P$ is a loop-free pasting scheme when it moreover satisfies the following:

(S2) $P$ has no direct loops;

(S3) for $u \in P$, $R(u) \in WF(P)$;

(S4) for $k, n \in \mathbb{N}$ with $k < n$, $X \in WF(P)_k$ and $u \in P_n$,

- if $\partial_k^-(R(u)) \subseteq X$, then $\langle u \rangle_{k,-}$ is a segment for $\langle X_{k,-} \rangle$,

- if $\partial_k^+(R(u)) \subseteq X$, then $\langle u \rangle_{k,+}$ is a segment for $\langle X_{k,+} \rangle$;

(S5) for $n \in \mathbb{N}$, $X \in WF(P)_n$ and $u \in P_{n+1}$ with $\partial_n^-(R(u)) \subseteq X$, the following hold:

(a) $X \cap E(u) = \emptyset$,

(b) for $y \in X$, if $B(u) \cap R(y) \neq \emptyset$, then $y \in B(u)$.

**Axiom (S1)** enforces basic globular properties on generators and forbids, the $\omega$-hypergraph (7) for example. **Axiom (S2)** forbids $\omega$-hypergraphs with loops like (5), (11) and (12). **Axiom (S3)** enforces fork-freeness on the iterated sources and targets of a generator (for example, it forbids the $\omega$-hypergraph (8)). **Axiom (S4)** relates to **Axiom (C5)** of parity complexes and prevent situations in the spirit of the $\omega$-hypergraph (14) (even though the latter does not satisfy **Axiom (S2)** in the first place). We motivate this axiom in Section 1.7 when we discuss a similar axiom for torsion-free complexes. **Axiom (S5)** can be deduced from the other axioms (c.f. [12, Theorem 3.7]) but it simplifies the proofs of [13]. An example of a sensible pre-pasting scheme that satisfy **Axioms (S0)** to (S3), but neither **Axiom (S4)** nor **Axiom (S5)**, exists in dimension four (see [19, Example 3.11]).

**A counter-example to the freeness property** The main result claimed in [13] is similar to the one of [23]: given a loop-free pasting scheme $P$, the globular set $WF(P)$ together with the source, target, identity and composition operations has the structure of an $\omega$-category, which is freely generated by the wfs's $R(u)$ for $u \in P$ ([13, Theorem 13]). Using the terminology of Section 2.3, this amounts to say that there exist an $\omega$-polygraph $Q$ and an $\omega$-functor $F: Q^* \to WF(P)$ such that

- $Q_k = P_k$ for $k \in \mathbb{N}$,

- $F(u) = R(u)$ for $u \in Q$,

- $F$ is an isomorphism.

But the same flaw as for parity complexes is present here too, which makes the freeness result wrong. In fact, the counter-example to the freeness property of parity complexes, introduced in Section 1.4, is also a counter-example to the freeness property of pasting schemes: the $\omega$-hypergraph $P$ is a loop-free pasting scheme and the canonical morphism $F: Q^* \to WF(P)$ sends $H_1$ and $H_2$ to the same 3-wfs $X = \{x, y, z, \alpha, \beta, \gamma, \delta, \alpha', \beta', \gamma', \delta', A, B\}$ refuting the freeness property [13, Theorem 13].

**1.6 Augmented directed complexes** Augmented directed complexes, designed by Steiner in [21], are not directly based on $\omega$-hypergraphs, but on chain complexes. Under the conditions required by Steiner, it happens that the data of a chain complex is equivalent to the data of an $\omega$-hypergraph. The definition of cells for this formalism strongly resembles the one of parity complexes. The only difference is that the cells are tuples of group elements instead of subsets of an $\omega$-hypergraph.

**Augmented directed complex** A pre-augmented directed complexes $(K, \mathrm{d}, \mathrm{e})$ (abbreviated pre-adc) is the data of
- for $n \in \mathbb{N}$, an abelian group $K_n$ together with a distinguished submonoid $K_n^* \subseteq K_n$,

- for $n \in \mathbb{N}$, group morphisms called *boundary operators*

$$d_n: K_{n+1} \to K_n,$$

- an *augmentation*, that is, a group morphism

$$e: K_0 \to \mathbb{Z}.$$

An *augmented directed complex*, abbreviated *adc*, is a pre-adc $(K, d, e)$ such that

$$e \circ d_0 = 0 \quad \text{and} \quad d_n \circ d_{n+1} = 0 \text{ for } n \in \mathbb{N}.$$

**Bases for pre-adc's** Given a pre-adc $(K, d, e)$, a *basis* of $(K, d, e)$ is the data of a graded set $P \subseteq \bigsqcup_{n \in \mathbb{N}} K_n$ such that each $K_n^*$ is the free commutative monoid on $P_n$ and each $K_n$ is the free abelian group on $K_n^*$. Given a basis $P$ of $(K, d, e)$, every element $u \in K_n$ can be uniquely written as $u = \sum_{g \in P_n} u_g g$, with $u_g \in \mathbb{Z}$ such that $u_g \neq 0$ for a finite number of $g \in P_n$. This representation induces a partial order $\leq$ where, for $n \in \mathbb{N}$ and $u, v \in K_n$, $u \leq v$ when $u_g \leq v_g$ for all $g \in P_n$. Given $n \in \mathbb{N}$ and $u, v \in K_n$ we can define a *greatest lower bound* $u \wedge v$ of $u$ and $v$ by

$$u \wedge v = \sum_{g \in P_n} \min(u_g, v_g)g.$$

Given $n \in \mathbb{N}$ and $u \in K_{n+1}$, we write $u^\ddagger$, $u^\pm \in K_n^*$ for the unique elements which satisfies that $d_n(u) = u^\pm - u^\ddagger$ and $u^\ddagger \wedge u^\pm = 0$. Moreover, we write $u^-$, $u^+$ for

$$u^- = \sum_{g \in P_{n+1}} u_g g^\ddagger \quad \text{and} \quad u^+ = \sum_{g \in P_{n+1}} u_g g^\pm.$$

*Remark 1.6.1.* The elements $u^\ddagger$ and $u^\pm$ are respectively denoted by $\partial^-(u)$ and $\partial^+(u)$ in [21]. We adopt the former notation for consistency with those of Section 1.4.

**From $\omega$-hypergraphs to pre-adc's with basis** Given an $\omega$-hypergraph $P$, we define the pre-adc $(K, d, e)$ associated to $P$ as follows. For $n \in \mathbb{N}$, $K_n^*$ is defined as the free commutative monoid on $P_n$ and $K_n$ as the free abelian group on $K_n^*$. The augmentation $e: K_0 \to \mathbb{Z}$ is defined as the unique morphism such that $e(x) = 1$ for $x \in P_0$. Given $n \in \mathbb{N}$ and a finite subset $U \subseteq P_n$, we write $\Sigma_n(U)$ for $\sum_{u \in U} u \in K_n$. Then, $d_n: K_{n+1} \to K_n$ is defined as the unique morphism such that $d_n(u) = \Sigma_n(u^+) - \Sigma_n(u^-)$ for $u \in P_{n+1}$. Then, $K$ canonically admits $P$ as a basis. We say that $P$ is an *adc* when $K$ is an *adc*.

**Example 1.6.2.** We explicitly describe the pre-adc associated to the $\omega$-hypergraph (13) as follows. Writing $S^*$ for the free commutative monoid on a set $S$, we put

$$K_0^* = \{x, y_1, y_2, z\}^*, \quad K_1^* = \{a_1, a_2, b, c_1, c_2\}^*, \quad K_2^* = \{\alpha, \beta\}^*$$

and $K_n^* = \{0\}$ for $n \geq 3$. $K_0$, $K_1$, $K_2$ and $K_n$ for $n \geq 3$ are then the induced free abelian groups on these monoids. The operations $e$ and $d$ are defined by universal property to be the unique morphisms such that $e(x) = e(y_1) = e(y_2) = e(z) = 1$ and

$$d_0(a_1) = y_1 - x, \quad d_0(a_2) = z - y_1, \quad d_0(b) = z - x,$$

$$d_0(c_1) = y_2 - x, \quad d_0(c_2) = z - y_2,$$

$$d_1(\alpha) = b - (a_1 + a_2), \quad d_1(\beta) = (c_1 + c_2) - b.$$
- for $n \in \mathbb{N}$, an abelian group $K_n$ together with a distinguished submonoid $K_n^* \subseteq K_n$,

- for $n \in \mathbb{N}$, group morphisms called *boundary operators*

$$d_n: K_{n+1} \to K_n,$$

- an *augmentation*, that is, a group morphism

$$e: K_0 \to \mathbb{Z}.$$

An *augmented directed complex*, abbreviated *adc*, is a pre-adc $(K, d, e)$ such that

$$e \circ d_0 = 0 \quad \text{and} \quad d_n \circ d_{n+1} = 0 \text{ for } n \in \mathbb{N}.$$

**Bases for pre-adc's** Given a pre-adc $(K, d, e)$, a *basis* of $(K, d, e)$ is the data of a graded set $P \subseteq \bigsqcup_{n \in \mathbb{N}} K_n$ such that each $K_n^*$ is the free commutative monoid on $P_n$ and each $K_n$ is the free abelian group on $K_n^*$. Given a basis $P$ of $(K, d, e)$, every element $u \in K_n$ can be uniquely written as $u = \sum_{g \in P_n} u_g g$, with $u_g \in \mathbb{Z}$ such that $u_g \neq 0$ for a finite number of $g \in P_n$. This representation induces a partial order $\leq$ where, for $n \in \mathbb{N}$ and $u, v \in K_n$, $u \leq v$ when $u_g \leq v_g$ for all $g \in P_n$. Given $n \in \mathbb{N}$ and $u, v \in K_n$ we can define a greatest lower bound $u \wedge v$ of $u$ and $v$ by

$$u \wedge v = \sum_{g \in P_n} \min(u_g, v_g)g.$$

Given $n \in \mathbb{N}$ and $u \in K_{n+1}$, we write $u^\mp$, $u^\pm \in K_n^*$ for the unique elements which satisfies that $d_n(u) = u^\pm - u^\mp$ and $u^\mp \wedge u^\pm = 0$. Moreover, we write $u^-, u^+$ for

$$u^- = \sum_{g \in P_{n+1}} u_g g^\mp \quad \text{and} \quad u^+ = \sum_{g \in P_{n+1}} u_g g^\pm.$$

*Remark 1.6.1.* The elements $u^\mp$ and $u^\pm$ are respectively denoted by $\partial^-(u)$ and $\partial^+(u)$ in [21]. We adopt the former notation for consistency with those of Section 1.4.

**From $\omega$-hypergraphs to pre-adc's with basis** Given an $\omega$-hypergraph $P$, we define the pre-adc $(K, d, e)$ associated to $P$ as follows. For $n \in \mathbb{N}$, $K_n^*$ is defined as the free commutative monoid on $P_n$ and $K_n$ as the free abelian group on $K_n^*$. The augmentation $e: K_0 \to \mathbb{Z}$ is defined as the unique morphism such that $e(x) = 1$ for $x \in P_0$. Given $n \in \mathbb{N}$ and a finite subset $U \subseteq P_n$, we write $\Sigma_n(U)$ for $\sum_{u \in U} u \in K_n$. Then, $d_n: K_{n+1} \to K_n$ is defined as the unique morphism such that $d_n(u) = \Sigma_n(u^+) - \Sigma_n(u^-)$ for $u \in P_{n+1}$. Then, $K$ canonically admits $P$ as a basis. We say that $P$ is an *adc* when $K$ is an *adc*.

**Example 1.6.2.** We explicitly describe the pre-adc associated to the $\omega$-hypergraph (13) as follows. Writing $S^*$ for the free commutative monoid on a set $S$, we put

$$K_0^* = \{x, y_1, y_2, z\}^*, \quad K_1^* = \{a_1, a_2, b, c_1, c_2\}^*, \quad K_2^* = \{\alpha, \beta\}^*$$

and $K_n^* = \{0\}$ for $n \geq 3$. $K_0, K_1, K_2$ and $K_n$ for $n \geq 3$ are then the induced free abelian groups on these monoids. The operations $e$ and $d$ are defined by universal property to be the unique morphisms such that $e(x) = e(y_1) = e(y_2) = e(z) = 1$ and

$$\begin{align*}
d_0(a_1) &= y_1 - x, & d_0(a_2) &= z - y_1, & d_0(b) &= z - x, \\
d_0(c_1) &= y_2 - x, & d_0(c_2) &= z - y_2, & & \\
d_1(\alpha) &= b - (a_1 + a_2), & d_1(\beta) &= (c_1 + c_2) - b. & &
\end{align*}$$
We can now give some examples for the operations $(-)^{\mp}$ and $(-)^{\pm}$ operations defined above:

$$ (a_1 + a_2)^{\mp} = x, \quad (a_1 + a_2)^{\pm} = z, \quad (\alpha + \beta)^{\mp} = a_1 + a_2, \quad (\alpha + \beta)^{\pm} = c_1 + c_2. $$

We moreover illustrate the operations $(-)^{-}$ and $(-)^{+}$:

$$ (a_1 + a_2)^{-} = x + y_1, \quad (a_1 + a_2)^{+} = y_1 + z, \quad (\alpha + \beta)^{-} = a_1 + a_2 + b, \quad (\alpha + \beta)^{+} = b + c_1 + c_2. $$

Thus, the operations $(-)^{\mp}$ and $(-)^{\pm}$ compute the source and target “borders” of an element of $K_n$, whereas the operations $(-)^{-}$ and $(-)^{+}$ compute the sum of the “inner” sources and targets of an element of $K_n$. They are the analogues of the operations defined for $\omega$-hypergraph in Section 1.3.

**Cells** Let $K$ be a pre-adc. Given $n \in \mathbb{N}$, an $n$-pre-cell of $K$ is given by an $(2n+1)$-tuple

$$ X = (X_{0,-}, X_{0,+}, \dots, X_{n-1,-}, X_{n-1,+}, X_n) $$

with $X_n \in K_n^*$ and $X_{i,-}, X_{i,+} \in K_i^*$ for $i \in \mathbb{N}_{n-1}$. For the sake of conciseness, we often refer to $X_n$ by $X_{n,-}$ or $X_{n,+}$. We write $PCell^*(K)$ for the graded set of pre-cells of $K$. When $n > 0$, given $\epsilon \in \{-, +\}$, we define the $n$-pre-cell $\partial_n^\epsilon(X)$ as

$$ \partial_n^\epsilon(X) = (X_{0,-}, X_{0,+}, \dots, X_{n-1,-}, X_{n-1,+}, X_{n,\epsilon}). $$

The functions $\partial^{-}, \partial^{+}$ then equip $PCell^*(K)$ with a structure of $\omega$-globular set.

Given $n \in \mathbb{N}$, an $n$-cell of $K$ is an $n$-pre-cell $X$ of $K$ such that

$$ (i) \text{ for } i \in \mathbb{N}_{n-1}, d_i(X_{i+1,-}) = d_i(X_{i+1,+}) = X_{i,+} - X_{i,-}, $$

$$ (ii) e(X_{0,-}) = e(X_{0,+}) = 1. $$

We denote by $Cell^*(K)$ the graded set of cells of $K$, which inherits the $\omega$-globular structure from $PCell^*(K)$.

**Remark 1.6.3.** The condition (i) is analogous to the moving condition (i) of parity complex cells, and the condition (ii) is related to the fork-freeness condition (ii) of parity complex cells instantiated in dimension 0.

**Identity and composition operations** Let $K$ be a pre-adc. Given $n \in \mathbb{N}$ and an $n$-pre-cell $X$ of $K$, we define the identity of $X$ as the $(n+1)$-pre-cell $\mathrm{id}^{n+1}(X)$ of $K$ such that

$$ \mathrm{id}^{n+1}(X) = (X_{0,-}, X_{0,+}, \dots, X_{n-1,-}, X_{n-1,+}, X_n, X_n, 0). $$

Given $i, n \in \mathbb{N}$ with $i < n$ and $i$-composable $n$-cells $X, Y$, we define the $i$-composition $X *_i Y$ of $X$ and $Y$ as the $n$-pre-cell $Z$ such that, for $j \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$,

$$ Z_{j,\epsilon} = \begin{cases} X_{j,\epsilon} + Y_{j,\epsilon} & \text{when } j > i, \\ X_{i,-} & \text{when } j = i \text{ and } \epsilon = -, \\ Y_{i,+} & \text{when } j = i \text{ and } \epsilon = +, \\ X_{j,\epsilon} \text{ (or equivalently } Y_{j,\epsilon}) & \text{when } j < i. \end{cases} $$

We then easily verify that $Z \in \mathrm{Cell}^*(K)$.
We can now give some examples for the operations $(-)^{\mp}$ and $(-)^{\pm}$ operations defined above:

$$ (a_1 + a_2)^{\mp} = x, \quad (a_1 + a_2)^{\pm} = z, \quad (\alpha + \beta)^{\mp} = a_1 + a_2, \quad (\alpha + \beta)^{\pm} = c_1 + c_2. $$

We moreover illustrate the operations $(-)^{-}$ and $(-)^{+}$:

$$ (a_1 + a_2)^{-} = x + y_1, \quad (a_1 + a_2)^{+} = y_1 + z, \quad (\alpha + \beta)^{-} = a_1 + a_2 + b, \quad (\alpha + \beta)^{+} = b + c_1 + c_2. $$

Thus, the operations $(-)^{\mp}$ and $(-)^{\pm}$ compute the source and target “borders” of an element of $K_n$, whereas the operations $(-)^{-}$ and $(-)^{+}$ compute the sum of the “inner” sources and targets of an element of $K_n$. They are the analogues of the operations defined for $\omega$-hypergraph in Section 1.3.

**Cells** Let $K$ be a pre-adc. Given $n \in \mathbb{N}$, an $n$-pre-cell of $K$ is given by an $(2n+1)$-tuple

$$ X = (X_{0,-}, X_{0,+}, \dots, X_{n-1,-}, X_{n-1,+}, X_n) $$

with $X_n \in K_n^*$ and $X_{i,-}, X_{i,+} \in K_i^*$ for $i \in \mathbb{N}_{n-1}$. For the sake of conciseness, we often refer to $X_n$ by $X_{n,-}$ or $X_{n,+}$. We write $PCell^*(K)$ for the graded set of pre-cells of $K$. When $n > 0$, given $\epsilon \in \{-, +\}$, we define the $n$-pre-cell $\partial_n^\epsilon(X)$ as

$$ \partial_n^\epsilon(X) = (X_{0,-}, X_{0,+}, \dots, X_{n-1,-}, X_{n-1,+}, X_{n,\epsilon}). $$

The functions $\partial^{-}, \partial^{+}$ then equip $PCell^*(K)$ with a structure of $\omega$-globular set.

Given $n \in \mathbb{N}$, an $n$-cell of $K$ is an $n$-pre-cell $X$ of $K$ such that

$$ (i) \text{ for } i \in \mathbb{N}_{n-1}, d_i(X_{i+1,-}) = d_i(X_{i+1,+}) = X_{i,+} - X_{i,-}, $$

$$ (ii) e(X_{0,-}) = e(X_{0,+}) = 1. $$

We denote by $Cell^*(K)$ the graded set of cells of $K$, which inherits the $\omega$-globular structure from $PCell^*(K)$.

**Remark 1.6.3.** The condition (i) is analogous to the moving condition (i) of parity complex cells, and the condition (ii) is related to the fork-freeness condition (ii) of parity complex cells instantiated in dimension 0.

**Identity and composition operations** Let $K$ be a pre-adc. Given $n \in \mathbb{N}$ and an $n$-pre-cell $X$ of $K$, we define the identity of $X$ as the $(n+1)$-pre-cell $\mathrm{id}^{n+1}(X)$ of $K$ such that

$$ \mathrm{id}^{n+1}(X) = (X_{0,-}, X_{0,+}, \dots, X_{n-1,-}, X_{n-1,+}, X_n, X_n, 0). $$

Given $i, n \in \mathbb{N}$ with $i < n$ and $i$-composable $n$-cells $X, Y$, we define the $i$-composition $X *_i Y$ of $X$ and $Y$ as the $n$-pre-cell $Z$ such that, for $j \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$,

$$ Z_{j,\epsilon} = \begin{cases} X_{j,\epsilon} + Y_{j,\epsilon} & \text{when } j > i, \\ X_{i,-} & \text{when } j = i \text{ and } \epsilon = -, \\ Y_{i,+} & \text{when } j = i \text{ and } \epsilon = +, \\ X_{j,\epsilon} \text{ (or equivalently } Y_{j,\epsilon} \text{)} & \text{when } j < i. \end{cases} $$

We then easily verify that $Z \in \mathrm{Cell}^*(K)$.
**Atoms** Let $K$ be a pre-adc equipped with a basis $P$. Given $n \in \mathbb{N}$ and $u \in P_n$, we define $[u]_{i,\epsilon} \subseteq P_i$ for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$ using a downward induction by $[u]_{n,-} = [u]_{n,+} = u$ and, for $j \in \mathbb{N}_{n-1}$, $[u]_{j,-} = [u]_{j+1,-}^{\pm}$ and $[u]_{j,+} = [u]_{j+1,+}^{\pm}$. For simplicity, we sometimes write $[u]_{n,-}$ or $[u]_{n,+}$ for $[u]_n$. The atom associated to $u$ is then the $n$-pre-cell of $K$

$$[u] = ([u]_{0,-}, [u]_{0,+}, \dots, [u]_{n-1,-}, [u]_{n-1,+}, [u]_n).$$

**Example 1.6.4.** In the pre-adc associated to the $\omega$-hypergraph (13), the atom $[\alpha]$ associated to $\alpha$ is defined by

$$[\alpha]_2 = \alpha,$$

$$[\alpha]_{1,-} = a_1 + a_2,$$

$$[\alpha]_{1,+} = b,$$

$$[\alpha]_{0,-} = x,$$

$$[\alpha]_{0,+} = z.$$

**Unital loop-free basis** Let $K$ be a pre-adc equipped with a basis $B$. Given $i \in \mathbb{N}$, we define a relation $<_i$ on $B$ as the smallest transitive relation such that, for $k, l \in \mathbb{N}$ with $i < \min(k, l)$, and $u \in B_k, v \in B_l$ with $[u]_{i,+} \wedge [v]_{i,-} \neq 0$, we have $u <_i v$. The basis $B$ is then said

- unital when for all $u \in B$, $e([u]_{0,-}) = e([u]_{0,+}) = 1$,

- loop-free when, for all $i \in \mathbb{N}$, $<_i$ is irreflexive.

**Example 1.6.5.** Consider the pre-adc $K$ with basis $B$ derived from the hypergraph (5). The basis $B$ is then unital but not loop-free since $a <_0 b <_0 a$. Now, consider the pre-adc with basis $B$ derived from the hypergraph (8). The basis $B$ is then not unital since $e([a]_{0,-}) = e(x + y) = 2$, but it is loop-free. Now consider the pre-adc $K$ with basis $B$ derived from the hypergraph (6). We have, among others, the relations

$$a <_0 b <_0 c <_0 d <_0 e <_0 f, \quad a <_0 \alpha <_0 \delta <_0 f, \quad \beta <_1 \alpha <_1 \gamma \quad \text{and} \quad \beta <_1 \delta <_1 \gamma.$$

It can be verified that $B$ is unital and loop-free.

**The freeness property** In [21], the author shows that, given an adc $K$ with a loop-free unital basis $B$, the globular set $\text{Cell}^*(K)$, together with identity and composition operations, has a structure of an $\omega$-category which is freely generated by the atoms $[u]$ for $u \in B$. Using the terminology of Section 2.3, this amounts to say that there exist an $\omega$-polygraph $\mathbb{Q}$ and an $\omega$-functor $F: \mathbb{Q}^* \to \text{Cell}^*(K)$ such that

- $Q_k = B_k$ for $k \in \mathbb{N}$,

- $F(u) = [u]$ for $u \in \mathbb{Q}$,

- $F$ is an isomorphism.

Contrary to parity complexes and pasting schemes, the pre-adc with basis associated to the $\omega$-hypergraph (10) is not a loop-free adc. Indeed, it is an adc with unital basis, but the basis is not loop-free since $A <_1 B <_1 A$. Thus, augmented directed complexes are, to the best of our knowledge, the only formalism of pasting diagrams among the three that we have already introduced for which the freeness property holds.

**1.7 Torsion-free complexes** In this section, we introduce *torsion-free complexes*. They are a new formalism for pasting diagrams based on parity complexes. More precisely, torsion-free complexes rely on the same notion of cell than parity complexes, but satisfy different axioms, namely the axioms (T0) to (T4).
**Atoms** Let $K$ be a pre-adc equipped with a basis $P$. Given $n \in \mathbb{N}$ and $u \in P_n$, we define $[u]_{i,\epsilon} \subseteq P_i$ for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$ using a downward induction by $[u]_{n,-} = [u]_{n,+} = u$ and, for $j \in \mathbb{N}_{n-1}$, $[u]_{j,-} = [u]_{j+1,-}^{\pm}$ and $[u]_{j,+} = [u]_{j+1,+}^{\pm}$. For simplicity, we sometimes write $[u]_{n,-}$ or $[u]_{n,+}$ for $[u]_n$. The atom associated to $u$ is then the $n$-pre-cell of $K$

$$[u] = ([u]_{0,-}, [u]_{0,+}, \dots, [u]_{n-1,-}, [u]_{n-1,+}, [u]_n).$$

**Example 1.6.4.** In the pre-adc associated to the $\omega$-hypergraph (13), the atom $[\alpha]$ associated to $\alpha$ is defined by

$$[\alpha]_2 = \alpha,$$

$$[\alpha]_{1,-} = a_1 + a_2,$$

$$[\alpha]_{1,+} = b,$$

$$[\alpha]_{0,-} = x,$$

$$[\alpha]_{0,+} = z.$$

**Unital loop-free basis** Let $K$ be a pre-adc equipped with a basis $B$. Given $i \in \mathbb{N}$, we define a relation $<_i$ on $B$ as the smallest transitive relation such that, for $k, l \in \mathbb{N}$ with $i < \min(k, l)$, and $u \in B_k, v \in B_l$ with $[u]_{i,+} \wedge [v]_{i,-} \neq 0$, we have $u <_i v$. The basis $B$ is then said

- unital when for all $u \in B$, $e([u]_{0,-}) = e([u]_{0,+}) = 1$,

- loop-free when, for all $i \in \mathbb{N}$, $<_i$ is irreflexive.

**Example 1.6.5.** Consider the pre-adc $K$ with basis $B$ derived from the hypergraph (5). The basis $B$ is then unital but not loop-free since $a <_0 b <_0 a$. Now, consider the pre-adc with basis $B$ derived from the hypergraph (8). The basis $B$ is then not unital since $e([a]_{0,-}) = e(x + y) = 2$, but it is loop-free. Now consider the pre-adc $K$ with basis $B$ derived from the hypergraph (6). We have, among others, the relations

$$a <_0 b <_0 c <_0 d <_0 e <_0 f, \quad a <_0 \alpha <_0 \delta <_0 f, \quad \beta <_1 \alpha <_1 \gamma \quad \text{and} \quad \beta <_1 \delta <_1 \gamma.$$

It can be verified that $B$ is unital and loop-free.

**The freeness property** In [21], the author shows that, given an adc $K$ with a loop-free unital basis $B$, the globular set $\text{Cell}^*(K)$, together with identity and composition operations, has a structure of an $\omega$-category which is freely generated by the atoms $[u]$ for $u \in B$. Using the terminology of Section 2.3, this amounts to say that there exist an $\omega$-polygraph $\mathbb{Q}$ and an $\omega$-functor $F: \mathbb{Q}^* \to \text{Cell}^*(K)$ such that

- $Q_k = B_k$ for $k \in \mathbb{N}$,

- $F(u) = [u]$ for $u \in \mathbb{Q}$,

- $F$ is an isomorphism.

Contrary to parity complexes and pasting schemes, the pre-adc with basis associated to the $\omega$-hypergraph (10) is not a loop-free adc. Indeed, it is an adc with unital basis, but the basis is not loop-free since $A <_1 B <_1 A$. Thus, augmented directed complexes are, to the best of our knowledge, the only formalism of pasting diagrams among the three that we have already introduced for which the freeness property holds.

**1.7 Torsion-free complexes** In this section, we introduce *torsion-free complexes*. They are a new formalism for pasting diagrams based on parity complexes. More precisely, torsion-free complexes rely on the same notion of cell than parity complexes, but satisfy different axioms, namely the axioms (T0) to (T4).
<figure><img src="image_8.png" /><figcaption>Figure 2: A problematic $\omega$-hypergraph</figcaption></figure>

**Definitions** Let $P$ be an $\omega$-hypergraph. Given $k \in \mathbb{N}$ and $u \in P_k$, we say that $u$ satisfies the segment condition when, for all $n \in \mathbb{N}_{k-1}$ and every $n$-cell $X$ such that $\langle u \rangle_{n,-} \subseteq X_n$, it holds that both $\langle u \rangle_{n,-}$ and $\langle u \rangle_{n,+}$ are segments for $\triangleleft_{X_n}$. Given $n, k, l \in \mathbb{N}$ with $0 < n < \min(k, l)$, $u \in P_k$, $v \in P_l$ and an $n$-cell $X$, $u$ and $v$ are said to be in torsion with respect to $X$ when

$$ \langle u \rangle_{n,+} \subseteq X_n, \quad \langle v \rangle_{n,-} \subseteq X_n, \quad \langle u \rangle_{n,+} \cap \langle v \rangle_{n,-} = \emptyset \quad \text{and} \quad \langle u \rangle_{n,+} \triangleleft_{X_n} \langle v \rangle_{n,-} \triangleleft_{X_n} \langle u \rangle_{n,+}. $$

The $\omega$-hypergraph $P$ is then a torsion-free complex when it satisfies the following axioms:

(T0) (non-emptiness) for all $u \in P$, $u^{-} \neq \emptyset$ and $u^{+} \neq \emptyset$;

(T1) (acyclicity) $P$ is acyclic;

(T2) (relevance) for all $u \in P$, $u$ is relevant;

(T3) (segment condition) for $u \in P$, $u$ satisfies the segment condition;

(T4) (torsion-freeness) for all $n, k, l \in \mathbb{N}^*$ with $n < \min(k, l)$, $u \in P_k$, $v \in P_l$ and every $n$-cell $X$, $u$ and $v$ are not in torsion with respect to $X$.

**Axiom (T1)** enforces the same notion of acyclicity than for parity complexes, forbidding loops like (5). **Axiom (T2)** requires that the generators of the $\omega$-hypergraph induce cells, forbidding $\omega$-hypergraphs like (7) and (8). It can be shown that **Axiom (T2)** entails **Axioms (C1)** and **(C2)** of parity complexes. The last axioms deserve their own paragraphs.

**The segment Axiom (T3)** Our goal is to find conditions on $\omega$-hypergraphs $P$ so that the freeness property holds, i.e., the $\omega$-category of cells $\text{Cell}(P)$ is freely generated by the atoms. In particular, a technical result states that every cell should be decomposable as a sequence of “whiskered atoms” (c.f. Proposition B.3.4). But there are cells of $\omega$-hypergraphs satisfying **Axioms (T0)** to **(T2)** that cannot be decomposed this way, because of the constraints that $\triangleleft$ requires on the composition order. We illustrate this with an example. Consider the $\omega$-hypergraph $P$ represented on Figure 2 where, more precisely,

$$ \begin{align*} A^{-} &= \{\alpha_1, \alpha_4\}, & A^{+} &= \{\alpha'_1, \alpha'_4\}, \\ \alpha'_1 &= \alpha_1^{-} = \{a\}, & \alpha'_1 &= \alpha_1^{+} = \{a'\}, \\ \alpha_4^{-} &= \alpha'_4^{-} = \{d\}, & \alpha_4^{+} &= \alpha'_4^{+} = \{d'\}, \quad \text{etc.} \end{align*} $$
<figure><img src="image_9.png" /><figcaption>Figure 2: A problematic $\omega$-hypergraph</figcaption></figure>

**Definitions** Let $P$ be an $\omega$-hypergraph. Given $k \in \mathbb{N}$ and $u \in P_k$, we say that $u$ satisfies the segment condition when, for all $n \in \mathbb{N}_{k-1}$ and every $n$-cell $X$ such that $\langle u \rangle_{n,-} \subseteq X_n$, it holds that both $\langle u \rangle_{n,-}$ and $\langle u \rangle_{n,+}$ are segments for $\triangleleft_{X_n}$. Given $n, k, l \in \mathbb{N}$ with $0 < n < \min(k, l)$, $u \in P_k$, $v \in P_l$ and an $n$-cell $X$, $u$ and $v$ are said to be in torsion with respect to $X$ when

$$ \langle u \rangle_{n,+} \subseteq X_n, \quad \langle v \rangle_{n,-} \subseteq X_n, \quad \langle u \rangle_{n,+} \cap \langle v \rangle_{n,-} = \emptyset \quad \text{and} \quad \langle u \rangle_{n,+} \triangleleft_{X_n} \langle v \rangle_{n,-} \triangleleft_{X_n} \langle u \rangle_{n,+}. $$

The $\omega$-hypergraph $P$ is then a torsion-free complex when it satisfies the following axioms:

(T0) (non-emptiness) for all $u \in P$, $u^{-} \neq \emptyset$ and $u^{+} \neq \emptyset$;

(T1) (acyclicity) $P$ is acyclic;

(T2) (relevance) for all $u \in P$, $u$ is relevant;

(T3) (segment condition) for $u \in P$, $u$ satisfies the segment condition;

(T4) (torsion-freeness) for all $n, k, l \in \mathbb{N}^*$ with $n < \min(k, l)$, $u \in P_k$, $v \in P_l$ and every $n$-cell $X$, $u$ and $v$ are not in torsion with respect to $X$.

**Axiom (T1)** enforces the same notion of acyclicity than for parity complexes, forbidding loops like (5). **Axiom (T2)** requires that the generators of the $\omega$-hypergraph induce cells, forbidding $\omega$-hypergraphs like (7) and (8). It can be shown that **Axiom (T2)** entails **Axioms (C1)** and **(C2)** of parity complexes. The last axioms deserve their own paragraphs.

**The segment Axiom (T3)** Our goal is to find conditions on $\omega$-hypergraphs $P$ so that the freeness property holds, i.e., the $\omega$-category of cells $\text{Cell}(P)$ is freely generated by the atoms. In particular, a technical result states that every cell should be decomposable as a sequence of “whiskered atoms” (c.f. Proposition B.3.4). But there are cells of $\omega$-hypergraphs satisfying Axioms (T0) to (T2) that cannot be decomposed this way, because of the constraints that $\triangleleft$ requires on the composition order. We illustrate this with an example. Consider the $\omega$-hypergraph $P$ represented on Figure 2 where, more precisely,

$$ \begin{align*} A^{-} &= \{\alpha_1, \alpha_4\}, & A^{+} &= \{\alpha'_1, \alpha'_4\}, \\ \alpha'_1 &= \alpha_1^{-} = \{a\}, & \alpha'_1 &= \alpha_1^{+} = \{a'\}, \\ \alpha_4^{-} &= \alpha'_4^{-} = \{d\}, & \alpha_4^{+} &= \alpha'_4^{+} = \{d'\}, \quad \text{etc.} \end{align*} $$
One can verify that $P$ satisfies Axioms (T0), (T1) and (T2). In this $\omega$-hypergraph, there is a 2-cell $X$ given by

$$
\begin{align*}
X_2 &= \{\alpha_1, \alpha_2, \alpha_3, \alpha_4\}, \\
X_{1,-} &= \{a, b\}, & X_{1,+} &= \{c, d', e\}, \\
X_{0,-} &= \{x\}, & X_{0,+} &= \{z\},
\end{align*}
$$

and a 3-cell $Y$ uniquely defined by $\partial_2^-(Y) = X$ and $Y_3 = \{A\}$. Suppose by contradiction that $\text{Cell}(P)$ is an $\omega$-category which is freely generated by the atoms. Then, $Y$ can be written

$$
Y = \mathrm{id}_{\lambda} *_{1} (\mathrm{id}_{l}^{2} *_{0} \langle A \rangle *_{0} \mathrm{id}_{r}^{2}) *_{1} \mathrm{id}_{\rho}
$$

for some 1-cells $l, r$ and 2-cells $\lambda, \rho$. Thus, $X = \mathrm{id}_{\lambda} *_{1} X' *_{1} \mathrm{id}_{\rho}$ where $X'$ is a 2-cell such that $X'_2 = A^-$. Since $\mathrm{Cell}(P)_{\le 2} \simeq \mathrm{Cell}(P \setminus \{A\})_{\le 2}$ and $P \setminus \{A\}$ is a torsion-free complex, using Lemma 2.2.1 introduced later, the existence of the decomposition of $X$ implies that

the sets $\lambda_2$, $X'_2$ and $\rho_2$ form a partition of $X_2 = \{\alpha_1, \alpha_2, \alpha_3, \alpha_4\}$,

and, using Lemma B.5.4 introduced later, we have that

$$
\text{for } (\beta, \gamma) \in (\lambda_2 \times (X'_2 \cup \rho_2)) \cup ((\rho_2 \cup X'_2) \times \rho_2), \text{ we have } \neg(\gamma \triangleleft_{X_2} \beta),
$$

or, more simply put, the partition $\lambda_2, X'_2, \rho_2$ respects the relation $\triangleleft_{X_2}$. But this cannot be
possible since $X'_2 = \{\alpha_1, \alpha_4\}$ and $\alpha_1 \triangleleft \alpha_2 \triangleleft \alpha_3 \triangleleft \alpha_4$. Hence, $\mathrm{Cell}(P)$ is not an $\omega$-category freely
generated by the atoms. Since $\langle A \rangle_{2,-}$ is not a segment for $\triangleleft_{X_2}$, Axiom (T3) prevents this kind
of problem.

**The torsion-freeness Axiom (T4)** The notion of torsion captures the essence of the counter-example to the freeness property of parity complexes and pasting schemes presented in **Section 1.4**. Indeed, considering the $\omega$-hypergraph $P$ represented by (10), there is a 2-cell $X$ defined by

$$
\begin{align*}
X_2 &= \{\alpha', \beta, \gamma, \delta'\}, \\
X_{1,-} &= \{a, d\}, & X_{1,+} &= \{c, f\}, \\
X_{0,-} &= \{x\}, & X_{0,+} &= \{z\},
\end{align*}
$$

which is induced by the pasting diagram

<figure><img src="image_9.png" /></figure>

Then, one can verify that $A$ and $B$ are in torsion with respect to $X$, so that $P$ does not satisfy
**Axiom (T4)** (on the other hand, it satisfies Axioms (T0) to (T3)).

Intuitively, the situations with torsion are the minimal cases where the freeness property fails for a parity complex $P$ (and similarly for a pasting scheme $P$). When $u, v \in P$ are in torsion with respect to a cell $X$ of $P$, there are two possible order to compose $u$ and $v$: first $u$ then $v$, or first $v$ then $u$. And both composites produce equal cells in $\mathrm{Cell}(P)$. However, this equality can not be deduced from an exchange law, since the torsion says basically that $u$ and $v$ cross each other, preventing to use the exchange law to swap them.
One can verify that $P$ satisfies Axioms (T0), (T1) and (T2). In this $\omega$-hypergraph, there is a 2-cell $X$ given by

$$ 
\begin{align*}
X_2 &= \{\alpha_1, \alpha_2, \alpha_3, \alpha_4\}, \\
X_{1,-} &= \{a, b\}, & X_{1,+} &= \{c, d', e\}, \\
X_{0,-} &= \{x\}, & X_{0,+} &= \{z\},
\end{align*}
$$

and a 3-cell $Y$ uniquely defined by $\partial_2^-(Y) = X$ and $Y_3 = \{A\}$. Suppose by contradiction that $\text{Cell}(P)$ is an $\omega$-category which is freely generated by the atoms. Then, $Y$ can be written

$$ Y = \text{id}_\lambda *_1 (\text{id}_l^2 *_0 \langle A \rangle *_0 \text{id}_r^2) *_1 \text{id}_\rho $$

for some 1-cells $l, r$ and 2-cells $\lambda, \rho$. Thus, $X = \text{id}_\lambda *_1 X' *_1 \text{id}_\rho$ where $X'$ is a 2-cell such that $X'_2 = A^-$. Since $\text{Cell}(P)_{\le 2} \simeq \text{Cell}(P \setminus \{A\})_{\le 2}$ and $P \setminus \{A\}$ is a torsion-free complex, using Lemma 2.2.1 introduced later, the existence of the decomposition of $X$ implies that

the sets $\lambda_2, X'_2$ and $\rho_2$ form a partition of $X_2 = \{\alpha_1, \alpha_2, \alpha_3, \alpha_4\}$,

and, using Lemma B.5.4 introduced later, we have that

$$ \text{for } (\beta, \gamma) \in (\lambda_2 \times (X'_2 \cup \rho_2)) \cup ((\rho_2 \cup X'_2) \times \rho_2), \text{ we have } \neg(\gamma \triangleleft_{X_2} \beta), $$

or, more simply put, the partition $\lambda_2, X'_2, \rho_2$ respects the relation $\triangleleft_{X_2}$. But this cannot be possible since $X'_2 = \{\alpha_1, \alpha_4\}$ and $\alpha_1 \triangleleft \alpha_2 \triangleleft \alpha_3 \triangleleft \alpha_4$. Hence, $\text{Cell}(P)$ is not an $\omega$-category freely generated by the atoms. Since $\langle A \rangle_{2,-}$ is not a segment for $\triangleleft_{X_2}$, Axiom (T3) prevents this kind of problem.

**The torsion-freeness Axiom (T4)** The notion of torsion captures the essence of the counter-example to the freeness property of parity complexes and pasting schemes presented in Section 1.4. Indeed, considering the $\omega$-hypergraph $P$ represented by (10), there is a 2-cell $X$ defined by

$$ 
\begin{align*}
X_2 &= \{\alpha', \beta, \gamma, \delta'\}, \\
X_{1,-} &= \{a, d\}, & X_{1,+} &= \{c, f\}, \\
X_{0,-} &= \{x\}, & X_{0,+} &= \{z\},
\end{align*}
$$

which is induced by the pasting diagram

<figure><img src="image_10.png" /></figure>

Then, one can verify that $A$ and $B$ are in torsion with respect to $X$, so that $P$ does not satisfy Axiom (T4) (on the other hand, it satisfies Axioms (T0) to (T3)).

Intuitively, the situations with torsion are the minimal cases where the freeness property fails for a parity complex $P$ (and similarly for a pasting scheme $P$). When $u, v \in P$ are in torsion with respect to a cell $X$ of $P$, there are two possible order to compose $u$ and $v$: first $u$ then $v$, or first $v$ then $u$. And both composites produce equal cells in $\text{Cell}(P)$. However, this equality can not be deduced from an exchange law, since the torsion says basically that $u$ and $v$ cross each other, preventing to use the exchange law to swap them.
**More computable axioms** Axioms (T3) and (T4) happen to be hard to check in practice, since they both involve a quantification on all the cells of an $\omega$-hypergraph. Here, we give stronger axioms that are simpler to verify.

Given an $\omega$-hypergraph $P$, for $n \in \mathbb{N}$, $u, v \in P_n$, we write $u \frown v$ when there exists $w \in P_{n+1}$ such that $u \in w^-$ and $v \in w^+$ and we write $\frown^*$ for the reflexive transitive closure of $\frown$. For $U, V \subseteq P_n$, we write $U \frown^* V$ when there exist $u \in U$ and $v \in V$ such that $u \frown^* v$. Consider the following axiom on an $\omega$-hypergraph $P$:

(T3') for $k, n \in \mathbb{N}^*$ with $k < n$ and $u \in P_n$, we do not have $\langle u \rangle_{k,+} \frown^* \langle u \rangle_{k,-}$.

Then, Axiom (T3') can be replaced by Axiom (T3') in the axioms of torsion-free complexes:

**Proposition 1.7.1.** Let $P$ be an $\omega$-hypergraph satisfying Axioms (T0), (T1) and (T2). If $P$ satisfies Axiom (T3'), then it satisfies Axiom (T3).

*Proof.* Suppose that $P$ satisfies Axiom (T3'). Let $n, k \in \mathbb{N}$ with $n < k$, $X$ be an $n$-cell and $u \in P_k$ such that $\langle u \rangle_{n,-} \subseteq X_n$. If $n = 0$, there is nothing to prove, so we can assume $n > 0$. By contradiction, suppose that $\langle u \rangle_{n,-}$ is not a segment for $\triangleleft_{X_n}$. So there are $r \in \mathbb{N}$ with $r > 2$ and $u_1, \dots, u_r \in X_n$ such that $u_1, u_r \in \langle u \rangle_{n,-}$, $u_2, \dots, u_{r-1} \notin \langle u \rangle_{n,-}$ and $u_i \triangleleft_{X_n} u_{i+1}$ for $i \in \mathbb{N}_{r-1}^*$. In particular, there are $v_1, \dots, v_{r-1} \in P_{n-1}$ such that $v_i \in u_i^+ \cap u_{i+1}^-$ for $i \in \mathbb{N}_{r-1}^*$. Given $w \in X_n$ such that $v_1 \in w^-$, since $X_n$ is fork-free, we have $w = u_2 \notin \langle u \rangle_{n,-}$. Thus, since $u$ is relevant by Axiom (T2), $v_1 \in \langle u \rangle_{n,-}^\pm = \langle u \rangle_{n-1,+}$. Similarly, we have that $v_{r-1} \in \langle u \rangle_{n-1,-}$. So, $\langle u \rangle_{n-1,+} \frown^* \langle u \rangle_{n-1,-}$, contradicting Axiom (T3'). Hence, $P$ satisfies Axiom (T3). $\square$

Now, consider the following axiom on an $\omega$-hypergraph $P$:

(T4') for $n, k, l \in \mathbb{N}^*$ with $n < \min(k, l)$, $u \in P_k$ and $v \in P_l$, if $\langle u \rangle_{n,+} \cap \langle v \rangle_{n,-} = \emptyset$, then at most one of the following holds:

$$ - \langle u \rangle_{n-1,+} \frown^* \langle v \rangle_{n-1,-}, $$

$$ - \langle v \rangle_{n-1,+} \frown^* \langle u \rangle_{n-1,-}. $$

Then, Axiom (T4') can be replaced by Axiom (T4') in the axioms of torsion-free complexes:

**Proposition 1.7.2.** Let $P$ be an $\omega$-hypergraph satisfying Axioms (T0), (T1) and (T2). If $P$ satisfies Axiom (T4'), then it satisfies Axiom (T4).

*Proof.* Suppose that $P$ satisfies Axiom (T4'). By contradiction, assume that $P$ does not satisfy Axiom (T4). So there are $n, k, l \in \mathbb{N}^*$ with $n < \min(k, l)$, $u \in P_k$, $v \in P_l$ and an $n$-cell $X$ such that $u$ and $v$ are in torsion with respect to $X$. That is, $\langle u \rangle_{n,+} \subseteq X_n$, $\langle v \rangle_{n,-} \subseteq X_n$ $\langle u \rangle_{n,+} \cap \langle v \rangle_{n,-} = \emptyset$ and $\langle u \rangle_{n,+} \triangleleft_{X_n} \langle v \rangle_{n,-} \triangleleft_{X_n} \langle u \rangle_{n,+}$. By the last condition, there are $r \in \mathbb{N}$ with $r > 1$, and $w_1, \dots, w_r \in X_n$ such that

$$ w_1 \in \langle u \rangle_{n,+}, \quad w_r \in \langle v \rangle_{n,-}, \quad w_2, \dots, w_{r-1} \notin \langle u \rangle_{n,+} \cup \langle v \rangle_{n,-}, \quad \text{and} \quad w_i \triangleleft_{X_n} w_{i+1} $$

for $i \in \mathbb{N}_{r-1}^*$. Thus, there are $\bar{w}_1, \dots, \bar{w}_{r-1} \in P_{n-1}$ such that $\bar{w}_i \in w_i^+ \cap w_{i+1}^-$ for $i \in \mathbb{N}_{r-1}^*$. Given $w \in X_n$ with $\bar{w}_1 \in w^-$, we have $w = w_2 \notin \langle u \rangle_{n,+}$ since $X_n$ is fork-free. Thus, $\bar{w}_1 \in \langle u \rangle_{n,+}^\pm = \langle u \rangle_{n-1,+}$. Similarly, $\bar{w}_{r-1} \in \langle v \rangle_{n-1,-}$, so $\langle u \rangle_{n-1,+} \frown^* \langle v \rangle_{n-1,-}$. Likewise, we have $\langle v \rangle_{n-1,+} \frown^* \langle u \rangle_{n-1,-}$, which contradicts Axiom (T4'). Hence, $P$ satisfies Axiom (T4). $\square$

## 2. The free $\omega$-category of cells

In this section, we show that the cells on a torsion-free complex have a structure of a free $\omega$-category. For the $\omega$-categorical structure, we essentially have to prove that the composition of two
**More computable axioms** Axioms (T3) and (T4) happen to be hard to check in practice, since they both involve a quantification on all the cells of an $\omega$-hypergraph. Here, we give stronger axioms that are simpler to verify.

Given an $\omega$-hypergraph $P$, for $n \in \mathbb{N}$, $u, v \in P_n$, we write $u \frown v$ when there exists $w \in P_{n+1}$ such that $u \in w^-$ and $v \in w^+$ and we write $\frown^*$ for the reflexive transitive closure of $\frown$. For $U, V \subseteq P_n$, we write $U \frown^* V$ when there exist $u \in U$ and $v \in V$ such that $u \frown^* v$. Consider the following axiom on an $\omega$-hypergraph $P$:

(T3') for $k, n \in \mathbb{N}^*$ with $k < n$ and $u \in P_n$, we do not have $\langle u \rangle_{k,+} \frown^* \langle u \rangle_{k,-}$.

Then, Axiom (T3') can be replaced by Axiom (T3') in the axioms of torsion-free complexes:

**Proposition 1.7.1.** Let $P$ be an $\omega$-hypergraph satisfying Axioms (T0), (T1) and (T2). If $P$ satisfies Axiom (T3'), then it satisfies Axiom (T3).

*Proof.* Suppose that $P$ satisfies Axiom (T3'). Let $n, k \in \mathbb{N}$ with $n < k$, $X$ be an $n$-cell and $u \in P_k$ such that $\langle u \rangle_{n,-} \subseteq X_n$. If $n = 0$, there is nothing to prove, so we can assume $n > 0$. By contradiction, suppose that $\langle u \rangle_{n,-}$ is not a segment for $\triangleleft_{X_n}$. So there are $r \in \mathbb{N}$ with $r > 2$ and $u_1, \dots, u_r \in X_n$ such that $u_1, u_r \in \langle u \rangle_{n,-}$, $u_2, \dots, u_{r-1} \notin \langle u \rangle_{n,-}$ and $u_i \triangleleft_{X_n}^1 u_{i+1}$ for $i \in \mathbb{N}_{r-1}^*$. In particular, there are $v_1, \dots, v_{r-1} \in P_{n-1}$ such that $v_i \in u_i^+ \cap u_{i+1}^-$ for $i \in \mathbb{N}_{r-1}^*$. Given $w \in X_n$ such that $v_1 \in w^-$, since $X_n$ is fork-free, we have $w = u_2 \notin \langle u \rangle_{n,-}$. Thus, since $u$ is relevant by Axiom (T2), $v_1 \in \langle u \rangle_{n,-}^\pm = \langle u \rangle_{n-1,+}$. Similarly, we have that $v_{r-1} \in \langle u \rangle_{n-1,-}$. So, $\langle u \rangle_{n-1,+} \frown^* \langle u \rangle_{n-1,-}$, contradicting Axiom (T3'). Hence, $P$ satisfies Axiom (T3). $\square$

Now, consider the following axiom on an $\omega$-hypergraph $P$:

(T4') for $n, k, l \in \mathbb{N}^*$ with $n < \min(k, l)$, $u \in P_k$ and $v \in P_l$, if $\langle u \rangle_{n,+} \cap \langle v \rangle_{n,-} = \emptyset$, then at most one of the following holds:

$$- \langle u \rangle_{n-1,+} \frown^* \langle v \rangle_{n-1,-},$$

$$- \langle v \rangle_{n-1,+} \frown^* \langle u \rangle_{n-1,-}.$$

Then, Axiom (T4') can be replaced by Axiom (T4') in the axioms of torsion-free complexes:

**Proposition 1.7.2.** Let $P$ be an $\omega$-hypergraph satisfying Axioms (T0), (T1) and (T2). If $P$ satisfies Axiom (T4'), then it satisfies Axiom (T4).

*Proof.* Suppose that $P$ satisfies Axiom (T4'). By contradiction, assume that $P$ does not satisfy Axiom (T4). So there are $n, k, l \in \mathbb{N}^*$ with $n < \min(k, l)$, $u \in P_k$, $v \in P_l$ and an $n$-cell $X$ such that $u$ and $v$ are in torsion with respect to $X$. That is, $\langle u \rangle_{n,+} \subseteq X_n$, $\langle v \rangle_{n,-} \subseteq X_n$ and $\langle u \rangle_{n,+} \cap \langle v \rangle_{n,-} = \emptyset$ and $\langle u \rangle_{n,+} \triangleleft_{X_n} \langle v \rangle_{n,-} \triangleleft_{X_n} \langle u \rangle_{n,+}$. By the last condition, there are $r \in \mathbb{N}$ with $r > 1$, and $w_1, \dots, w_r \in X_n$ such that

$$w_1 \in \langle u \rangle_{n,+}, \quad w_r \in \langle v \rangle_{n,-}, \quad w_2, \dots, w_{r-1} \notin \langle u \rangle_{n,+} \cup \langle v \rangle_{n,-}, \quad \text{and} \quad w_i \triangleleft_{X_n}^1 w_{i+1}$$

for $i \in \mathbb{N}_{r-1}^*$. Thus, there are $\bar{w}_1, \dots, \bar{w}_{r-1} \in P_{n-1}$ such that $\bar{w}_i \in w_i^+ \cap w_{i+1}^-$ for $i \in \mathbb{N}_{r-1}^*$. Given $w \in X_n$ with $\bar{w}_1 \in w^-$, we have $w = w_2 \notin \langle u \rangle_{n,+}$ since $X_n$ is fork-free. Thus, $\bar{w}_1 \in \langle u \rangle_{n,+}^\pm = \langle u \rangle_{n-1,+}$. Similarly, $\bar{w}_{r-1} \in \langle v \rangle_{n-1,-}$, so $\langle u \rangle_{n-1,+} \frown^* \langle v \rangle_{n-1,-}$. Likewise, we have $\langle v \rangle_{n-1,+} \frown^* \langle u \rangle_{n-1,-}$, which contradicts Axiom (T4'). Hence, $P$ satisfies Axiom (T4). $\square$

## 2. The free $\omega$-category of cells

In this section, we show that the cells on a torsion-free complex have a structure of a free $\omega$-category. For the $\omega$-categorical structure, we essentially have to prove that the composition of two
cells is a cell. In order to show this, we adapt a result of [23] and prove a “gluing theorem”, which enables building an (n+1)-cell from an n-cell by gluing a set of (n+1)-generators. As a by-product, it also gives some properties satisfied by composable cells, from which one can derive that the composition of cells is well-defined. Then, for the freeness, we must first define the meaning of “freeness” that we use. The most natural notion for our context is the one of polygraph [22, 4], which is a set of generators for strict categories whose sources and targets are composites made of other generators, and from which a free strict category can be generated. This structure adequately encodes the fact that the source and target of each generator of a torsion-free complexes are themselves pasting diagrams of generators that must be composed.

In Section 2.1, we first state the gluing theorem (Theorem 2.1.1), after introducing the adequate terminology for the “gluing” of a set of generators on a cell. Then, in Section 2.2, we use this property to prove that the cells of a torsion-free complex have a structure of an $\omega$-category (Theorem 2.2.3). In Section 2.3, we introduce the definition of “freeness” for strict categories that we are going to use for the $\omega$-category of cells by recalling the definition of polygraphs and the associated free strict category construction. In Section 2.4, we prove that the $\omega$-category of cells of a torsion-free complex is free in this sense, i.e., is the free $\omega$-category on a canonical polygraph (Corollary 2.4.2).

## 2.1 Gluing sets on cells

**Gluings and activations** Let $P$ be an $\omega$-hypergraph. Given $n \in \mathbb{N}$, an $n$-pre-cell $X$ of $P$ and a finite set $G \subseteq P_{n+1}$, we say that $G$ is *glueable on $X$ if $G^\mp \subseteq X_n$*. If so, we call *gluing of $G$ on $X$ the $(n+1)$-pre-cell $Y$ of $P$* defined by

$$Y_{n+1} = G, \quad Y_{n,-} = X_n, \quad Y_{n,+} = (X_n \cup G^+) \setminus G^- \quad \text{and} \quad Y_{i,\epsilon} = X_{i,\epsilon}$$

for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$. We denote $Y$ by $\text{Glue}(X, G)$. Moreover, we call *activation of $G$ on $X$* the $n$-pre-cell $\text{Act}(X, G)$ defined by

$$\text{Act}(X, G) = \partial_n^+(\text{Glue}(X, G))$$

We say that $G$ is *dually glueable on $X$* when $G^\pm \subseteq X_n$, and we define the dual gluing $\overline{\text{Glue}}(X, G)$ and the dual activation $\overline{\text{Act}}(X, G)$ similarly. For example, consider the $\omega$-hypergraph (14) from Section 1.7 and recall there the definitions of $X$ and $Y$. Then $\{A\}$ is glueable on $X$ and $\text{Glue}(X, \{A\}) = Y$, and $\text{Act}(X, \{A\})$ is the 2-pre-cell $\bar{X}$ with

$$\begin{align*}
\bar{X}_2 &= \{\alpha_1, \alpha'_2, \alpha'_3, \alpha_4\}, \\
\bar{X}_{1,-} &= \{a, b\}, & \bar{X}_{1,+} &= \{c, d', e\}, \\
\bar{X}_{0,-} &= \{x\}, & \bar{X}_{0,+} &= \{z\}.
\end{align*}$$

Conversely, $\{A\}$ is dually glueable on $\bar{X}$, and we have $\overline{\text{Glue}}(\bar{X}, \{A\}) = Y$, and $\overline{\text{Act}}(\bar{X}, \{A\}) = X$.

**The gluing theorem** We now state the “gluing theorem”. It is an adapted version of [23, Lemma 3.2] which enables building new cells using the gluing and activation operations. The theorem moreover gives additional results concerning intersections with the source and the target sets of gluing sets.
cells is a cell. In order to show this, we adapt a result of [23] and prove a “gluing theorem”, which enables building an (n+1)-cell from an n-cell by gluing a set of (n+1)-generators. As a by-product, it also gives some properties satisfied by composable cells, from which one can derive that the composition of cells is well-defined. Then, for the freeness, we must first define the meaning of “freeness” that we use. The most natural notion for our context is the one of polygraph [22, 4], which is a set of generators for strict categories whose sources and targets are composites made of other generators, and from which a free strict category can be generated. This structure adequately encodes the fact that the source and target of each generator of a torsion-free complexes are themselves pasting diagrams of generators that must be composed.

In Section 2.1, we first state the gluing theorem (Theorem 2.1.1), after introducing the adequate terminology for the “gluing” of a set of generators on a cell. Then, in Section 2.2, we use this property to prove that the cells of a torsion-free complex have a structure of an $\omega$-category (Theorem 2.2.3). In Section 2.3, we introduce the definition of “freeness” for strict categories that we are going to use for the $\omega$-category of cells by recalling the definition of polygraphs and the associated free strict category construction. In Section 2.4, we prove that the $\omega$-category of cells of a torsion-free complex is free in this sense, i.e., is the free $\omega$-category on a canonical polygraph (Corollary 2.4.2).

## 2.1 Gluing sets on cells

**Gluings and activations** Let $P$ be an $\omega$-hypergraph. Given $n \in \mathbb{N}$, an $n$-pre-cell $X$ of $P$ and a finite set $G \subseteq P_{n+1}$, we say that $G$ is *glueable on $X$ if $G^\mp \subseteq X_n$*. If so, we call *gluing of $G$ on $X$ the $(n+1)$-pre-cell $Y$ of $P$* defined by

$$Y_{n+1} = G, \quad Y_{n,-} = X_n, \quad Y_{n,+} = (X_n \cup G^+) \setminus G^- \quad \text{and} \quad Y_{i,\epsilon} = X_{i,\epsilon}$$

for $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$. We denote $Y$ by $\text{Glue}(X, G)$. Moreover, we call *activation of $G$ on $X$* the $n$-pre-cell $\text{Act}(X, G)$ defined by

$$\text{Act}(X, G) = \partial_n^+(\text{Glue}(X, G))$$

We say that $G$ is *dually glueable on $X$* when $G^\pm \subseteq X_n$, and we define the dual gluing $\overline{\text{Glue}}(X, G)$ and the dual activation $\overline{\text{Act}}(X, G)$ similarly. For example, consider the $\omega$-hypergraph (14) from Section 1.7 and recall there the definitions of $X$ and $Y$. Then $\{A\}$ is glueable on $X$ and $\text{Glue}(X, \{A\}) = Y$, and $\text{Act}(X, \{A\})$ is the 2-pre-cell $\bar{X}$ with

$$\begin{align*}
\bar{X}_2 &= \{\alpha_1, \alpha'_2, \alpha'_3, \alpha_4\}, \\
\bar{X}_{1,-} &= \{a, b\}, & \bar{X}_{1,+} &= \{c, d', e\}, \\
\bar{X}_{0,-} &= \{x\}, & \bar{X}_{0,+} &= \{z\}.
\end{align*}$$

Conversely, $\{A\}$ is dually glueable on $\bar{X}$, and we have $\overline{\text{Glue}}(\bar{X}, \{A\}) = Y$, and $\overline{\text{Act}}(\bar{X}, \{A\}) = X$.

**The gluing theorem** We now state the “gluing theorem”. It is an adapted version of [23, Lemma 3.2] which enables building new cells using the gluing and activation operations. The theorem moreover gives additional results concerning intersections with the source and the target sets of gluing sets.
<figure><img src="image_10.png" /><figcaption>Figure 3: Cells involved and their movements in Theorem 2.1.1</figcaption></figure>

**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 $X$.

A representation of the cells of the statement is shown in Figure 3.

*Proof.* See Appendix A. $\square$

<figure><img src="image_11.png" /></figure>

**2.2 Structure of $\omega$-category** Here, we prove that the cells on a torsion-free complex have a structure of an $\omega$-category. For this purpose, we first show that the composite of two cells is a cell using Theorem 2.1.1 shown above. Then, we quickly verify that the axioms of $\omega$-categories are satisfied, which is almost immediate by the definitions of the operations on cells.

We first handle the case of compositions of cells in codimension 1 with the following result:

**Lemma 2.2.1.** Let $P$ be an $\omega$-hypergraph satisfying Axioms (T0), (T1), (T2) and (T3). Given $n \in \mathbb{N}^*$ and two $n$-cells $X, Y$ of $P$ that are $(n-1)$-composable, the following hold:

(a) $X_n^- \cap Y_n^+ = \emptyset$,

(b) $X_n \cap Y_n = \emptyset$,

(c) $X *_{n-1} Y$ is an $n$-cell of $P$.

*Proof.* Using Theorem 2.1.1(c) with $\partial_{n-1}^+(X)$, $X_n$ and $Y_n$, we get $X_n^- \cap Y_n^+ = \emptyset$. Moreover,

$$ \begin{align*} X_n^+ \cap Y_n^+ &= X_n^\pm \cap Y_n^+ && (\text{since } X_n^- \cap Y_n^+ = \emptyset) \\ &\subseteq X_{n-1,+} \cap Y_n^+ \\ &= Y_{n-1,-} \cap Y_n^+ \\ &= \emptyset && (\text{by Theorem 2.1.1(a)}). \end{align*} $$
<figure><img src="image_11.png" /><figcaption>Figure 3: Cells involved and their movements in Theorem 2.1.1</figcaption></figure>

**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 $X$.

A representation of the cells of the statement is shown in Figure 3.

*Proof.* See Appendix A. $\square$

<figure><img src="image_12.png" /></figure>

**2.2 Structure of $\omega$-category** Here, we prove that the cells on a torsion-free complex have a structure of an $\omega$-category. For this purpose, we first show that the composite of two cells is a cell using Theorem 2.1.1 shown above. Then, we quickly verify that the axioms of $\omega$-categories are satisfied, which is almost immediate by the definitions of the operations on cells.

We first handle the case of compositions of cells in codimension 1 with the following result:

**Lemma 2.2.1.** Let $P$ be an $\omega$-hypergraph satisfying Axioms (T0), (T1), (T2) and (T3). Given $n \in \mathbb{N}^*$ and two $n$-cells $X, Y$ of $P$ that are $(n-1)$-composable, the following hold:

(a) $X_n^- \cap Y_n^+ = \emptyset$,

(b) $X_n \cap Y_n = \emptyset$,

(c) $X *_{n-1} Y$ is an $n$-cell of $P$.

*Proof.* Using Theorem 2.1.1(c) with $\partial_{n-1}^+(X)$, $X_n$ and $Y_n$, we get $X_n^- \cap Y_n^+ = \emptyset$. Moreover,

$$ \begin{align*} X_n^+ \cap Y_n^+ &= X_n^\pm \cap Y_n^+ && (\text{since } X_n^- \cap Y_n^+ = \emptyset) \\ &\subseteq X_{n-1,+} \cap Y_n^+ \\ &= Y_{n-1,-} \cap Y_n^+ \\ &= \emptyset && (\text{by Theorem 2.1.1(a)}). \end{align*} $$
By Axiom (T0), it implies that $X_n \cap Y_n = \emptyset$. Similarly, $X_n^- \cap Y_n^- = \emptyset$, so $X_n \cup Y_n$ is fork-free. For $X_{*n-1}Y$ to be a cell, $X_n \cup Y_n$ must move $X_{n-1,-}$ to $Y_{n-1,+}$. But, since $X$ and $Y$ are cells and are $(n-1)$-composable, we know that $X_n$ moves $X_{n-1,-}$ to $X_{n-1,+}$, $Y_n$ moves $Y_{n-1,-}$ to $Y_{n-1,+}$ and $X_{n-1,+} = Y_{n-1,-}$. Since $X_n^- \cap Y_n^+ = \emptyset$, using Lemma A.1.3, we get that $X_n \cup Y_n$ moves $X_{n-1,-}$ to $Y_{n-1,+}$. Hence, $X_{*n-1}Y$ is a cell. $\square$

We now handle the general case of compositions of cells:

**Lemma 2.2.2.** Let $P$ be an $\omega$-hypergraph satisfying Axioms (T0), (T1), (T2) and (T3). Let $i, n \in \mathbb{N}$ with $i < n$ and $X, Y$ be two $n$-cells of $P$ that are $i$-composable. Then,

(i) for $j \in \mathbb{N}$ with $i < j \le n$, $(X_{j,-}^- \cup X_{j,+}^-) \cap (Y_{j,-}^+ \cup Y_{j,+}^+) = \emptyset$,

(ii) $X_{*i}Y$ is a cell.

*Proof.* By induction on $n-i$. If $n-i = 1$, the properties follow from Lemma 2.2.1. So suppose that $n-i > 1$. For $\epsilon, \eta \in \{-, +\}$, by induction hypothesis with $\partial_{n-1}^\epsilon(X)$ and $\partial_{n-1}^\eta(Y)$, we get $X_{n-1,\epsilon}^- \cap Y_{n-1,\eta}^+ = \emptyset$. Therefore, $(X_{n-1,-}^- \cup X_{n-1,+}^-) \cap (Y_{n-1,-}^+ \cup Y_{n-1,+}^+) = \emptyset$. We moreover obtain that $(X_{j,-}^- \cup X_{j,+}^-) \cap (Y_{j,-}^+ \cup Y_{j,+}^+) = \emptyset$ for $j \in \mathbb{N}$ with $i < j < n-1$. Let $Z = \partial_{n-1}^+(X)_{*i} \partial_{n-1}^-(Y)$. By induction, $Z$ is a $(n-1)$-cell and $Z_{n-1} = X_{n-1,+} \cup Y_{n-1,-}$. Using Theorem 2.1.1(c), we get that $X_n^- \cap Y_n^+ = \emptyset$ which concludes the proof of (i).

For (ii), we already know that $\partial_{n-1}^-(X)_{*i} \partial_{n-1}^-(Y)$ and $\partial_{n-1}^+(X)_{*i} \partial_{n-1}^+(Y)$ are cells by induction. So, in order to prove that $X_{*i}Y$ is a cell, we just need to show that $X_n \cup Y_n$ is fork-free and moves $X_{n-1,-} \cup Y_{n-1,-}$ to $X_{n-1,+} \cup Y_{n-1,+}$. But

$$ 
\begin{align*}
X_n^+ \cap Y_n^+ &= X_n^\pm \cap Y_n^+ \tag{by (i)} \\
&\subseteq Z_{n-1} \cap Y_n^+ \tag{by Theorem 2.1.1(a)} \\
&= \emptyset
\end{align*}
$$

and similarly, $X_n^- \cap Y_n^- = \emptyset$, so $X_n \cup Y_n$ is fork-free. Using the dual of Theorem 2.1.1(a) with $Z$ and $X_n$, we get

$$ X_n^- \cap (X_{n-1,+} \cup Y_{n-1,-}) = X_n^- \cap Y_{n-1,-} = \emptyset. $$

Similarly, if $Z' = \partial_{n-1}^-(X)_{*i} \partial_{n-1}^-(Y)$ then $Z'_{n-1} = X_{n-1,-} \cup Y_{n-1,-}$. Using Theorem 2.1.1(a) with $Z'$ and $X_n$, we have

$$ X_n^+ \cap (X_{n-1,-} \cup Y_{n-1,-}) = X_n^+ \cap Y_{n-1,-} = \emptyset. $$

Since $X_n$ moves $X_{n-1,-}$ to $X_{n-1,+}$, using Lemma A.1.2, we deduce that $X_n$ moves $X_{n-1,-} \cup Y_{n-1,-}$ to $X_{n-1,+} \cup Y_{n-1,-}$. Similarly, $Y_n$ moves $X_{n-1,+} \cup Y_{n-1,-}$ to $X_{n-1,+} \cup Y_{n-1,+}$. Since $X_n^- \cap Y_n^+ = \emptyset$, by Lemma A.1.3, we have that $X_n \cup Y_n$ moves $X_{n-1,-} \cup Y_{n-1,-}$ to $X_{n-1,+} \cup Y_{n-1,+}$. Hence, $X_{*n-1}Y$ is a cell. $\square$

We can finally conclude that the $\omega$-category of cells has a structure of $\omega$-category given by the identity and composition operations on cells:

**Theorem 2.2.3.** Let $P$ be a torsion-free complex. (Cell($P$), $\partial^-$, $\partial^+$, id, *) is an $\omega$-category.

*Proof.* We already know that Cell($P$) is a $\omega$-globular set. By Lemma 2.2.2, the composition operation $*$ is well-defined on composable cells. Moreover, all the axioms of $\omega$-categories (given in Section 1.2), follow readily from the definitions of $\partial^-$, $\partial^+$, id and $*$. For example, consider the
By Axiom (T0), it implies that $X_n \cap Y_n = \emptyset$. Similarly, $X_n^- \cap Y_n^- = \emptyset$, so $X_n \cup Y_n$ is fork-free. For $X_{n-1}Y$ to be a cell, $X_n \cup Y_n$ must move $X_{n-1,-}$ to $Y_{n-1,+}$. But, since $X$ and $Y$ are cells and are $(n-1)$-composable, we know that $X_n$ moves $X_{n-1,-}$ to $X_{n-1,+}$, $Y_n$ moves $Y_{n-1,-}$ to $Y_{n-1,+}$ and $X_{n-1,+} = Y_{n-1,-}$. Since $X_n^- \cap Y_n^+ = \emptyset$, using Lemma A.1.3, we get that $X_n \cup Y_n$ moves $X_{n-1,-}$ to $Y_{n-1,+}$. Hence, $X_{n-1}Y$ is a cell. $\square$

We now handle the general case of compositions of cells:

**Lemma 2.2.2.** Let $P$ be an $\omega$-hypergraph satisfying Axioms (T0), (T1), (T2) and (T3). Let $i, n \in \mathbb{N}$ with $i < n$ and $X, Y$ be two $n$-cells of $P$ that are $i$-composable. Then,

(i) for $j \in \mathbb{N}$ with $i < j \le n$, $(X_{j,-}^- \cup X_{j,+}^-) \cap (Y_{j,-}^+ \cup Y_{j,+}^+) = \emptyset$,

(ii) $X_{n-1}Y$ is a cell.

*Proof.* By induction on $n-i$. If $n-i=1$, the properties follow from Lemma 2.2.1. So suppose that $n-i>1$. For $\epsilon, \eta \in \{-,+\}$, by induction hypothesis with $\partial_{n-1}^\epsilon(X)$ and $\partial_{n-1}^\eta(Y)$, we get $X_{n-1,\epsilon}^- \cap Y_{n-1,\eta}^+ = \emptyset$. Therefore, $(X_{n-1,-}^- \cup X_{n-1,+}^-) \cap (Y_{n-1,-}^+ \cup Y_{n-1,+}^+) = \emptyset$. We moreover obtain that $(X_{j,-}^- \cup X_{j,+}^-) \cap (Y_{j,-}^+ \cup Y_{j,+}^+) = \emptyset$ for $j \in \mathbb{N}$ with $i < j < n-1$. Let $Z = \partial_{n-1}^+(X)_{*i} \partial_{n-1}^-(Y)$. By induction, $Z$ is a $(n-1)$-cell and $Z_{n-1} = X_{n-1,+} \cup Y_{n-1,-}$. Using Theorem 2.1.1(c), we get that $X_n^- \cap Y_n^+ = \emptyset$ which concludes the proof of (i).

For (ii), we already know that $\partial_{n-1}^-(X)_{*i} \partial_{n-1}^-(Y)$ and $\partial_{n-1}^+(X)_{*i} \partial_{n-1}^+(Y)$ are cells by induction. So, in order to prove that $X_{n-1}Y$ is a cell, we just need to show that $X_n \cup Y_n$ is fork-free and moves $X_{n-1,-} \cup Y_{n-1,-}$ to $X_{n-1,+} \cup Y_{n-1,+}$. But

$$ 
\begin{align*}
X_n^+ \cap Y_n^+ &= X_n^\pm \cap Y_n^+ \tag{by (i)} \\
&\subseteq Z_{n-1} \cap Y_n^+ \tag{by Theorem 2.1.1(a)} \\
&= \emptyset
\end{align*}
$$

and similarly, $X_n^- \cap Y_n^- = \emptyset$, so $X_n \cup Y_n$ is fork-free. Using the dual of Theorem 2.1.1(a) with $Z$ and $X_n$, we get

$$ X_n^- \cap (X_{n-1,+} \cup Y_{n-1,-}) = X_n^- \cap Y_{n-1,-} = \emptyset. $$

Similarly, if $Z' = \partial_{n-1}^-(X)_{*i} \partial_{n-1}^-(Y)$ then $Z'_{n-1} = X_{n-1,-} \cup Y_{n-1,-}$. Using Theorem 2.1.1(a) with $Z'$ and $X_n$, we have

$$ X_n^+ \cap (X_{n-1,-} \cup Y_{n-1,-}) = X_n^+ \cap Y_{n-1,-} = \emptyset. $$

Since $X_n$ moves $X_{n-1,-}$ to $X_{n-1,+}$, using Lemma A.1.2, we deduce that $X_n$ moves $X_{n-1,-} \cup Y_{n-1,-}$ to $X_{n-1,+} \cup Y_{n-1,-}$. Similarly, $Y_n$ moves $X_{n-1,+} \cup Y_{n-1,-}$ to $X_{n-1,+} \cup Y_{n-1,+}$. Since $X_n^- \cap Y_n^+ = \emptyset$, by Lemma A.1.3, we have that $X_n \cup Y_n$ moves $X_{n-1,-} \cup Y_{n-1,-}$ to $X_{n-1,+} \cup Y_{n-1,+}$. Hence, $X_{n-1}Y$ is a cell. $\square$

We can finally conclude that the $\omega$-category of cells has a structure of $\omega$-category given by the identity and composition operations on cells:

**Theorem 2.2.3.** Let $P$ be a torsion-free complex. (Cell($P$), $\partial^-$, $\partial^+$, id, *) is an $\omega$-category.

*Proof.* We already know that Cell($P$) is a $\omega$-globular set. By Lemma 2.2.2, the composition operation $*$ is well-defined on composable cells. Moreover, all the axioms of $\omega$-categories (given in Section 1.2), follow readily from the definitions of $\partial^-$, $\partial^+$, id and $*$. For example, consider the
exchange law Axiom (S-v). Given $i, j, n \in \mathbb{N}$ with $i < j \le n$ and $X, X', Y, Y' \in \text{Cell}(P)_n$ such that $X, Y$ are $i$-composable, $X, X'$ are $j$-composable and $Y, Y'$ are $j$-composable, let

$$Z = (X *_j Y) *_i (X' *_j Y') \quad \text{and} \quad Z' = (X *_i X') *_j (Y *_i Y').$$

One then easily verifies that $Z_{k,\epsilon} = Z'_{k,\epsilon}$ for $k \le n$ and $\epsilon \in \{-, +\}$, so $Z = Z'$. Thus, $\text{Cell}(P)$ satisfies Axiom (S-v), and the other axioms are shown as easily. $\square$

**Remark 2.2.4.** For the proof of Theorem 2.2.3, we did not use Axiom (T4), so that the same property holds for an $\omega$-hypergraph which only satisfies Axioms (T0), (T1), (T2), (T3).

**2.3 The notion of freeness** Our aim is to show that the $\omega$-category of cells on a torsion-free complex is “free” on the generators of the $\omega$-hypergraph. We now give a precise sense to the notion of freeness that we want to use. It is based on the structure of polygraph [22, 4]: the latter describes generators of strict categories of multiple dimensions, whose sources and targets are composite of generators of lower dimensions. We recall its definition following [4], using the intermediate notion of cellular extension. The latter describes sets of $(n+1)$-generators specified on strict $n$-categories. A polygraph is then simply a tower of cellular extensions.

**Cellular extensions** Given $n \in \mathbb{N}$, an $n$-cellular extension is a pair $(C, S)$ where $C$ is an $n$-category and $S$ is a set, together with two functions

$$d_n^-, d_n^+ : S \to C_n$$

such that, when $n > 0$, $\partial_{n-1}^\epsilon \circ d_n^- = \partial_{n-1}^\epsilon \circ d_n^+$ for $\epsilon \in \{-, +\}$. The set $S$ is to be considered as a set of $(n+1)$-generators. Given two $n$-cellular extensions $(C, S)$ and $(C', S')$, a morphism between $(C, S)$ and $(C', S')$ is a pair $(F, f)$ where

$$F: C \to C' \in \mathbf{Cat}_n \quad \text{and} \quad f: S \to S' \in \mathbf{Set}$$

and such that $d_n^\epsilon \circ f = F_n \circ d_n^\epsilon$ for $\epsilon \in \{-, +\}$. We write $\mathbf{Cat}_n^+$ for the category of $n$-cellular extensions. There is a canonical functor

$$\mathcal{V}_n: \mathbf{Cat}_{n+1} \to \mathbf{Cat}_n^+$$

which forgets the operations on the $(n+1)$-cells except the globular ones, and we have that

**Proposition 2.3.1.** *The functor $\mathcal{V}_n$ admits a left adjoint.*

*Proof.* By the equational definitions of strict categories, the functor $\mathcal{V}_n$ is a functor induced by a morphism of sketches and thus has a left adjoint (see for example [1, Theorem 3.5]). $\square$

We write

$$-[\cdot]^n: \mathbf{Cat}_n^+ \to \mathbf{Cat}_{n+1}$$

for such a left adjoint, or even $-[\cdot]$ when there is no ambiguity on $n$. This functor maps an $n$-cellular extension $(C, S)$ to its free extension $C[S]$. In fact, the functor $-[\cdot]$ can be chosen so that the canonical morphism

$$C \to C[S]_{\le n}$$

is the identity for every $(C, S) \in \mathbf{Cat}_n^+$. The reason is that the theory of strict categories is *truncable* in the sense of [2] (see for example [7, Proposition 1.3.2.10]). Given $(C, S) \in \mathbf{Cat}_n^+$ and $g \in S$, we often abuse notation and write $g$ for the embedding of $g$ in $C[S]$.
exchange law Axiom (S-v). Given $i, j, n \in \mathbb{N}$ with $i < j \le n$ and $X, X', Y, Y' \in \text{Cell}(P)_n$ such that $X, Y$ are $i$-composable, $X, X'$ are $j$-composable and $Y, Y'$ are $j$-composable, let

$$Z = (X *_j Y) *_i (X' *_j Y') \quad \text{and} \quad Z' = (X *_i X') *_j (Y *_i Y').$$

One then easily verifies that $Z_{k,\epsilon} = Z'_{k,\epsilon}$ for $k \le n$ and $\epsilon \in \{-, +\}$, so $Z = Z'$. Thus, $\text{Cell}(P)$ satisfies Axiom (S-v), and the other axioms are shown as easily. $\square$

**Remark 2.2.4.** For the proof of Theorem 2.2.3, we did not use Axiom (T4), so that the same property holds for an $\omega$-hypergraph which only satisfies Axioms (T0), (T1), (T2), (T3).

**2.3 The notion of freeness** Our aim is to show that the $\omega$-category of cells on a torsion-free complex is “free” on the generators of the $\omega$-hypergraph. We now give a precise sense to the notion of freeness that we want to use. It is based on the structure of polygraph [22, 4]: the latter describes generators of strict categories of multiple dimensions, whose sources and targets are composite of generators of lower dimensions. We recall its definition following [4], using the intermediate notion of cellular extension. The latter describes sets of $(n+1)$-generators specified on strict $n$-categories. A polygraph is then simply a tower of cellular extensions.

**Cellular extensions** Given $n \in \mathbb{N}$, an $n$-cellular extension is a pair $(C, S)$ where $C$ is an $n$-category and $S$ is a set, together with two functions

$$d_n^-, d_n^+ : S \to C_n$$

such that, when $n > 0$, $\partial_{n-1}^\epsilon \circ d_n^- = \partial_{n-1}^\epsilon \circ d_n^+$ for $\epsilon \in \{-, +\}$. The set $S$ is to be considered as a set of $(n+1)$-generators. Given two $n$-cellular extensions $(C, S)$ and $(C', S')$, a morphism between $(C, S)$ and $(C', S')$ is a pair $(F, f)$ where

$$F: C \to C' \in \mathbf{Cat}_n \quad \text{and} \quad f: S \to S' \in \mathbf{Set}$$

and such that $d_n^\epsilon \circ f = F_n \circ d_n^\epsilon$ for $\epsilon \in \{-, +\}$. We write $\mathbf{Cat}_n^+$ for the category of $n$-cellular extensions. There is a canonical functor

$$\mathcal{V}_n: \mathbf{Cat}_{n+1} \to \mathbf{Cat}_n^+$$

which forgets the operations on the $(n+1)$-cells except the globular ones, and we have that

**Proposition 2.3.1.** The functor $\mathcal{V}_n$ admits a left adjoint.

*Proof.* By the equational definitions of strict categories, the functor $\mathcal{V}_n$ is a functor induced by a morphism of sketches and thus has a left adjoint (see for example [1, Theorem 3.5]). $\square$

We write

$$-[\cdot]^n: \mathbf{Cat}_n^+ \to \mathbf{Cat}_{n+1}$$

for such a left adjoint, or even $-[\cdot]$ when there is no ambiguity on $n$. This functor maps an $n$-cellular extension $(C, S)$ to its free extension $C[S]$. In fact, the functor $-[\cdot]$ can be chosen so that the canonical morphism

$$C \to C[S]_{\le n}$$

is the identity for every $(C, S) \in \mathbf{Cat}_n^+$. The reason is that the theory of strict categories is *truncable* in the sense of [2] (see for example [7, Proposition 1.3.2.10]). Given $(C, S) \in \mathbf{Cat}_n^+$ and $g \in S$, we often abuse notation and write $g$ for the embedding of $g$ in $C[S]$.
**Polygraphs** For $n \in \mathbb{N}$, we define inductively on $n$ the notion of an $n$-polygraph $P$ together with a free $n$-category $P^*$ on $P$:

- a 0-polygraph $P$ is a set $P_0$ and the free 0-category on $P^*$ is $P_0$ (seen as a 0-category),

- an $(n+1)$-polygraph $P$ is given by an $n$-polygraph $P_{\le n}$ together with an $n$-cellular extension $(P_{\le n}, P_{n+1})$ and the free $(n+1)$-category $P^*$ on $P$ is the free extension $(P_{\le n})^*[P_{n+1}]$.

By induction on $n$, we naturally define a notion of morphism between $n$-polygraphs: a morphism between 0-polygraphs is simply a function between sets, and a morphism of $(n+1)$-polygraphs is the data of a morphism between the underlying $n$-polygraphs together with a morphism between the underlying $n$-cellular extensions. Thus, for every $n \in \mathbb{N}$ we obtain a category $\mathbf{Pol}_n$ of $n$-polygraphs, and a functor

$$(-)^*: \mathbf{Pol}_n \to \mathbf{Cat}_n.$$

*Remark 2.3.2.* An $n$-polygraph $P$ can alternatively be described as a diagram in **Set** of the form

<figure><img src="image_12.png" /></figure>

where, for $i \in \mathbb{N}_{n-1}$, $P_i^*$ is the set of cells freely generated on the generators of dimensions $\le i$ with associated embedding $e_i: P_i \to P_i^*$, and such that

$$\partial_i^- \circ d_{i+1}^- = \partial_i^- \circ d_{i+1}^+ \quad \text{and} \quad \partial_i^+ \circ d_{i+1}^- = \partial_i^+ \circ d_{i+1}^+$$

for $i \in \mathbb{N}_{n-1}$. This description of polygraphs can already be found in the original paper of Burroni [4].

These constructions naturally extend to $\omega$: an $\omega$-polygraph $P$ is a sequence $(P^k)_{k \ge 0}$ where $P^k$ is a $k$-polygraph such that $(P^{k+1})_{\le k} = P^k$ and the free $\omega$-category on $P$ is defined by

$$P^* = \colim_{k \to \omega} ((P^k)^*)_{\uparrow \omega, k}.$$

where, for every $k \in \mathbb{N}$,

$$(-)_{\uparrow \omega, k}: \mathbf{Cat}_k \to \mathbf{Cat}_{\omega}$$

is the left adjoint to the truncation functor $(-)_{\le k}: \mathbf{Cat}_{\omega} \to \mathbf{Cat}_k$. The notion of morphism of $\omega$-polygraph is defined as expected and we obtain a category $\mathbf{Pol}_{\omega}$ of $\omega$-polygraphs together with a functor $(-)^*: \mathbf{Pol}_{\omega} \to \mathbf{Cat}_{\omega}$.

**2.4 Freeness of the $\omega$-category of cells** Here, we prove that the $\omega$-category of cells on a torsion-free complex is free in the sense introduced previously, i.e., that it is isomorphic to the free $\omega$-category on a certain polygraph. For this purpose, we introduce the canonical cellular extensions from which this polygraph is built from, and show inductively that the adequate restrictions of the $\omega$-category of cells are isomorphic to the free extensions on these cellular extensions.
**Polygraphs** For $n \in \mathbb{N}$, we define inductively on $n$ the notion of an $n$-polygraph $P$ together with a free $n$-category $P^*$ on $P$:

- a 0-polygraph $P$ is a set $P_0$ and the free 0-category on $P^*$ is $P_0$ (seen as a 0-category),

- an $(n+1)$-polygraph $P$ is given by an $n$-polygraph $P_{\le n}$ together with an $n$-cellular extension $(P_{\le n}, P_{n+1})$ and the free $(n+1)$-category $P^*$ on $P$ is the free extension $(P_{\le n})^*[P_{n+1}]$.

By induction on $n$, we naturally define a notion of morphism between $n$-polygraphs: a morphism between 0-polygraphs is simply a function between sets, and a morphism of $(n+1)$-polygraphs is the data of a morphism between the underlying $n$-polygraphs together with a morphism between the underlying $n$-cellular extensions. Thus, for every $n \in \mathbb{N}$ we obtain a category $\mathbf{Pol}_n$ of $n$-polygraphs, and a functor

$$(-)^*: \mathbf{Pol}_n \to \mathbf{Cat}_n.$$

*Remark 2.3.2.* An $n$-polygraph $P$ can alternatively be described as a diagram in **Set** of the form

<figure><img src="image_13.png" /></figure>

where, for $i \in \mathbb{N}_{n-1}$, $P_i^*$ is the set of cells freely generated on the generators of dimensions $\le i$ with associated embedding $e_i: P_i \to P_i^*$, and such that

$$\partial_i^- \circ d_{i+1}^- = \partial_i^- \circ d_{i+1}^+ \quad \text{and} \quad \partial_i^+ \circ d_{i+1}^- = \partial_i^+ \circ d_{i+1}^+$$

for $i \in \mathbb{N}_{n-1}$. This description of polygraphs can already be found in the original paper of Burroni [4].

These constructions naturally extend to $\omega$: an $\omega$-polygraph $P$ is a sequence $(P^k)_{k \ge 0}$ where $P^k$ is a $k$-polygraph such that $(P^{k+1})_{\le k} = P^k$ and the free $\omega$-category on $P$ is defined by

$$P^* = \colim_{k \to \omega} ((P^k)^*)_{\uparrow \omega, k}.$$

where, for every $k \in \mathbb{N}$,

$$(-)_{\uparrow \omega, k}: \mathbf{Cat}_k \to \mathbf{Cat}_\omega$$

is the left adjoint to the truncation functor $(-)_{\le k}: \mathbf{Cat}_\omega \to \mathbf{Cat}_k$. The notion of morphism of $\omega$-polygraph is defined as expected and we obtain a category $\mathbf{Pol}_\omega$ of $\omega$-polygraphs together with a functor $(-)^*: \mathbf{Pol}_\omega \to \mathbf{Cat}_\omega$.

**2.4 Freeness of the $\omega$-category of cells** Here, we prove that the $\omega$-category of cells on a torsion-free complex is free in the sense introduced previously, i.e., that it is isomorphic to the free $\omega$-category on a certain polygraph. For this purpose, we introduce the canonical cellular extensions from which this polygraph is built from, and show inductively that the adequate restrictions of the $\omega$-category of cells are isomorphic to the free extensions on these cellular extensions.
**The canonical cellular extension** Let $P$ be a torsion-free complex $P$. Given $n \in \mathbb{N}$, there is an $n$-cellular extension

$$ \mathrm{Cell}(P)_{\le n} \xleftarrow[\partial_n^+ \circ (-)]{\partial_n^- \circ (-)} P_{n+1} $$

where, for $x \in P_{n+1}$ and $\epsilon \in \{-, +\}$, $\partial_n^\epsilon \circ (-)(x) = \partial_n^\epsilon(\langle x \rangle)$, which is an $n$-cell by Axiom (T2). We write $\mathrm{Cell}(P)^{n+}$ for the $(n+1)$-category

$$ \mathrm{Cell}(P)^{n+} = \mathrm{Cell}(P)_{\le n}[P_{n+1}] $$

i.e., the image of $(\mathrm{Cell}(P)_{\le n}, P_{n+1}) \in \mathrm{Cat}_n^+$ by the functor $-[\cdot]^n: \mathrm{Cat}_n^+ \to \mathrm{Cat}_{n+1}$. There is a morphism of $n$-cellular extension

$$ (\mathrm{Cell}(P)_{\le n}, P_{n+1}) \xrightarrow{(\mathrm{id}_{\mathrm{Cell}(P)_{\le n}}, (-))} (\mathrm{Cell}(P)_{\le n}, \mathrm{Cell}(P)_{n+1}) $$

which maps $x \in P_{n+1}$ to $(-)(x) = \langle x \rangle$. By the universal property of $\mathrm{Cell}(P)^{n+}$ as a free extension, it induces a unique $(n+1)$-functor

$$ \mathrm{eval}^n: \mathrm{Cell}(P)^{n+} \to \mathrm{Cell}(P)_{\le n+1} $$

often written eval for conciseness, such that $\mathrm{eval}_{\le n}^n = \mathrm{id}_{\mathrm{Cell}(P)_{\le n}}$ and eval(g) = g for all g \in P_{n+1}$.

**Freeness of Cell($P$)** We can now assert the freeness of the $\omega$-category $\mathrm{Cell}(P)$. First, we show that it is inductively built from the canonical free extensions:

**Theorem 2.4.1.** Given a torsion-free complex $P$, for $n \in \mathbb{N}$, the $(n+1)$-functor eval$^n$ is an isomorphism between $\mathrm{Cell}(P)^{n+}$ and $\mathrm{Cell}(P)_{\le n+1}$.

*Proof.* See the proof in Appendix B.5. $\square$

<figure><img src="image_13.png" /></figure>

By an inductive argument, we conclude that the $\omega$-category of cells is freely generated on the $\omega$-polygraph made from the atoms:

**Corollary 2.4.2.** Given a torsion-free complex $P$, there are unique polygraph $\mathbb{Q} \in \mathrm{Pol}_\omega$ and $\omega$-functor

$$ F: \mathbb{Q}^* \to \mathrm{Cell}(P) \in \mathrm{Cat}_\omega $$

such that $\mathbb{Q}_n = P_n$ for $n \in \mathbb{N}$ and $F(g) = \langle g \rangle$ for $g \in P$. Moreover, $F$ is an isomorphism.

*Proof.* We show by induction on $n \in \mathbb{N}$ that there are unique $n$-polygraph $\mathbb{Q}^n$ and morphism

$$ F^n: (\mathbb{Q}^n)^* \to \mathrm{Cell}(P)_{\le n} $$

such that $\mathbb{Q}_k^n = P_k$ for $k \in \mathbb{N}$ and $F^n(g) = \langle g \rangle$ for $g \in \mathbb{Q}^n$, and that $F^n$ is moreover an isomorphism. This is clear for $n = 0$. So suppose that $n > 0$. If $\mathbb{Q}^n$ and $F^n$ as above exist, then, by the unicity property of the induction hypothesis, we have $\mathbb{Q}_{\le n-1}^n = \mathbb{Q}^{n-1}$ and $F_{\le n-1}^n = F^{n-1}$. The $n$-functor $F^n$ is then uniquely defined by the universal property of $(\mathbb{Q}^n)^* = (\mathbb{Q}^{n-1})^*[\mathbb{Q}^n]$ given by Proposition 2.3.1 knowing that $F^n(g) = \langle g \rangle$ for $g \in \mathbb{Q}_n^n$. Moreover, the $n$-polygraph structure on $\mathbb{Q}^n$ is unique since

$$ d_{n-1}^{\epsilon}(g) = (F^{n-1})^{-1} \circ \partial_{n-1}^{\epsilon}(\langle g \rangle) \quad (15) $$
**The canonical cellular extension** Let $P$ be a torsion-free complex $P$. Given $n \in \mathbb{N}$, there is an $n$-cellular extension

$$ \mathrm{Cell}(P)_{\le n} \xleftarrow[\partial_n^+ \circ (-)]{\partial_n^- \circ (-)} P_{n+1} $$

where, for $x \in P_{n+1}$ and $\epsilon \in \{-, +\}$, $\partial_n^\epsilon \circ (-)(x) = \partial_n^\epsilon(\langle x \rangle)$, which is an $n$-cell by Axiom (T2). We write $\mathrm{Cell}(P)^{n+}$ for the $(n+1)$-category

$$ \mathrm{Cell}(P)^{n+} = \mathrm{Cell}(P)_{\le n}[P_{n+1}] $$

i.e., the image of $(\mathrm{Cell}(P)_{\le n}, P_{n+1}) \in \mathrm{Cat}_n^+$ by the functor $-[\cdot]^n: \mathrm{Cat}_n^+ \to \mathrm{Cat}_{n+1}$. There is a morphism of $n$-cellular extension

$$ (\mathrm{Cell}(P)_{\le n}, P_{n+1}) \xrightarrow{(\mathrm{id}_{\mathrm{Cell}(P)_{\le n}}, (-))} (\mathrm{Cell}(P)_{\le n}, \mathrm{Cell}(P)_{n+1}) $$

which maps $x \in P_{n+1}$ to $(-)(x) = \langle x \rangle$. By the universal property of $\mathrm{Cell}(P)^{n+}$ as a free extension, it induces a unique $(n+1)$-functor

$$ \mathrm{eval}^n: \mathrm{Cell}(P)^{n+} \to \mathrm{Cell}(P)_{\le n+1} $$

often written eval for conciseness, such that $\mathrm{eval}^n_{\le n} = \mathrm{id}_{\mathrm{Cell}(P)_{\le n}}$ and eval($g$) = $g$ for all $g \in P_{n+1}$.

**Freeness of Cell($P$)** We can now assert the freeness of the $\omega$-category $\mathrm{Cell}(P)$. First, we show that it is inductively built from the canonical free extensions:

**Theorem 2.4.1.** Given a torsion-free complex $P$, for $n \in \mathbb{N}$, the $(n+1)$-functor eval$^n$ is an isomorphism between $\mathrm{Cell}(P)^{n+}$ and $\mathrm{Cell}(P)_{\le n+1}$.

*Proof.* See the proof in Appendix B.5. $\square$

<figure><img src="image_14.png" /></figure>

By an inductive argument, we conclude that the $\omega$-category of cells is freely generated on the $\omega$-polygraph made from the atoms:

**Corollary 2.4.2.** Given a torsion-free complex $P$, there are unique polygraph $\mathbb{Q} \in \mathrm{Pol}_\omega$ and $\omega$-functor

$$ F: \mathbb{Q}^* \to \mathrm{Cell}(P) \in \mathrm{Cat}_\omega $$

such that $\mathbb{Q}_n = P_n$ for $n \in \mathbb{N}$ and $F(g) = \langle g \rangle$ for $g \in P$. Moreover, $F$ is an isomorphism.

*Proof.* We show by induction on $n \in \mathbb{N}$ that there are unique $n$-polygraph $\mathbb{Q}^n$ and morphism

$$ F^n: (\mathbb{Q}^n)^* \to \mathrm{Cell}(P)_{\le n} $$

such that $\mathbb{Q}_k^n = P_k$ for $k \in \mathbb{N}$ and $F^n(g) = \langle g \rangle$ for $g \in \mathbb{Q}^n$, and that $F^n$ is moreover an isomorphism. This is clear for $n = 0$. So suppose that $n > 0$. If $\mathbb{Q}^n$ and $F^n$ as above exist, then, by the unicity property of the induction hypothesis, we have $\mathbb{Q}_{\le n-1}^n = \mathbb{Q}^{n-1}$ and $F_{\le n-1}^n = F^{n-1}$. The $n$-functor $F^n$ is then uniquely defined by the universal property of $(\mathbb{Q}^n)^* = (\mathbb{Q}^{n-1})^*[\mathbb{Q}^n]$ given by Proposition 2.3.1 knowing that $F^n(g) = \langle g \rangle$ for $g \in \mathbb{Q}_n^n$. Moreover, the $n$-polygraph structure on $\mathbb{Q}^n$ is unique since

$$ d_{n-1}^\epsilon(g) = (F^{n-1})^{-1} \circ \partial_{n-1}^\epsilon(\langle g \rangle) \quad (15) $$
for $g \in \mathbb{Q}_n^n$ and $\epsilon \in \{-, +\}$. Finally, $F^n$ is an isomorphism since, by Theorem 2.4.1, the func- tor $(\mathrm{eval}^{n-1})^{-1} \circ F^n$ is the image by $-[\cdot]^{n-1}$ of the isomorphism

$$ (F^{n-1}, 1_{P_n}): ((\mathbb{Q}^{n-1})^*, \mathbb{Q}_n^n) \to (\mathrm{Cell}(P)_{\le n-1}, P_n) \in \mathrm{Cat}_{n-1}^+ $$

so that the unicity of $\mathbb{Q}^n$ and $F^n$, and the fact that $F^n$ is an isomorphism are proved. For existence, one defines the $n$-polygraph structure on $\mathbb{Q}^n$ from the one on $\mathbb{Q}^{n-1}$ and with (15), and the $n$-functor $F^n$ is then defined by extending $F^{n-1}$, using the universal property of $(\mathbb{Q}^n)^*$. By the definition of $\mathrm{Pol}_\omega$, we obtain unique $\omega$-polygraph $\mathbb{Q}$ together with a unique $\omega$-functor $F: \mathbb{Q}^* \to \mathrm{Cell}(P)$ as wanted. $\square$

### 3. Relating formalisms

In this section, we relate all the introduced formalisms together. In particular, we show that the formalism of torsion-free complexes is a Rosetta stone which can express the other ones (after correcting the defect of parity complexes and pasting schemes). Embedding parity complexes into torsion-free complexes is almost direct, since they share the same definition of cells and several axioms. However, additional developments are needed for translating pasting schemes and augmented directed complexes into torsion-free complexes. Indeed, in the first case, one needs to show that a definition of cells analogous to the ones of pasting schemes can be used for torsion-free complexes before being able to relate the axioms of the two formalisms. In the second case, one needs to link the abelian group setting of augmented directed complexes to the set setting of torsion-free complexes.

We first introduce two other set-based definitions of cells for torsion-free complexes: closed-well-formed fgs's and maximal-well-formed fgs's (Section 3.1). The former is similar to the well-formed fgs of pasting schemes, while the latter is a convenient intermediate between the cells of torsion-free complexes and closed-well-formed fgs's. The $\omega$-categories of cells induced by these two other definitions is then isomorphic to the one obtained with the initial definition (Theorems 3.1.18 and 3.1.21). Using the more natural definition of cells as closed-well-formed fgs's, we give a characterization of polygraphs that can be represented by torsion-free complex (Theorem 3.1.22). Next, we show the embeddings of parity complexes (Section 3.2) and pasting schemes (Section 3.3) into torsion-free complexes. Then, we develop the relation between the set-based and group-based definitions of cells before showing the embedding augmented directed complexes into torsion-free complexes (Section 3.4). Finally, we illustrate that those are the only embeddings between the formalisms by providing counter-examples to the other ones (Section 3.5).

#### 3.1 Closed and maximal cells

In this section, we introduce two other set-based definitions of cells for torsion-free complexes, namely closed-well-formed fgs's and maximal-well-formed fgs's, together with identity and compositions operations for them. We moreover provide translation functions between the different definitions of cells, and show that the $\omega$-categories of cells with the new definitions are isomorphic to the one with the original definition of cells (Theorems 3.1.18 and 3.1.21). Finally, using this different representation, we characterize the polygraphs that can be represented by torsion-free complexes (Theorem 3.1.22).
for $g \in \mathbb{Q}_n^n$ and $\epsilon \in \{-, +\}$. Finally, $F^n$ is an isomorphism since, by Theorem 2.4.1, the func- tor $(\mathrm{eval}^{n-1})^{-1} \circ F^n$ is the image by $-[\cdot]^{n-1}$ of the isomorphism

$$ (F^{n-1}, 1_{P_n}): ((\mathbb{Q}^{n-1})^*, \mathbb{Q}_n^n) \to (\mathrm{Cell}(P)_{\le n-1}, P_n) \in \mathrm{Cat}_{n-1}^+ $$

so that the unicity of $\mathbb{Q}^n$ and $F^n$, and the fact that $F^n$ is an isomorphism are proved. For existence, one defines the $n$-polygraph structure on $\mathbb{Q}^n$ from the one on $\mathbb{Q}^{n-1}$ and with (15), and the $n$-functor $F^n$ is then defined by extending $F^{n-1}$, using the universal property of $(\mathbb{Q}^n)^*$. By the definition of $\mathrm{Pol}_\omega$, we obtain unique $\omega$-polygraph $\mathbb{Q}$ together with a unique $\omega$-functor $F: \mathbb{Q}^* \to \mathrm{Cell}(P)$ as wanted. $\square$

### 3. Relating formalisms

In this section, we relate all the introduced formalisms together. In particular, we show that the formalism of torsion-free complexes is a Rosetta stone which can express the other ones (after correcting the defect of parity complexes and pasting schemes). Embedding parity complexes into torsion-free complexes is almost direct, since they share the same definition of cells and several axioms. However, additional developments are needed for translating pasting schemes and augmented directed complexes into torsion-free complexes. Indeed, in the first case, one needs to show that a definition of cells analogous to the ones of pasting schemes can be used for torsion-free complexes before being able to relate the axioms of the two formalisms. In the second case, one needs to link the abelian group setting of augmented directed complexes to the set setting of torsion-free complexes.

We first introduce two other set-based definitions of cells for torsion-free complexes: closed-well-formed fgs's and maximal-well-formed fgs's (Section 3.1). The former is similar to the well-formed fgs of pasting schemes, while the latter is a convenient intermediate between the cells of torsion-free complexes and closed-well-formed fgs's. The $\omega$-categories of cells induced by these two other definitions is then isomorphic to the one obtained with the initial definition (Theorems 3.1.18 and 3.1.21). Using the more natural definition of cells as closed-well-formed fgs's, we give a characterization of polygraphs that can be represented by torsion-free complex (Theorem 3.1.22). Next, we show the embeddings of parity complexes (Section 3.2) and pasting schemes (Section 3.3) into torsion-free complexes. Then, we develop the relation between the set-based and group-based definitions of cells before showing the embedding augmented directed complexes into torsion-free complexes (Section 3.4). Finally, we illustrate that those are the only embeddings between the formalisms by providing counter-examples to the other ones (Section 3.5).

#### 3.1 Closed and maximal cells

In this section, we introduce two other set-based definitions of cells for torsion-free complexes, namely closed-well-formed fgs's and maximal-well-formed fgs's, together with identity and compositions operations for them. We moreover provide translation functions between the different definitions of cells, and show that the $\omega$-categories of cells with the new definitions are isomorphic to the one with the original definition of cells (Theorems 3.1.18 and 3.1.21). Finally, using this different representation, we characterize the polygraphs that can be represented by torsion-free complexes (Theorem 3.1.22).
**Definitions** Let $P$ be an $\omega$-hypergraph. Recall the definitions of fgs and closed fgs from Section 1.5. We write

$$ \text{Closed}(P) $$

for the graded set of closed fgs's of $P$. Given an $n$-fgs $X$ of $P$, $x \in X$ is said to be maximal in $X$ when for all $y \in P$ such that $x \R y$ and $x \neq y$, it holds that $y \notin X$. We write $\max(X)$ for the $n$-fgs of $P$ made of the maximal elements of $X$. The $n$-fgs $X$ is then said to be maximal when $\max(X) = X$. We write

$$ \text{Max}(P) $$

for the graded set of maximal fgs. Given $n \in \mathbb{N}$ and $X$ an $n$-pre-cell of $P$, we write $\cup X$ for the $n$-fgs of $P$ given by

$$ \cup X = \bigcup_{i \in \mathbb{N}_n} (X_{i,-} \cup X_{i,+}). $$

**Maximality lemma** Let $P$ be an $\omega$-hypergraph. In order to relate the cells of $\text{Cell}(P)$ with the fgs's of $\text{Max}(P)$, we give here a simple criterion to characterize the maximal elements in a cell of $\text{Cell}(P)$:

**Lemma 3.1.1 (Maximality lemma).** Suppose that $P$ satisfies Axioms (T0), (T1), (T2) and (T3). Let $k, n \in \mathbb{N}$ with $k < n$ and $X \in \text{Cell}(P)_n$. For $x \in X_{k,-}$ (resp. $x \in X_{k,+}$) with $x$ not maximal in $\cup X$, we have $x \in X_{k+1,-}^{\pm}$ (resp. $x \in X_{k+1,+}^{\pm}$).

*Proof.* We prove this property by induction on $l = n - k$. By symmetry, we only prove the case where $x \in X_{k,-}$. Since $x$ is not maximal, by definition of $\mathbb{R}$, there exist

$$ p \in \mathbb{N}^*, \quad \eta \in \{-, +\}, \quad x_0, x_1, \dots, x_p \in P \quad \text{and} \quad \epsilon_1, \dots, \epsilon_p \in \{-, +\} $$

such that

$$ x_0 = x, \quad x_p \in X_{k+p,\eta} \quad \text{and} \quad x_i \in x_{i+1}^{\epsilon_{i+1}} \quad \text{for } i \in \mathbb{N}_{p-1}. $$

Suppose that $p = 1$. By Lemma A.1.1, we have $X_{k,-} \cap X_{k+1,\eta}^{+} = \emptyset$. Since $x \in x_1^{\epsilon_1}$ and $x_1 \in X_{k+1,\eta}$, we have $\epsilon_1 = -$ and $x \in X_{k+1,\eta}^{\pm}$. Hence, by Lemma A.1.5, $x \in X_{k+1,-}^{\pm}$.

Otherwise, suppose that $p > 1$. Let $y \in X_{k+p,\eta}$ be the smallest of $X_{k+p,\eta}$ for $\langle X_{k+p,\eta} \rangle$ such that $y \R x_{p-1}$. If $x_{p-1} \in y^{-}$, then, by minimality of $y$, there is no $\bar{y} \in X_{k+p,\eta}$ such that $x_{p-1} \in \bar{y}^{+}$. Therefore, $x_{p-1} \in X_{k+p,\eta}^{\pm} \subseteq X_{k+p-1,-}$. Hence, $x$ is not minimal in $\partial_{k+p-1}^{-}(X)$ and we conclude by induction. We now consider the case $x_{p-1} \in y^{+}$. Let

$$ G = \{z \in X_{k+p,\eta} \mid z \trianglelefteq_{X_{k+p,\eta}} y\} \cup \{y\} \quad \text{and} \quad Y = \text{Act}(\partial_{k+p-1}^{-}(X), G). $$

We have $x \in Y_{k,-}$ and $x_{p-1} \in Y_{k+p-1}$. Moreover, by Theorem 2.1.1, $Y$ is a cell. By induction hypothesis, we have $x \in Y_{k+1,-}^{\pm}$. Since $X_{k+1,-}$ and $Y_{k+1,-}$ both move $X_{k,-}$ to $X_{k,+}$, by Lemma A.1.5, we have $x \in X_{k+1,-}^{\pm}$ which concludes the proof. $\square$

We then have a simple description of the set of maximal elements of a cell of $\text{Cell}(P)$:

**Lemma 3.1.2.** Suppose that $P$ satisfies Axioms (T0), (T1), (T2) and (T3). Let $k, n \in \mathbb{N}$ with $k < n$, an $n$-cell $X \in \text{Cell}(P)_n$ and $\epsilon \in \{-, +\}$. Then, $\max(\cup X) \cap P_k = X_{k,-} \cap X_{k,+}$.

*Proof.* By Lemmas 3.1.1 and A.1.6,

$$ \max(\cup X) \cap P_k = (X_{k,-} \setminus X_{k+1,-}^{\pm}) \cup (X_{k,+} \setminus X_{k+1,+}^{\pm}) = X_{k,-} \cap X_{k,+}. \quad \square $$
**Definitions** Let $P$ be an $\omega$-hypergraph. Recall the definitions of fgs and closed fgs from Section 1.5. We write

$$ \mathrm{Closed}(P) $$

for the graded set of closed fgs's of $P$. Given an $n$-fgs $X$ of $P$, $x \in X$ is said to be maximal in $X$ when for all $y \in P$ such that $x \R y$ and $x \neq y$, it holds that $y \notin X$. We write $\max(X)$ for the $n$-fgs of $P$ made of the maximal elements of $X$. The $n$-fgs $X$ is then said to be maximal when $\max(X) = X$. We write

$$ \mathrm{Max}(P) $$

for the graded set of maximal fgs. Given $n \in \mathbb{N}$ and $X$ an $n$-pre-cell of $P$, we write $\cup X$ for the $n$-fgs of $P$ given by

$$ \cup X = \bigcup_{i \in \mathbb{N}_n} (X_{i,-} \cup X_{i,+}). $$

**Maximality lemma** Let $P$ be an $\omega$-hypergraph. In order to relate the cells of $\mathrm{Cell}(P)$ with the fgs's of $\mathrm{Max}(P)$, we give here a simple criterion to characterize the maximal elements in a cell of $\mathrm{Cell}(P)$:

**Lemma 3.1.1 (Maximality lemma).** Suppose that $P$ satisfies Axioms (T0), (T1), (T2) and (T3). Let $k, n \in \mathbb{N}$ with $k < n$ and $X \in \mathrm{Cell}(P)_n$. For $x \in X_{k,-}$ (resp. $x \in X_{k,+}$) with $x$ not maximal in $\cup X$, we have $x \in X_{k+1,-}^{\pm}$ (resp. $x \in X_{k+1,+}^{\pm}$).

*Proof.* We prove this property by induction on $l = n - k$. By symmetry, we only prove the case where $x \in X_{k,-}$. Since $x$ is not maximal, by definition of $\mathbb{R}$, there exist

$$ p \in \mathbb{N}^*, \quad \eta \in \{-, +\}, \quad x_0, x_1, \dots, x_p \in P \quad \text{and} \quad \epsilon_1, \dots, \epsilon_p \in \{-, +\} $$

such that

$$ x_0 = x, \quad x_p \in X_{k+p,\eta} \quad \text{and} \quad x_i \in x_{i+1}^{\epsilon_{i+1}} \quad \text{for } i \in \mathbb{N}_{p-1}. $$

Suppose that $p = 1$. By Lemma A.1.1, we have $X_{k,-} \cap X_{k+1,\eta}^{+} = \emptyset$. Since $x \in x_1^{\epsilon_1}$ and $x_1 \in X_{k+1,\eta}$, we have $\epsilon_1 = -$ and $x \in X_{k+1,\eta}^{\pm}$. Hence, by Lemma A.1.5, $x \in X_{k+1,-}^{\pm}$.

Otherwise, suppose that $p > 1$. Let $y \in X_{k+p,\eta}$ be the smallest of $X_{k+p,\eta}$ for $\langle X_{k+p,\eta} \rangle$ such that $y \R x_{p-1}$. If $x_{p-1} \in y^{-}$, then, by minimality of $y$, there is no $\bar{y} \in X_{k+p,\eta}$ such that $x_{p-1} \in \bar{y}^{+}$. Therefore, $x_{p-1} \in X_{k+p,\eta}^{\pm} \subseteq X_{k+p-1,-}$. Hence, $x$ is not minimal in $\partial_{k+p-1}^{-}(X)$ and we conclude by induction. We now consider the case $x_{p-1} \in y^{+}$. Let

$$ G = \{z \in X_{k+p,\eta} \mid z \trianglelefteq_{X_{k+p,\eta}} y\} \cup \{y\} \quad \text{and} \quad Y = \mathrm{Act}(\partial_{k+p-1}^{-}(X), G). $$

We have $x \in Y_{k,-}$ and $x_{p-1} \in Y_{k+p-1}$. Moreover, by Theorem 2.1.1, $Y$ is a cell. By induction hypothesis, we have $x \in Y_{k+1,-}^{\pm}$. Since $X_{k+1,-}$ and $Y_{k+1,-}$ both move $X_{k,-}$ to $X_{k,+}$, by Lemma A.1.5, we have $x \in X_{k+1,-}^{\pm}$ which concludes the proof. $\square$

We then have a simple description of the set of maximal elements of a cell of $\mathrm{Cell}(P)$:

**Lemma 3.1.2.** Suppose that $P$ satisfies Axioms (T0), (T1), (T2) and (T3). Let $k, n \in \mathbb{N}$ with $k < n$, an $n$-cell $X \in \mathrm{Cell}(P)_n$ and $\epsilon \in \{-, +\}$. Then, $\max(\cup X) \cap P_k = X_{k,-} \cap X_{k,+}$.

*Proof.* By Lemmas 3.1.1 and A.1.6,

$$ \max(\cup X) \cap P_k = (X_{k,-} \setminus X_{k+1,-}^{\pm}) \cup (X_{k,+} \setminus X_{k+1,+}^{\pm}) = X_{k,-} \cap X_{k,+}. \quad \square $$
**The translation functions** We now provide translation functions between the three graded sets Cell($P$), Max($P$) and Closed($P$) and introduce several properties on them. The functions we introduce are the ones represented on the diagram

<figure><img src="image_14.png" /></figure>

and are defined as follows:

- $T_M^{PC}$: PCell($P$) → Max($P$) is defined by $T_M^{PC}(X) = \max(\cup X)$ for $X \in PCell(P)$,

- $T_{PC}^M$: Max($P$) → PCell($P$) is such that, for $n \in \mathbb{N}$ and $X \in \text{Max}(P)$, $T_{PC}^M(X)$ is the $n$-pre-cell $Y$ of $P$ defined by $Y_n = X_n$, and, for $i \in \mathbb{N}_{n-1}$,

$$Y_{i,-} = X_i \cup Y_{i+1,-}^{\mp} \qquad Y_{i,+} = X_i \cup Y_{i+1,+}^{\pm},$$

- $T_{Cl}^M$: Max($P$) → Closed($P$) is defined by $T_{Cl}^M(X) = R(X)$ for $X \in \text{Max}(P)$,

- $T_M^{Cl}$: Closed($P$) → Max($P$) is defined by $T_M^{Cl}(X) = \max(X)$ for $X \in \text{Closed}(P)$,

- $T_{Cl}^{PC}$: PCell($P$) → Closed($P$) is defined by $T_{Cl}^{PC}(X) = R(\cup X)$ for $X \in PCell(P)$,

- $T_{PC}^{Cl}$: Closed($P$) → PCell($P$) is defined by $T_{PC}^{Cl} = T_{PC}^M \circ T_M^{Cl}$.

These operations can be related to each other, as state the following lemmas.

**Proposition 3.1.3.** We have $T_{Cl}^M \circ T_M^{Cl} = 1_{\text{Closed}(P)}$ and $T_M^{Cl} \circ T_M^M = 1_{\text{Max}(P)}$.

*Proof.* Let $X \in \text{Closed}(P)$. By the definitions, we have $T_{Cl}^M \circ T_M^{Cl}(X) \subseteq X$. Moreover, given $x \in X$, since $X$ is finite, there exists $y \in \max(X)$ with $y \nmid x$. It implies that $y \in T_M^{Cl}(X)$ and $x \in T_M^M \circ T_M^{Cl}(X)$. Therefore, $X \subseteq T_{Cl}^M \circ T_M^{Cl}(X)$.

For the other equality, note that, for all $n$-fgs $X$ of $P$, $R(X)$ has the same maximal elements as $X$. Thus, $T_M^{Cl} \circ T_M^M = 1_{\text{Max}(P)}$. $\square$

**Lemma 3.1.4.** Suppose that $P$ satisfies Axioms (T0), (T1), (T2) and (T3). Let $n \in \mathbb{N}$, $X \in \text{Cell}(P)_n$ and $Y = T_M^{PC}(X)$. Then, $Y_n = X_n$ and $Y_i = X_{i,-} \cap X_{i,+}$ for $i \in \mathbb{N}_{n-1}$.

*Proof.* This is a direct consequence of Lemma 3.1.2. $\square$

**Proposition 3.1.5.** Suppose that $P$ satisfies Axioms (T0), (T1), (T2) and (T3). Then, given a cell $X \in \text{Cell}(P)$, we have $T_{PC}^M \circ T_M^{PC}(X) = X$.

*Proof.* Let $n \in \mathbb{N}$, $X \in \text{Cell}(P)_n$, $Y = T_M^{PC}(X)$ and $Z = T_{PC}^M(Y)$. For $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$, we show that $X_{i,\epsilon} = Z_{i,\epsilon}$ by a decreasing induction on $i$. By Lemma 3.1.4, we have $Z_n = Y_n = X_n$ and, for $i \in \mathbb{N}_{n-1}$, by Lemma A.1.7, we have

$$Z_{i,-} = Y_i \cup Z_{i+1,-}^{\mp} = (X_{i,-} \cap X_{i,+}) \cup X_{i+1,-}^{\mp} = X_{i,-}.$$

Similarly, $Z_{i,+} = X_{i,+}$, so $X = Z$. Hence, $T_{PC}^M \circ T_M^{PC}(X) = X$. $\square$

**Proposition 3.1.6.** We have $T_{Cl}^M \circ T_M^{PC} = T_{Cl}^{PC}$.

*Proof.* It readily follows from the definitions. $\square$
**The translation functions** We now provide translation functions between the three graded sets Cell($P$), Max($P$) and Closed($P$) and introduce several properties on them. The functions we introduce are the ones represented on the diagram

<figure><img src="image_15.png" /></figure>

and are defined as follows:

- $T_M^{PC}$: PCell($P$) → Max($P$) is defined by $T_M^{PC}(X) = \max(\cup X)$ for $X \in PCell(P)$,

- $T_{PC}^M$: Max($P$) → PCell($P$) is such that, for $n \in \mathbb{N}$ and $X \in \text{Max}(P)$, $T_{PC}^M(X)$ is the $n$-pre-cell $Y$ of $P$ defined by $Y_n = X_n$, and, for $i \in \mathbb{N}_{n-1}$,

$$Y_{i,-} = X_i \cup Y_{i+1,-}^{\mp} \qquad Y_{i,+} = X_i \cup Y_{i+1,+}^{\pm},$$

- $T_{Cl}^M$: Max($P$) → Closed($P$) is defined by $T_{Cl}^M(X) = R(X)$ for $X \in \text{Max}(P)$,

- $T_M^{Cl}$: Closed($P$) → Max($P$) is defined by $T_M^{Cl}(X) = \max(X)$ for $X \in \text{Closed}(P)$,

- $T_{Cl}^{PC}$: PCell($P$) → Closed($P$) is defined by $T_{Cl}^{PC}(X) = R(\cup X)$ for $X \in PCell(P)$,

- $T_{PC}^{Cl}$: Closed($P$) → PCell($P$) is defined by $T_{PC}^{Cl} = T_{PC}^M \circ T_M^{Cl}$.

These operations can be related to each other, as state the following lemmas.

**Proposition 3.1.3.** We have $T_{Cl}^M \circ T_M^{Cl} = 1_{\text{Closed}(P)}$ and $T_M^{Cl} \circ T_M^M = 1_{\text{Max}(P)}$.

*Proof.* Let $X \in \text{Closed}(P)$. By the definitions, we have $T_{Cl}^M \circ T_M^{Cl}(X) \subseteq X$. Moreover, given $x \in X$, since $X$ is finite, there exists $y \in \max(X)$ with $y \nmid x$. It implies that $y \in T_M^{Cl}(X)$ and $x \in T_M^M \circ T_M^{Cl}(X)$. Therefore, $X \subseteq T_{Cl}^M \circ T_M^{Cl}(X)$.

For the other equality, note that, for all $n$-fgs $X$ of $P$, $R(X)$ has the same maximal elements as $X$. Thus, $T_M^{Cl} \circ T_M^M = 1_{\text{Max}(P)}$. $\square$

**Lemma 3.1.4.** Suppose that $P$ satisfies Axioms (T0), (T1), (T2) and (T3). Let $n \in \mathbb{N}$, $X \in \text{Cell}(P)_n$ and $Y = T_M^{PC}(X)$. Then, $Y_n = X_n$ and $Y_i = X_{i,-} \cap X_{i,+}$ for $i \in \mathbb{N}_{n-1}$.

*Proof.* This is a direct consequence of Lemma 3.1.2. $\square$

**Proposition 3.1.5.** Suppose that $P$ satisfies Axioms (T0), (T1), (T2) and (T3). Then, given a cell $X \in \text{Cell}(P)$, we have $T_{PC}^M \circ T_M^{PC}(X) = X$.

*Proof.* Let $n \in \mathbb{N}$, $X \in \text{Cell}(P)_n$, $Y = T_M^{PC}(X)$ and $Z = T_{PC}^M(Y)$. For $i \in \mathbb{N}_n$ and $\epsilon \in \{-, +\}$, we show that $X_{i,\epsilon} = Z_{i,\epsilon}$ by a decreasing induction on $i$. By Lemma 3.1.4, we have $Z_n = Y_n = X_n$ and, for $i \in \mathbb{N}_{n-1}$, by Lemma A.1.7, we have

$$Z_{i,-} = Y_i \cup Z_{i+1,-}^{\mp} = (X_{i,-} \cap X_{i,+}) \cup X_{i+1,-}^{\mp} = X_{i,-}.$$

Similarly, $Z_{i,+} = X_{i,+}$, so $X = Z$. Hence, $T_{PC}^M \circ T_M^{PC}(X) = X$. $\square$

**Proposition 3.1.6.** We have $T_{Cl}^M \circ T_M^{PC} = T_{Cl}^{PC}$.

*Proof.* It readily follows from the definitions. $\square$
**Sources and targets** Let $P$ be an $\omega$-hypergraph. We now define source and target operations for Closed($P$) and Max($P$). Given $n \in \mathbb{N}^*$ and $X \in \text{Closed}(P)$, we define the source $\bar{\partial}_{n-1}^{-}(X)$ (resp. target $\bar{\partial}_{n-1}^{+}(X)$) of $X$ as the closed $(n-1)$-fgs $Y$ defined by

$$Y = \mathrm{R}(X \setminus (X_n \cup \mathrm{R}(X_n^{+}))) \quad (\text{resp. } \mathrm{R}(X \setminus (X_n \cup \mathrm{R}(X_n^{-}))))$$

Respectively, given $n \in \mathbb{N}^*$ and a maximal $n$-fgs $X$, we define the source $\tilde{\partial}_{n-1}^{-}(X)$ (resp. target $\tilde{\partial}_{n-1}^{+}(X)$) of $X$ as the maximal $(n-1)$-fgs $Y$ such that

$$Y_{n-1} = X_{n-1} \cup X_n^{\mp} \quad (\text{resp. } Y_{n-1} = X_{n-1} \cup X_n^{\pm}) \quad \text{and} \quad Y_i = X_i \quad \text{for } i \in \mathbb{N}_{n-2}.$$

We have the following compatibility results between the source and target operations and the translation functions:

**Proposition 3.1.7.** If $P$ satisfies Axioms (T0), (T1), (T2) and (T3), then, for $n \in \mathbb{N}^*$, $\epsilon \in \{-, +\}$ and $X \in \text{Cell}(P)_n$, we have $\mathrm{T}_{\mathrm{M}}^{\mathrm{PC}}(\partial_{n-1}^{\epsilon}(X)) = \tilde{\partial}_{n-1}^{\epsilon}(\mathrm{T}_{\mathrm{M}}^{\mathrm{PC}}(X))$.

*Proof.* Let $Y = \mathrm{T}_{\mathrm{M}}^{\mathrm{PC}}(\partial_{n-1}^{\epsilon}(X))$, $X' = \mathrm{T}_{\mathrm{M}}^{\mathrm{PC}}(X)$ and $Z = \tilde{\partial}_{n-1}^{\epsilon}(X')$. By Lemma 3.1.4, we have $Y_{n-1} = X_{n-1,\epsilon}$ and $Y_i = X_{i,-} \cap X_{i,+}$ for $i \in \mathbb{N}_{n-1}$. Moreover, $X'_n = X_n$ and $X'_i = X_{i,-} \cap X_{i,+}$ for $i \in \mathbb{N}_{n-1}$. If $\epsilon = -$, then, by Lemma A.1.7,

$$Z_{n-1} = (X_{n-1,-} \cap X_{n-1,+}) \cup X_n^{\mp} = X_{n-1,-}$$

and $Z_i = X'_i = X_{i,-} \cap X_{i,+}$ for $i \in \mathbb{N}_{n-1}$, so $Y = Z$. Similarly, if $\epsilon = +$, we have $Y = Z$. $\square$

**Proposition 3.1.8.** For $n \in \mathbb{N}^*$, $\epsilon \in \{-, +\}$ and $X \in \text{Max}(P)_n$, we have

$$\mathrm{T}_{\mathrm{Cl}}^{\mathrm{M}}(\tilde{\partial}_{n-1}^{\epsilon}(X)) = \bar{\partial}_{n-1}^{\epsilon}(\mathrm{T}_{\mathrm{Cl}}^{\mathrm{M}}(X)).$$

*Proof.* By symmetry, it is sufficient to handle the case $\epsilon = -$. Let $Y = \mathrm{T}_{\mathrm{Cl}}^{\mathrm{M}}(\tilde{\partial}_{n-1}^{-}(X))$ and $Z = \bar{\partial}_{n-1}^{-}(\mathrm{T}_{\mathrm{Cl}}^{\mathrm{M}}(X))$. By unfolding the definitions, we have

$$Y = \mathrm{R}((X \setminus X_n) \cup X_n^{\mp}) \quad \text{and} \quad Z = \mathrm{R}(\mathrm{R}(X) \setminus (X_n \cup \mathrm{R}(X_n^{+}))).$$

In order to show that $Y \subseteq Z$, we only need to prove that $Y' \subseteq Z$ where $Y' = (X \setminus X_n) \cup X_n^{\mp}$. First, we have that $Y' \subseteq \mathrm{R}(X)$. Moreover,

$$\begin{align*}
Y' \cap (X_n \cup \mathrm{R}(X_n^{+})) &= ((X \setminus X_n) \cup X_n^{\mp}) \cap (X_n \cup \mathrm{R}(X_n^{+})) \\
&= ((X \setminus X_n) \cup X_n^{\mp}) \cap \mathrm{R}(X_n^{+}) \\
&= (X \setminus X_n) \cap \mathrm{R}(X_n^{+}) \\
&= X \cap \mathrm{R}(X_n^{+}) = \emptyset \tag*{(\text{since } X \text{ is maximal})}.
\end{align*}$$

So $Y' \subseteq Z$, which implies that $Y \subseteq Z$. Similarly, in order to show that $Z \subseteq Y$, we only need to prove that $Z' \subseteq Y$ where $Z' = \mathrm{R}(X) \setminus (X_n \cup \mathrm{R}(X_n^{+}))$. But

$$Z' \subseteq Y \Leftrightarrow \mathrm{R}(X) \subseteq Y \cup X_n \cup \mathrm{R}(X_n^{+})$$

and

$$\begin{align*}
Y \cup X_n \cup \mathrm{R}(X_n^{+}) &= \mathrm{R}((X \setminus X_n) \cup X_n^{\mp}) \cup X_n \cup \mathrm{R}(X_n^{+}) \\
&= \mathrm{R}((X \setminus X_n) \cup X_n^{\mp} \cup X_n^{+}) \cup X_n \\
&= \mathrm{R}((X \setminus X_n) \cup X_n^{-} \cup X_n^{+}) \cup X_n \\
&= \mathrm{R}((X \setminus X_n) \cup X_n^{-} \cup X_n^{+} \cup X_n) = \mathrm{R}(X).
\end{align*}$$

So $Z' \subseteq Y$, which implies that $Z \subseteq Y$. Hence, $Y = Z$, which concludes the proof. $\square$
**Sources and targets** Let $P$ be an $\omega$-hypergraph. We now define source and target operations for Closed($P$) and Max($P$). Given $n \in \mathbb{N}^*$ and $X \in \text{Closed}(P)$, we define the source $\bar{\partial}_{n-1}^{-}(X)$ (resp. target $\bar{\partial}_{n-1}^{+}(X)$) of $X$ as the closed $(n-1)$-fgs $Y$ defined by

$$Y = \mathrm{R}(X \setminus (X_n \cup \mathrm{R}(X_n^{+}))) \quad (\text{resp. } \mathrm{R}(X \setminus (X_n \cup \mathrm{R}(X_n^{-}))))$$

Respectively, given $n \in \mathbb{N}^*$ and a maximal $n$-fgs $X$, we define the source $\tilde{\partial}_{n-1}^{-}(X)$ (resp. target $\tilde{\partial}_{n-1}^{+}(X)$) of $X$ as the maximal $(n-1)$-fgs $Y$ such that

$$Y_{n-1} = X_{n-1} \cup X_n^{\mp} \quad (\text{resp. } Y_{n-1} = X_{n-1} \cup X_n^{\pm}) \quad \text{and} \quad Y_i = X_i \quad \text{for } i \in \mathbb{N}_{n-2}.$$

We have the following compatibility results between the source and target operations and the translation functions:

**Proposition 3.1.7.** If $P$ satisfies Axioms (T0), (T1), (T2) and (T3), then, for $n \in \mathbb{N}^*$, $\epsilon \in \{-, +\}$ and $X \in \text{Cell}(P)_n$, we have $\mathrm{T}_{\mathrm{M}}^{\mathrm{PC}}(\partial_{n-1}^{\epsilon}(X)) = \tilde{\partial}_{n-1}^{\epsilon}(\mathrm{T}_{\mathrm{M}}^{\mathrm{PC}}(X))$.

*Proof.* Let $Y = \mathrm{T}_{\mathrm{M}}^{\mathrm{PC}}(\partial_{n-1}^{\epsilon}(X))$, $X' = \mathrm{T}_{\mathrm{M}}^{\mathrm{PC}}(X)$ and $Z = \tilde{\partial}_{n-1}^{\epsilon}(X')$. By Lemma 3.1.4, we have $Y_{n-1} = X_{n-1, \epsilon}$ and $Y_i = X_{i, -} \cap X_{i, +}$ for $i \in \mathbb{N}_{n-1}$. Moreover, $X'_n = X_n$ and $X'_i = X_{i, -} \cap X_{i, +}$ for $i \in \mathbb{N}_{n-1}$. If $\epsilon = -$, then, by Lemma A.1.7,

$$Z_{n-1} = (X_{n-1, -} \cap X_{n-1, +}) \cup X_n^{\mp} = X_{n-1, -}$$

and $Z_i = X'_i = X_{i, -} \cap X_{i, +}$ for $i \in \mathbb{N}_{n-1}$, so $Y = Z$. Similarly, if $\epsilon = +$, we have $Y = Z$. $\square$

**Proposition 3.1.8.** For $n \in \mathbb{N}^*$, $\epsilon \in \{-, +\}$ and $X \in \text{Max}(P)_n$, we have

$$\mathrm{T}_{\mathrm{Cl}}^{\mathrm{M}}(\tilde{\partial}_{n-1}^{\epsilon}(X)) = \bar{\partial}_{n-1}^{\epsilon}(\mathrm{T}_{\mathrm{Cl}}^{\mathrm{M}}(X)).$$

*Proof.* By symmetry, it is sufficient to handle the case $\epsilon = -$. Let $Y = \mathrm{T}_{\mathrm{Cl}}^{\mathrm{M}}(\tilde{\partial}_{n-1}^{-}(X))$ and $Z = \bar{\partial}_{n-1}^{-}(\mathrm{T}_{\mathrm{Cl}}^{\mathrm{M}}(X))$. By unfolding the definitions, we have

$$Y = \mathrm{R}((X \setminus X_n) \cup X_n^{\mp}) \quad \text{and} \quad Z = \mathrm{R}(\mathrm{R}(X) \setminus (X_n \cup \mathrm{R}(X_n^{+}))).$$

In order to show that $Y \subseteq Z$, we only need to prove that $Y' \subseteq Z$ where $Y' = (X \setminus X_n) \cup X_n^{\mp}$. First, we have that $Y' \subseteq \mathrm{R}(X)$. Moreover,

$$\begin{aligned}
Y' \cap (X_n \cup \mathrm{R}(X_n^{+})) &= ((X \setminus X_n) \cup X_n^{\mp}) \cap (X_n \cup \mathrm{R}(X_n^{+})) \\
&= ((X \setminus X_n) \cup X_n^{\mp}) \cap \mathrm{R}(X_n^{+}) \\
&= (X \setminus X_n) \cap \mathrm{R}(X_n^{+}) \\
&= X \cap \mathrm{R}(X_n^{+}) = \emptyset \qquad (\text{since } X \text{ is maximal}).
\end{aligned}$$

So $Y' \subseteq Z$, which implies that $Y \subseteq Z$. Similarly, in order to show that $Z \subseteq Y$, we only need to prove that $Z' \subseteq Y$ where $Z' = \mathrm{R}(X) \setminus (X_n \cup \mathrm{R}(X_n^{+}))$. But

$$Z' \subseteq Y \Leftrightarrow \mathrm{R}(X) \subseteq Y \cup X_n \cup \mathrm{R}(X_n^{+})$$

and

$$\begin{aligned}
Y \cup X_n \cup \mathrm{R}(X_n^{+}) &= \mathrm{R}((X \setminus X_n) \cup X_n^{\mp}) \cup X_n \cup \mathrm{R}(X_n^{+}) \\
&= \mathrm{R}((X \setminus X_n) \cup X_n^{\mp} \cup X_n^{+}) \cup X_n \\
&= \mathrm{R}((X \setminus X_n) \cup X_n^{-} \cup X_n^{+}) \cup X_n \\
&= \mathrm{R}((X \setminus X_n) \cup X_n^{-} \cup X_n^{+} \cup X_n) = \mathrm{R}(X).
\end{aligned}$$

So $Z' \subseteq Y$, which implies that $Z \subseteq Y$. Hence, $Y = Z$, which concludes the proof. $\square$
**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. $*^{M}$) for $*_{i}^{\text{Cl}}$ (resp. $*_{i}^{M}$). 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

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