A coherence theorem for pseudonatural transformations

We prove coherence theorems for bicategories, pseudofunctors and pseudonatural transformations. These theorems boil down to proving the coherence of some free $(4,2)$-categories. In the case of bicategories and pseudofunctors, existing rewriting techniques based on Squier's Theorem allow us to conclude. In the case of pseudonatural transformations this approach only proves the coherence of part of the structure, and we use a new rewriting result to conclude. To this end, we introduce the notions of white-categories and partial coherence.


Introduction
The intended purpose of this relation is that, between any two bracketings of A 1 ⊗A 2 ⊗. . .⊗A n−1 ⊗A n , there exists a unique isomorphism constructed from the isomorphisms α A,B,C . This statement was made precise and proved by Mac Lane in the case of monoidal categories [12]. In general a coherence theorem contains a description of a certain class of diagrams that are to commute. Coherence theorems exist for various other structures, e.g. bicategories [13], or V -natural transformations for a symmetric monoidal closed category V [10].
Coherence results are often a consequence of (arguably more essential [9]) strictification theorems. A strictification theorem states that a "weak" structure is equivalent to a "strict" (or at least "stricter") one. For example any bicategory is biequivalent to a 2-category, and the same is true for pseudofunctors (this is a consequence of this general strictification result [15]). It does not hold however for pseudonatural transformations.

Free categories and rewriting
Coherence theorems can also be proven through rewriting techniques. The link between coherence and rewriting goes back to Squier's homotopical Theorem [16], and has since been expanded upon [6]. Squier's theory is constructive, which means that the coherence conditions can be calculated from the relations, in a potentially automatic way. It can also be expanded to higher dimensions [8], a feature that may prove useful when studying weaker structures. In [7], the authors use Squier's theory to prove the coherence of monoidal categories. Let us give an outline of the proof in the case of categories equipped with an associative tensor product.
Polygraphs are presentations for higher-dimensional categories and were introduced by Burroni [3], and by Street under the name of computads [17] [18]. In this paper we use Burroni's terminology. For example, a 1-polygraph is given by a graph G, and the free 1-category it generates is the category of paths on G. If Σ is an n-polygraph, we denote by Σ * the free n-category generated by Σ.
An (n, p)-category is a category where all k-cells are invertible, for k > p. In particular, (n, 0)categories are commonly called n-groupoids, and (n, n)-categories are just n-categories. There is a corresponding notion of (n, p)-polygraph. If Σ is an (n, p)-polygraph, we denote by Σ * (p) the free (n, p)category generated by Σ.
The structure of category equipped with an associative tensor product is encoded into a 4-polygraph Assoc, which generates a free (4, 2)-category Assoc * (2) . The 4-polygraph Assoc contains one generating

2-cell
coding for product, one generating 3-cell : coding for associativity and one generating 4-cell corresponding to Mac Lane's pentagon: The coherence result for categories equipped with an associative product is now reduced to showing that, between every parallel 3-cells A, B in Assoc * (2) , there exists a 4-cell α : A 1 c B in Assoc * (2) . A 4-category satisfying this property is said to be 3-coherent.
Let us denote by Assoc * the free 4-category generated by Assoc. We have the following properties: • Starting from any given 2-cell in Assoc * , it is impossible to form an infinite sequence of non-identity 3-cells. This property is known as 3-termination.
• If A and B are two 3-cells in Assoc * with the same source, there exists 3-cells A and B in Assoc * such that the composites A 2 A and B 2 B are well-defined and have the same target. This property is known as 3-confluence.
The conjunction of these two properties make Assoc into a 3-convergent 4-polygraph.

Squier's theory and coherence
A generating 3-cell composed with some lower dimensional context is called a rewriting step of Assoc. A local branching in Assoc is a pair of rewriting step of same source. Local branching are ordered by adjunction of context, that is a branching (f, g) is smaller than a branching (u i f i v, u i g i v) for any 2-cells u and v and i = 0, 1. There are three types of local branchings: • A branching of the form (f, f ) is called aspherical .
• A branching of the form (f i s(g), s(f ) i g) for i = 0 or 1 is called a Peiffer branching.
• Otherwise, (f, g) is called an overlapping branching.
Overlapping branchings that are also minimal are called critical branchings.
There is exactly one critical branching in Assoc, of source . Note that the critical pair appears as the source of the generating 4-cell of Assoc. In particular there is a one-to-one correspondence between 4-cells and critical pairs. A 3-convergent 4-polygraph that satisfies this property is said to satisfy the 3-Squier condition. Proposition 4.3.4 in [6] states that a 4-polygraph satisfying the 3-Squier condition is 3-coherent (and more generally, that any (n+1)-polygraph satisfying the n-Squier condition is n-coherent). In particular, the 4-polygraph Assoc satisfies the 3-Squier condition, so it is 3-coherent.
In Section 3, we exhibit, for any sets C and D and any application f : C → D two 4-polygraphs BiCat [C] and PFonct[f ] presenting respectively the structures of bicategory and pseudofunctor. Applying the reasoning we just presented, we prove our first two results: Theorem 3.1.6 (Coherence for bicategories). Let C be a set.

The 2-Squier condition of depth 2
In order to circumvent this difficulty, we introduce the notion of 2-Squier condition of depth 2. We say that a (4, 2)-polygraph Σ satisfies the 2-Squier condition of depth 2 if it satisfies the 2-Squier condition, and if the 4-cells of Σ correspond to the critical triples induced by the 2-cells (with a prescribed shape).
For example, the 4-polygraph Assoc satisfies the 2-Squier condition of depth 2: its underlying 2polygraph is both 2-terminating and 2-confluent. Moreover the only critical pair corresponds to the associativity 3-cell. Finally, Mac Lane's pentagon can be written as follows, which shows that it corresponds to the only critical triple: We prove the following result about (4, 2)-polygraph satisfying the 2-Squier condition of depth 2: Theorem 1.4.9. Let Σ be a (4, 2)-polygraph satisfying the 2-Squier condition of depth 2. For every parallel 3-cells A, B ∈ Σ * (2) 3 whose 1-target is a normal form, there exists a 4-cell α : A 1 c B in the free (4, 2)-category Σ * (2) 4 .
Note in particular that the 2-Squier condition of depth 2 does not imply the 3-coherence of the (4, 2)-category generated by the polygraph, but only a partial coherence, "above the normal forms". For example in the case of Assoc, the only normal form is the 1-cell . So Theorem 1.4.9 only expresses the coherence of the 3-cells of Assoc * (2) whose 1-target is . On the other hand, Squier's Theorem as extended in [6] concerns all the 3-cells of Assoc * (2) , regardless of their 1-target.
The (4, 2)-polygraph PNTrans[f , g] does not satisfy the 2-Squier condition. However, we identify in Section 4.2 a sub-(4, 2)-polygraph PNTrans ++ [f , g] of PNTrans[f , g] that does. By Theorem 1.4.9, we get a partial coherence result in PNTrans ++ [f , g] * (2) . The rest of Section 4 is spent extending this partial coherence result to the rest of PNTrans[f , g] * (2) . To do so, we define a weight application from PNTrans[f , g] * (2) to N to keep track of the condition on the 1-targets of the 3-cells considered. We thereby prove the following result: Theorem 3.3.8 (Coherence for pseudonatural transformations). Let C and D be sets, and f , g : C → D applications.
Let A, B ∈ PNTrans[f , g] * (2) 3 be two parallel 3-cells whose 1-target is of weight 1. There is a 4-cell α : As a result, 0-composition is not defined for k-cells, for k > 1. The notion of 2-white-category coincides with the notion of sesquicategory (see [19]).
Most concepts from rewriting have a straightforward transcription in the setting of white-categories. In particular in Section 2.1, we define the notions of (n, k)-white-category and (n, k)-white-polygraph. We also give an explicit description of the free (n, k)-white-category Σ w(k) generated by an (n, k)-whitepolygraph Σ.
In this setting, we give a precise definition to the notion of partial coherence. Let C be a (4, 3)-whitecategory and S be a set of distinguished 2-cells of C. We call such a pair a pointed (4, 3)-white-category.
We say that C is S-coherent if for any parallel 2-cells f, g ∈ S and any 3-cells A, B : f g ∈ C, there exists a 4-cell α : A 1 c B ∈ C. In particular any (4, 3)-white category is ∅-coherent, and a (4, 3)-white category C is C 2 -coherent if and only if it is 3-coherent (where C 2 i sthe set of all the 2-cells of C). Theorem 1.4.9 amounts to showing that the free (4, 2)-category Σ * (2) is S Σ -coherent, where S Σ is the set of all 2-cells whose target is a normal form.
Finally, we give a way to modify partially coherent categories while retaining information about the partial coherence. Let (C, S) and (C , S ) be pointed (4, 3)-white-categories. We define a relation of strength between pointed (4, 3)-white-categories. We show that if (C, S) is stronger than (C , S ), then the S-coherence of C implies the S -coherence of C .
Sketch of the proof of Theorem 1.4.9 We now give an overview of the proof of Theorem 1.4.9. Let us fix a (4, 2)-polygraph A satisfying the 2-Squier condition of depth 2, and denote by S A the set of 2-cells whose target is a normal form. In particular, (A * (2) , S A ) is a pointed (4, 3)-white-category. The first half of the proof (Section 5) consists in applying to (A * (2) , S A ) a series of transformations. At each step, we verify that the new pointed (4, 3)-white-category we obtain is stronger than the previous one. In the end, we get a pointed (4, 3)white-category (F w (3) , S E ), where F is a 4-white-polygraph. In dimension 2, the 2-cells of F consists of the union of the 2-cells of A together with their formal inverses. We denote byf the formal inverse of a 2-cell f ∈ A * . Let F 3 be the set of 3-cells of F. It contains 3-cells C f,g for any minimal local branching (f, g), and cells η f for any 2-cell f ∈ A of the following shape: The purpose of this transformation is that in F w (3) , for any 2-cells f, g ∈ S E , 3-cells of the form f g (and 4-cells between them) are in one-to-one correspondence with 3-cells of the formḡ 1 f 1û (and 4-cells between them), whereû is the common target of f and g. More generally we study cells of the form h 1û, and 4-cells between them. We start by studying the rewriting system induced by the 3-cells. Note that the 4-white-polygraph F is not 3-terminating, so we cannot use a Squier-like Theorem to conclude. However, let N[A * 1 ] be the free monoid on A * 1 , the set of 1-cells of A * . There is a well-founded ordering on A * 1 induced by the fact that A is 2-terminating. This order induces a well-founded ordering on N[A * 1 ] called the multiset order. We define an application p : which induces a well-founded ordering on F w 2 , the set of 2-cells of F w , and show that the cells C f,g are compatible with this ordering (that is, the target of a cell C f,g is always smaller than the source). Thus, the fragment of F 3 consisting of the cells C f,g is 3-terminating.
Thus the η f cells constitute the non-terminating part of F w 3 . To control their behaviour, we introduce a weight application w η : , that essentially counts the number of η f cells present in a 3-cell. In section 6.3, using the applications p and w η , we prove that for any h ∈ F w 2 whose source and target are normal forms (for A 2 ), and for any 3-cells . Finally, we prove that this implies that F w(3) is S E -coherent, which concludes the proof.

Organisation
In Section 1, we recall some classical definitions and results from rewriting theory, and we enunciate (without proof) Theorem 1.4.9. Section 2 contains the definitions of white-categories and white-polygraphs, together with the study of the notion of partial coherence. In Section 3, we construct the free categories encoding the structures we want to study, and prove the coherence Theorems for bicategories (Theorem 3.1.6) and pseudofunctors (Theorem 3.2.7). The proof uses a lot of notions defined in Section 1 and relies in particular on Squier's Theorem to conclude. There remains to show the coherence of pseudonatural transformations (Theorem 3.3.8), which is done in Section 4. To prove Theorem 3.3.8, we show that a fragment of the structure of pseudonatural transformations satisfies the hypotheses of Squier's Theorem while an other satisfies the hypotheses of Theorem 1.4.9, which is temporarily admitted. The following sections contain the proof of Theorem 1.4.9. The first half of the proof is contained in Section 5 and consists in applying a series of transformations to a (4, 3)-polygraph satisfying the hypotheses of Theorem 1.4.9. The combinatorics of the result of these transformations is analysed in Section 6 where we conclude the proof.

Higher-dimensional rewriting
We recall definitions and results from rewriting theory. Section 1.1 is devoted to polygraphs, which are presentations of higher-dimensional categories. In Section 1.2, we define termination and enunciate Theorem 1.2.4 which we will use throughout Sections 3 and 4 in order to prove the 3-termination of polygraphs. In Section 1.3 we define the notion of branchings and classify them, which allows for a simple criterion to prove the n-confluence of a polygraph. Finally in Section 1.4, we define the n-Squier condition, and recall Squier's homotopical theorem, in a generalized form proven in [6]. We conclude this section by enunciating Theorem 1.4.9, whose proof will occupy Sections 5 and 6. Except for Theorem 1.4.9, the proof of every result in this section can be found in [6].

Polygraphs
Definition 1.1.1. Let n be a natural number. Let C be a (strict, globular) n-category. For k ≤ n, we denote by C k both the set of k-cells of C and the k-category obtained by deleting the cells of dimension greater than k. For x ∈ C k and i < k, we denote by s i (x) and t i (x) respectively the i-source and i-target of x. Finally we write s(x) and t(x) respectively for s k−1 (x) and t k−1 (x).
For C a 2-category, we denote by C op the 2-category obtained by reversing the direction of the 1-cells, and by C co the 2-category obtained by reversing the direction of the 2-cells.
We recall the definition of Polygraphs from [3]. For n ∈ N, we denote by Cat n the category of n-categories and by Graph n the category of n-graphs. The category of n-categories equipped with a cellular extension, denoted by Cat + n , is the limit of the following diagram: where the functor Cat n → Graph n forgets the categorical structure and the functor Graph n+1 → Graph n deletes the top-dimensional cells.
Hence an object of Cat + n is a couple (C, G) where C is an n-category and G is a graph such that for any u, v ∈ S n+1 , the following equations are verified: Let R n be the functor from Cat n+1 to Cat + n that sends an (n + 1)-category C on the couple . This functor admits a left-adjoint L n : Cat + n → Cat n+1 (see [14]). We now define by induction on n the category Pol n of n-polygraphs together with a functor Q n : Pol n → Cat n .
• The category Pol 0 is the category of sets, and Q 0 is the identity functor.
• Assume Q n : Pol n → Cat n is defined. Then Pol n+1 is the limit of the following diagram: and Q n+1 is the composite Given an n-polygraph Σ, the n-category Q n (Σ) is denoted by Σ * and is called the free n-category generated by Σ. Definition 1.1.3. Let C be an n-category, and 0 ≤ i < n and A ∈ C i+1 . If it exists, we denote by A −1 the inverse of A for the i-composition.
For k ≤ n, an (n, k)-category is an n-category which has every (i + 1)-cell invertible for the icomposition, for i ≥ k. We denote by Cat (k) n the full subcategory of Cat n whose objects are the (n, k)-categories.
In particular Cat (0) n is the category of n-groupoids, and Cat (n) n = Cat n . The functor R n restricts to a functor R (n) n from Cat (n) n+1 to Cat + n . Once again this functor admits a left-adjoint L (n) n . We define categories Pol (k) n of (n, k)-polygraphs and functors Q (k) n in a similar way to Pol n and Q n . See 2.2.3 in [8] for an explicit description of this construction. Definition 1.1.4. Given an (n, k)-polygraph Σ, the (n, k)-category Q (k) n (Σ) is denoted by Σ * (k) and is called the free (n, k)-category generated by Σ. For j ≤ n, we denote by Σ * (k) j both the the of j-cells of Σ * (k) and the (j, k)-category generated by Σ. Hence an (n, k)-polygraph Σ consists of the following data: Remark 1.1.5. Let n, j and k be integers, with j ≤ k ≤ n. Since an (n, k)-category is also an (n, j)category, an (n, k)-polygraph gives rise to an (n, j)-polygraph. In particular, if Σ is an (n, k)-polygraph, we denote by Σ * (j) the (n, j)-category it generates.

Termination
k is a preorder on Σ * k−1 (transitivity is given by composition, and reflexivity by the units). We say that the n-polygraph Σ is k-terminating if → * k is a well-founded ordering. We denote by → + k the strict ordering associated to → * k . We recall Theorem 4.2.1 from [6], which we will use in order to show the 3-termination of some polygraphs. Definition 1.2.2. Let sOrd be the 2-category with one object, whose 1-cells are partially ordered sets, whose 2-cells are monotonic functions and which 0-composition is the cartesian product. Definition 1.2.3. Let C be a 2-category, X : C 2 → sOrd and Y : C co 2 → sOrd two 2-functors, and M a commutative monoid. An (X, Y, M )-derivation on C is given by, for every 2-cell f ∈ C 2 , an application such that for every 2-cells f 1 , f 2 ∈ C 2 , every x, y, z and t respectively in X(s(f 1 )), Y (t(f 1 )), X(s(f 2 )) and Y (t(f 2 )), the following equalities hold: In order to show the 3-termination of some polygraphs, we are going to use the following result (Theorem 4.2.1 from [6]). Theorem 1.2.4. Let Σ be an n-polygraph, X : Σ * 2 → sOrd and Y : (Σ * 2 ) co → sOrd two 2-functors, and M be a commutative monoid equipped with a well-founded ordering ≥, and whose addition is strictly monotonous in both arguments.
Suppose that for every 3-cell A ∈ Σ 3 , the following inequalities hold: Then the n-polygraph Σ is 3-terminating.

Branchings and Confluence
. . , f k ) of n-cells in Σ * such that every f i has the same source u, which is called the source of the branching. The symmetric group S k acts on the set of all k-fold branchings of Σ. The equivalence class of a branching (f 1 , f 2 , . . . , f k ) under this action is denoted by [f 1 , f 2 , . . . , f k ]. Such an equivalence class is called a k-fold symmetrical branching, and Let Σ be an n-polygraph. We denote by N the n-category with exactly one k-cell for every k < n, whose n-cells are the natural numbers and whose compositions are given by addition.
We define an application : Σ * → N by setting (f ) = 1 for every f ∈ Σ n . For f ∈ Σ * n , we call (f ) the length of a f .
An n-cell of length 1 in Σ * n is also called a rewriting step.
. . , f k ) of source u is a strict aspherical branching if there exists an integer i such that f i = f i+1 . We say that it is an aspherical branching if it is in the equivalence class of a strict aspherical branching.
A k-fold local branching (f 1 , . . . , f k ) is a strict Peiffer branching if it is not aspherical and there exist v 1 , v 2 ∈ Σ * n−1 such that u = v 1 i v 2 , an integer m < n and f 1 , . . . , f k ∈ Σ * n such that for every j ≤ m, f j = f j i v 2 and for every j > m, f j = v 1 i f j . It is a Peiffer branching if it is in the equivalence class of a strict Peiffer branching.
A local branching that is neither aspherical nor Peiffer is overlapping. Proposition 1.3.9. Let Σ be a k-convergent n-polygraph. For every u ∈ Σ * k−1 , there exists a unique v ∈ Σ * k−1 such that u → * k v and v is minimal for → * k . Definition 1.3.10. Let Σ be an n-polygraph. A normal form for Σ is an (n − 1)-cell minimal for → * n . If Σ is n-convergent, for every u ∈ Σ * n−1 , the unique normal form v such that u → * n v is denoted bŷ u and is called the normal form of u. An (n + 1)-category C is n-coherent if, for each pair (f, g) of parallel n-cells in C n , there exists an (n + 1)-cell A : f → g in C n+1 . Definition 1.4.2. Let Σ be an (n + 1)-polygraph, and (f, g) be a local branching of Σ n . A filling of (f, g) is an (n + 1)-cell A ∈ Σ * (n) n+1 of the shape:

Coherence
3. An (n + 1)-polygraph Σ satisfies the n-Squier condition if: • it is n-convergent, • there is a bijective application from Σ n+1 to the set of all critical pairs of Σ n that associates to every A ∈ Σ n+1 , a critical pair b of Σ n such that A is a filling of a representative of b.
In the proof of this Theorem appears the following result (Lemma 4.3.3 in [6]). Proposition 1.4.5. Let Σ be an (n + 1)-polygraph satisfying the n-Squier condition.
For every parallel n-cells f, g ∈ Σ * n whose target is a normal form, there exists an (n+1)-cell A : f → g in Σ * (n) n+1 .
Let us compare those two last results. Let Σ be an (n + 1)-polygraph satifying the n-Squier relation, and let f, g ∈ Σ * n be two parallel n-cells whose target is a normal form. According to Theorem 1.4.4, there exists an (n + 1)-cell A : f → g in the free (n + 1, n − 1)-category Σ * (n−1) n+1 . Proposition 1.4.5 shows that such an A can be chosen in the free (n + 1, n)-category Σ * (n) n+1 , where the n-cells are not invertible.
Hence for cells f, g ∈ Σ * n whose target is a normal form, Proposition 1.4.5 is more precise than Theorem 1.4.4. Definition 1.4.6. Let Σ be an (n + 1)-polygraph, and (f, g) a local branching in Σ n . Depending on the nature of (f, g), we define the notion of canonical filling of (f, g).
• If (f, g) is an aspherical branching, then its canonical filling is the identity 1 f .
• Assume that Σ satisfies the n-Squier condition, and let (f, g) be a critical pair. Let A be the (n + 1)-cell associated to [f, g]. If A is a filling of (f, g) then the canonical filling of (f, g) is A.
Otherwise, A is a filling of (g, f ) and the canonical filling of (f, g) is A −1 .
• Assume that the branching (f, g) admits a canonical filler A. Then the canonical filler of (u Definition 1.4.7. Let Σ be an (n + 2, n)-polygraph satisfying the n-Squier condition, and (f, g, h) be a local branching of Σ n . A filling of (f, g, h) is an (n + 2)-cell α ∈ Σ * (n) n+2 of the shape: / / ?
, and A f,g , A g,h and A f,h are the canonical fillings of respectively (f, g), (g, h) and (f, h). • it satisfies the n-Squier condition, • there is a bijective application from Σ n+2 to the set of all critical triples of Σ n that associates to every α ∈ Σ n+2 a critical triple b of Σ n such that α is a filling of a representative of b.
We now enunciate the theorem whose proof will occupy Sections 5 and 6. . This Theorem can be compared with Proposition 4.4.4 in [8]. There, for every parallel A, B ∈ Σ * (1) 3 , a 4-cell α is constructed in the free (4, 1)-category Σ In Section 2.1 we define the notion of white-category together with the associated notion of whitepolygraph. The 2-white-categories are also known as sesquicategories (see [19]). White-categories are strict categories in which the interchange law between the compositions 0 and i need not hold, for every i > 0. That is, strict n-categories are exactly the n-white-categories satisfying the additional condition that for every i-cells f and g of 1-sources (resp. 1-targets) u and v (resp. u and v ): In Section 2.2, we define a notion of partial coherence for (4, 3)-white-categories. We show a simple criterion in order to deduce the partial coherence of a (4, 3)-white-category from that of an other one. This criterion will be used throughout Section 5. We also adapt the notion of Tietze-transformation from [5] to our setting of partial coherence in white-categories, in preparation for Section 5.5.
In Section 2.3, we study injective functors between free white-categories. In particular, we give a sufficient condition for a morphism of white-polygraphs to yield an injective functor between the whitecategories they generate. This result will be used in Section 5.3.
Note that, although Sections 2.2 and 2.3 are expressed in terms of white-categories (since this is how they will be used throughout Section 5), all the definitions and results in these Sections also hold in terms of strict categories, mutatis mutandis.

White-categories and White-polygraphs
Definition 2.1.1. Let n ∈ N. An (n + 1)-white-category is given by: • a set C 0 , • for every x, y ∈ C 0 , an n-category C(x, y). We denote by k+1 the k-composition in this category, • for every z ∈ C 0 and every u : x → y ∈ C 1 , functors u 0 _ : C(y, z) → C(x, z) and _ 0 u : C(z, x) → C(z, y), so that for every composable 1-cells u, v ∈ C 1 , their composite u 0 v is defined in a unique way, • for every x ∈ C 0 , a 1-cell 1 x ∈ C(x, x).
Moreover, this data must satisfy the following axioms: • For every x ∈ C 0 , and every y ∈ C 0 , the functors 1 x 0 _ : C(x, y) → C(x, y) and _ 0 1 y : C(x, y) → C(x, y) are identities.
• For every u, v ∈ C 1 , the following equalities hold: An (n, k)-white-category is an n-white-category in which every (i + 1)-cell is invertible for the icomposition, for every i ≥ k.
Let n be a natural number. Let C be an n-white-category. For k ≤ n, we denote by C k both the set of k-cells of C and the k-white-category obtained by deleting the cells of dimension greater than k. For x ∈ C k and i < k, we denote by s i (x) and t i (x) respectively the i-source and i-target of x. Finally we write s(x) and t(x) respectively for s k−1 (x) and t k−1 (x). Definition 2.1.2. Let C and D be n-white-categories. An n-white-functor is given by: • an application F 0 : C 0 → D 0 , • for every x, y ∈ C 0 , a functor F x,y : C(x, y) → D(F 0 (x), F 0 (y)).
Moreover, this data must satisfy the following axioms: • for every z ∈ C 0 and u : x → y ∈ C 1 , the following equalities hold between functors: This makes n-white-categories into a category, that we denote by WCat n .
Remark 2.1.3. Let us define a structure of monoidal category ⊗ on Cat n , in such a way that WCat n+1 is the category of categories enriched over (Cat n , ⊗).
Let C, D be two n-categories. The n-categories C × D 0 and C 0 × D are defined as follows: Let C 0 × D 0 be the n-category whose 0-cells are couples (x, y) ∈ C 0 × D 0 , and whose i-cells are identities for every i > 0. Let F : C 0 × D 0 → C × D 0 (resp. G : C 0 × D 0 → C 0 × D) be the n-functor which is the identity on 0-cells. Then C ⊗ D is the pushout (C × D 0 ) ⊕ C0×D0 (C 0 × D): The category of n-white-categories equipped with a cellular extension, denoted by WCat + n , is the limit of the following diagram: WCat n / / Graph n where the functor WCat n → Graph n forgets the white-categorical structure and the functor Graph n+1 → Graph n deletes the top-dimensional cells. Let R w n be the functor from WCat n+1 to WCat + n that sends an (n + 1)-white-category C on the couple (C n , C n C n+1 Proposition 2.1.4. The functor R w n admits a left-adjoint L w n : WCat + n → WCat n+1 . Proof. Let (C, Σ) ∈ WCat + n be an n-white-category equipped with a cellular extension. The construction of L w n (C, Σ) is split into three parts: • First, we define a formal language E Σ .
• Then, we define a typing system T C on E Σ . We denote by E T Σ the set of all typable expressions of E Σ .
• Finally, we define an equivalence relation ≡ * Σ on E T Σ . The set of (n + 1)-cell of L w n (C, Σ) is then Let E Σ be the formal language consisting of: • For every 1-cells u, v ∈ C 1 , and every (n + 1)-cell A ∈ Σ n+1 , such that t 0 (u) = s 0 (A) and t 0 (A) = s 0 (v), a constant symbol c uAv .
• For every n-cell f ∈ C n , a constant symbol i f .
• For every 0 < i ≤ n, a binary function symbol i .
Thus E Σ is the smallest set of expressions containing the constant symbols and such that e i f ∈ Σ whenever e, f ∈ E Σ . Let T C be the set of all n-spheres of C, that is of couples (f, g) in C n such that s(f ) = s(g) and t(f ) = t(g). For e ∈ E Σ and t ∈ T C , we define e : t (read as "e is of type t") as the smallest relation satisfying the following axioms: • For every 1-cells u and v in C 1 , and every (n + 1)-cell A ∈ Σ, such that t 0 (u) = s 0 (A) and • For every n-cell f ∈ C n i f : (f, f ) • For every e 1 , e 2 ∈ E Σ and i < n, if e 1 : (s 1 , t 1 ), e 2 : (s 2 , t 2 ) and t i (t 1 ) = s i (s 2 ), then • For every e 1 , e 2 ∈ E Σ , if e 1 : (s 1 , t 1 ), e 2 : (s 2 , t 2 ) and t 1 = s 2 , then e 1 n e 2 : (s 1 , t 2 ) An expression e ∈ E Σ is said to be typable if e : (s, t) for some n-sphere (s, t) ∈ T C . Moreover there is only one such n-sphere, so the operations s(e) := s and t(e) := t are well defined. We denote by E T Σ be the set of all typable expressions.
Let ≡ Σ be the symmetric relation generated by the following relations on E T Σ : • For every A, B, C, D ∈ E T Σ , and every i 1 , i 2 ≤ n non-zero distinct natural numbers, • For every A, B, C ∈ E T Σ , and every 0 < i ≤ n, • For every A ∈ E T Σ and f ∈ C n : • For every f 1 , f 2 ∈ C n and every i < n, The i-composition is given by the one of E T Σ , and identities by i f .
Definition 2.1.5. We now define by induction on n the category WPol n of n-white-polygraphs together with a functor Q w n : WPol n → WCat n .
• The category WPol 0 is the category of sets, and Q w 0 is the identity functor.
• Assume Q w n : WPol n → WCat n defined. Then WPol n+1 is the limit of the following diagram: and Q w n+1 is the composite Given an n-white-polygraph Σ, the n-white-category Q w n (Σ) is denoted by Σ w and is called the free n-white-category generated by Σ.
n+1 be the category of (n + 1, n)-white-categories. Once again we have a functor R w(n) n : WCat w(n) n+1 → WCat + n , and we are going to describe its left-adjoint L w(n) n+1 . Let (C, Σ) be an n-white-category together with a cellular extension. To construct L w(n) n+1 (C, Σ), we adapt the construction of the free n-white-categories as follows: • Let F Σ be the formal language E Σ∪Σ , whereΣ consists of formal inverses to the elements of Σ (that is their source and targets are reversed).
• The type system is extended by setting, for every 1-cells u, v in C 1 and every (n We denote by F T Σ the set of all typable expressions for this new typing system. • We extend ≡ Σ into a relation denoted by ∼ =Σ by adding the following relations: for every u, v in C 1 and every (n + 1)-cell A ∈ Σ, such that t 0 (u) = s 0 (A) and t 0 (A) = s 0 (v).
We define categories WPol (k) n of (n, k)-white-polygraphs and functors Q n in a similar way to Pol (k) n and Q (k) n .
Definition 2.1.7. Given an (n, k)-white-polygraph Σ, the (n, k)-white-category Q w(k) n (Σ) is denoted by Σ w(k) and is called the free (n, k)-white-category generated by Σ. For j ≤ n, we denote by Σ w(k) j both the the of j-cells of Σ w(k) and the (j, k)-category generated by Σ. Hence an (n, k)-polygraph Σ consists of the following data: 2.2 Partial coherence in pointed (4, 3)-white-categories where C is a 4-white-category, and S is a subset of C 2 .
Definition 2.2.2. Let (C, S) be a pointed (4, 3)-white-category. The restriction of C to S, denoted by C S, is the following (2, 1)-category: • its 0-cells are the 2-cells of C 2 that lie in S, • its 1-cells are the 3-cells of C 3 with source and target in S, • its 2-cells are the 4-cells of C 4 with 2-source and 2-target in S, • its 0-composition and 1-composition are respectively induced by the compositions 2 and 3 of C.
We now rephrase Theorem 1.4.9 in the setting of partial coherence.
Theorem 1.4.9. Let A be a (4, 2)-polygraph satisfying the 2-Squier condition of depth 2, and let S A be the set of all 2-cells whose target is a normal form.
Proof. The functor F induces a functorF : C S → C S. Since it is equal to F on objects, it is 0surjective. On 1-cellsF is the composition of F with the canonical projection associated to the quotient, hence it is 1-surjective, and so (C , S ) is stronger than (C, S). Lemma 2.2.8. Let (C, S), (C , S ) be two pointed (4, 3)-white-categories, and assume (C , S ) is stronger than (C, S).
If C is S -coherent, then C is S-coherent.
Proof. Let F : C S → C S be a functor that is 0-surjective and 1-surjective. Let A, B : f → g ∈ (C S) 1 be parallel 1-cells, andĀ,B be their projections in C S. Since F is 0-surjective, there exists f , g ∈ (C S ) 0 in the preimage of f and g under F . Since F is 1-surjective, there exists A , B ∈ (C S ) 1 of source f and of target g such that F (Ā ) =Ā and F (B ) =B.
Since C S is 2-coherent, there exists α : A ⇒ B ∈ (C S ) 2 . ThusĀ =B andĀ =B. Hence there exists α : A ⇒ B ∈ C S. This shows that C S is 1-coherent, and therefore that C is S-coherent.
We are going to define four families of Tietze-transformations on (4, 3)-white-polygraphs. Tietze transformations originates from combinatorial group theory [11], and was adapted for (3, 1)-categories in [5], as a way to modify a (3, 1)-polygraph without modifying the 2-categories it presents. In particular, they preserve the 2-coherence. Here we adapt these transformations to our setting of (4, 3)-whitepolygraphs and show that they preserve the partial coherence. This will be used in Section 5.5. We fix a 4-white-polygraph A.
. We define a 4-white-polygraph A(A) by adding to A a 3-cell B and a 4-cell α, whose sources and targets are given by:

The inclusion induces a functor between (4, 3)-white-categories ι
. We call this operation the adjunction of a 3-cell with its defining 4-cell.
Definition 2.2.10. Let α ∈ A 4 and A ∈ A 3 such that: , by sending t(α) on s(α) and that is the identity on the other cells of A 3 . This application extends into a 3-functor π α : , which sends α on the identity of s(α), and which is the identity on the other cells of A 4 . We call this operation the removal of a 3-cell with its defining 4-cell.
. We call this operation the adjunction of a superfluous 4-cell.
, that sends β on α and which is the identity on the other cells of A. We call this operation the removal of a superfluous 4-cell.
Remark 2.2.13. Note that, in those four cases, the set of 2-cells is left unchanged. In particular, let A be a 4-white-polygraph, and B a 4-white-polygraph constructed from A through a series of Tietzetransformations. If S is a sub-set of A w 2 , then S still is a subset of B w 2 .
Proposition 2.2.14. Let A be a 4-white-polygraph, S a sub-set of A w 2 , and B a 4-white-polygraph constructed from A through a series of Tietze-transformations.
Proof. We check that if B is constructed from A through a Tietze-transformation, then the 3-whitecategories presented by A and B are isomorphic. Suppose now that B is S-coherent, and let A, B ∈ A w 3 be parallel 3-cells, whose source and target are in S. Since B w(3) is S-coherent, the images of A and B in the 3-white-category presented by B are equal. Since it is isomorphic to the 3-white-category presented by A, there exists a 4-cell α : , which proves that A is S-coherent.

Injective functors between white-categories
Definition 2.3.1. Let Σ and Γ be two (n, k)-polygraphs (resp. (n, k)-white-polygraphs), and let F : Σ → Γ be a morphism of (n, k)-polygraphs (resp. (n, k)-white-polygraphs). We say that F is injective if for all j ≤ n it induces an injective application from Σ n to Γ n . Definition 2.3.2. Let C and D be two n-white-categories, and let F : C → D be a morphism of nwhite-categories. We say that F is injective if for all j ≤ n it induces an injective application from C to D.
Let Σ be the following 2-polygraph: They are distinct elements of Σ * (1) 2 . However using the exchange law, the following equality holds in Γ * (1) 2 , where denotes the inverse of : In what follows, we prove some sufficient conditions so that a morphism between two (n, k)-whitepolygraphs induces an injective fucntor between the (n, k)-categories they present. This is achieved in Proposition 2.3.8. This result will be used in Section 5.3.
To prove this result, we start by studying the more general case of an injective morphism I between (n, k)-white-categories equipped with a cellular extension. When its image is closed by divisors (see Definition 2.3.5), we show a simple sufficient condition so that I induces an injective (n + 1)-whitefunctor. We also show that the image of the (n + 1)-white-functor induced by I is then automatically closed by divisors. Hence this hypothesis disappears when we go back to morphisms of (n, k)-whitepolygraphs. In particular we show that every injective morphism of n-white-polygraphs induces an injective white-functor between n-white-categories.
For the rest of this section, we fix two n-white-categories equipped with cellular extensions (C, Σ), (C , Σ ) ∈ WCat + , and a morphism I : (C, Σ) → (C , Σ ) ∈ WCat + . That is, I is given by an n-white-functor I : C → C together with an application I n+1 : Σ → Σ such that the following squares are commute: We denote by I w (resp. I w(n) ) the (n + 1)-white-functor L w (I) (resp. L w(n) (I)). By definition, I w (resp. I w(n) ) is induced by an application from E T Σ to E T Σ (resp. from F T Σ to F T Σ ), that we again denote by I w (resp. I w(n) ).
Using their explicit definitions, the following properties of I w (resp. I w(n) ) hold: • Any element of E T Σ (resp. F T Σ ) whose image is a an identity is an identity.
Assume that the application I n+1 is injective, and that I induces an injection on C.
Then the applications I w : Proof. Let a 1 , a 2 ∈ E T Σ such that I w (a 1 ) = I w (a 2 ). We reason by induction on the structure of I w (a 1 ). If I w (a 1 ) = c u A v , with u , v ∈ C 1 and A ∈ Σ . Then there are u 1 , v 1 , u 2 , v 2 ∈ C 1 and A 1 , A 2 ∈ Σ such that a 1 = c u1A1v1 and a 2 = c u2A2v2 , and so: Since I and I n+1 are injective, we get: Then there exist f 1 , f 2 ∈ C n such that: Using the induction hypothesis, we get that A 1 = A 2 and B 1 = B 2 , and so a 1 = a 2 .
In the case of I w(n) , we reason as previously, and we have one more case to check: if I w(n) (a 1 ) = c u Ā v , with u , v ∈ C 1 and A ∈ Σ . Then there are u 1 , v 1 , u 2 , v 2 ∈ C 1 and A 1 , A 2 ∈ Σ such that a 1 = c u1Ā1v1 and a 2 = c u2Ā2v2 , and so: Using the injectivity of I and I n+1 , we get: and finally a 1 = a 2 .
Definition 2.3.5. Let C be an n-white-category, and E be a subset of C n . We say that E is closed by divisors if, for any f ∈ E, if f = f 1 i f 2 , then f 1 and f 2 are in E.
Lemma 2.3.6. Assume the image of I in C n is closed by divisors, and that I and I n+1 are injective. Then, for every a , b ∈ E T Σ such that a ≡ Σ b , and for every a ∈ E T Σ such that I w (a) = a , there Assume moreover that the application I n+1 is bijective and that I is bijective on the 1-cells of C.
Then, for every a , b ∈ F T Σ such that a ∼ =Σ b , and for every a ∈ F T Σ such that I w(n) (a) = a , there Proof. To show the result on I w we reason by induction on the structure of a .
The case where the roles of a and b are reversed is symmetrical.
If there exist A , B , C ∈ E T Σ , 0 < i ≤ n and a ∈ E T Σ such that: By construction, we have I w (b) = b and a ≡ Σ b. The case where the roles of a and b are reversed is symmetrical.
The case of the right-unit is symmetrical.
If there are f 1 , f 2 ∈ C n , i < n and a ∈ E T Σ such that: If there are f 1 , f 2 ∈ C n , i < n and a ∈ E T Σ such that: In the case of I w(n) , we reason as previously, and we have two more cases to check. If there exist u , v ∈ C 1 , A ∈ Σ and a ∈ F T Σ such that: then a = c u1A1v2 n c u2Ā2v2 , with u 1 , u 2 , v 1 , v 2 ∈ C 1 and A 1 , A 2 ∈ Σ such that: Assume that I n+1 and I are injective, and that the image of I in C n is closed by divisors. Then the functor I w : L w (C, Σ) → L w (C , Σ ) is injective, and its image is closed by divisors.
Assume moreover that I n+1 is bijective, and that I is bijective on the 1-cells of C. Then the functor I w(n) : L w(n) (C, Σ) → L w(n) (C , Σ ) is injective and its image is closed by divisors.
Proof. Let f 1 , f 2 ∈ L w (C, Σ) and a 1 , a 2 ∈ E T Σ such that: . Hence by definition, there exist n > 0 and t 1 , . . . , t n ∈ E T Σ such that: Applying Lemma 2.3.6 successively, we get t 1 , . . . , t n ∈ E T Σ such that: In particular a 1 ≡ * Σ t n and I w (t n ) = t n = I w (a 2 ). Using Lemma 2.3.4, this implies that t n = a 2 , and so a 1 ≡ * Σ a 2 , which proves that In particular, we have Using both Lemmas 2.3.4 and 2.3.6 as before, we get an Since the image of I w is closed by divisors, there : by construction we have: The case of I w(n) is identical, the only difference lying in the hypothesis needed to apply Lemma 2.3.6. Proposition 2.3.8. Let Σ and Γ be two (n, k)-white-polygraphs and I : Σ → Γ be an injective morphism of (n, k)-polygraphs. Then for every j ≤ k the functor I w j : Σ w j → Γ w j is injective, and its image is closed by divisors.
Assume moreover that I 0 and I 1 are bijections, and that for every j > k the application I j : Σ j → Γ j is bijective. Then for every j the functor I is injective, and its image is closed by divisors.
Proof. We reason by induction on j. The case j = 0 is true by hypothesis.
Let 1 ≤ j ≤ k. By hypothesis, the application I j is injective, and by induction hypothesis, the functor I w j−1 is injective with image closed by divisors. Hence I j satisfies the hypothesis of Lemma 2.3.7, and I w j is injective with image closed by divisors.
Let j > k. Again, using the hypothesis and induction hypothesis, we get that I j satisfies the hypotheses of Lemma 2.3.7. Hence I w(k) j is injective and its image is closed by divisors.
In what follows, we use the fact that the image of a functor generated by a morphism of polygraphs is closed by divisors in order to prove a characterisation of the image of such a functor. Definition 2.3.9. Let C, D be two n-white-categories, F : C → D be an n-functor and f be an n-cell of D. We say that F k-discriminates f if the following are equivalent: Assume that the image of I is closed by divisors, that the application I n is injective, and that I is n-discriminating on Σ .
Proof. Let us start with I w . Let E be the set all (n + 1)-cells of L w (C , Σ ) which I w discriminates. Let us show that E = L w (C , Σ ). Since I w commutes with the source and target applications, the implications ( The set E contains all units.
The set E contains all cells of length 1. Indeed, given such a cell A , there exist f k , g k ∈ C k and A 0 ∈ Σ such that Suppose that the source (resp. target) of A is in the image of I, and let us show that A is in the image of I w . Since the image of I is closed by divisors, we get first that f n , g n and s(A n−1 ) (resp. t(A n−1 )) are in the image of I. By iterating this reasoning, we get that, for all i, f i , g i and s(A i−1 ) (resp. t(A i−1 )) are in the image of I. Since I w discriminates Σ , there exist f k , g k ∈ C k and A 0 ∈ Σ such that: By induction on k we show that A k := f k k−1 A k−1 k−1 g k is well defined and that I w (A k ) = A k . Indeed, assume that it is true at rank k − 1. Then we have the equalities: Using the injectivity of I we get that t(f k ) = s k−1 (A k−1 ) and t k−1 (A k−1 ) = s(g k ), which shows that A k is well defined, and finally: In particular, we have A n = I w (A n ).
The set E is stable by n-composition. Indeed let A , B ∈ E, and assume that the source of A n B is in the image of I. Let us show that A n B is in the image of I w . The source of A n B is none other that the one of A . Since A is in E, there exists A ∈ L w (C, Σ) such that I w (A) = A . Hence the source of B is in the image of I, and since B ∈ E, there exists B ∈ L w (C, Σ) such that I w (B) = B . This concludes the proof for I w . Concerning I w(n) , the reasoning is the same except that we also have to show that E is stable under inversion. Indeed let A ∈ E and assume that the source (resp. target) of (A ) −1 is in the image of I. Then the target (resp. source) of A is in the image of I and since A is in E, there exists A ∈ L w(n) (C, Σ) such that I w(n) (A) = A , and so I w(n) (A −1 ) = (A ) −1 . Proposition 2.3.11. Let Σ and Γ be two (n, k)-white-polygraphs, and I : Σ → Γ be a morphism of polygraphs. Let k 0 such that for every j > k 0 , I j is a bijection.
Assume that I satisfies the hypothesis of Proposition 2.3.8, and that, for every j > k 0 , I j is k 0discriminating on Γ j . Then for every j ≥ k 0 , I Proof. Since I satisfies the hypotheses of Proposition 2.3.8, we know that for every j, the functor I w(k) j is injective, and that its image is closed by divisors.
We reason by induction on j > k 0 . For j = k 0 + 1, the result is a direct application of Lemma 2.3.10. Let j > k 0 + 1: let us show that I . Hence we can use Lemma 2.3.10, and we get that I

Application to the coherence of pseudonatural transformations
We now study the coherence problem successively for bicategories, pseudofunctors and pseudonatural transformations. In Section 3.1, we start by recalling the usual definition of bicategories (see [2]). We then give an alternative description of bicategories in terms of algebras over a certain 4-polygraph BiCat [C], and show that the two definitions coincide. The coherence problem for bicategories is now reduced to showing the 3-coherence of BiCat[C], and we use the techniques introduced in the previous section (especially Theorems 1. 2.4 and 1.4.4) to conclude. In Section 3.2 and 3.3, we apply the same reasoning to pseudofunctors and pseudonatural transformations. However in the case of pseudonatural transformations, we get a (4, 3)-polygraph PNTrans[f , g] which is not 3-confluent, and so we cannot directly apply Theorem 1.4.4. The proof of the coherence theorem for pseudonatural transformations will take place in Section 4 and will make use of Theorem 1.4.9.

Coherence for bicategories
Let Cat be the category of (small) categories. We denote by the terminal category in Cat. Let sCat be the 3-category with one 0-cell, (small) categories as 1-cells, functors as 2-cells, and natural transformations as 3-cells, where 0-composition is given by the cartesian product, 1-composition by functor composition, and 2-composition by composition of natural transformations. • For every a ∈ B 0 , a functor I a : → B(a, a), that is to say a 1-cell I a : a → a.
• For every a, b, c, d ∈ B 0 , a natural isomorphism α a,b,c,d : • For every a, b ∈ B 0 , natural isomorphisms R a,b and L a,b : This data must also satisfy the following axioms: • For every composable 2-cells f, g, h, i in B:  • For every a ∈ C, a 2-cell a : 1 a ⇒ a a .
Note that the indices are redundant with the source of a generating 2-cell. In what follows, we will therefore omit them when the context is clear. For example, the 2-cell of source a b c d designates the composite ( a b b,c,d ) 1 a,b,d . We will use the same notation for higher-dimensional cells.  • where C is a set, • where Φ is a functor from BiCat[C] to sCat.
Proposition 3.1.4. There is a one-to-one correspondence between (small) bicategories and Alg(BiCat).
Proof. The correspondence between a bicategory B and an algebra (C, Φ) over BiCat is given by: • At the level of sets: C = B 0 .
• For every a, b, c, d • The axioms that a bicategory must satisfy correspond to the fact that Φ is compatible with the quotient by the 4-cells and .
This correspondence between the structures of bicategory and of algebra over BiCat is summed up by the following table: Bicategory

3-cells Equalities
(1) (2) 4-cells Proof. In order to apply Theorem 1.2.4 we construct two functors X C : BiCat[C] * 2 → sOrd and Y C : 2 ) co → sOrd by setting, for every a, b ∈ C: and, for every i, j ∈ N * : We now define an (X C , Y C , N)-derivation d C on BiCat[C] * 2 by setting, for every i, j, k ∈ N * : It remains to show that the required inequalities are satisfied. Concerning X C and Y C , we have for every i, j, k ∈ N * : Concerning d C , we have for every i, j, k, l ∈ N * : The following Theorem is a rephrasing of Mac Lane's coherence Theorem ( [13]) in our setting. , which are constructed in a similar fashion as in the case of monoidal categories (see Proposition 3.5 in [7]).

Coherence for pseudofunctors
Definition 3.2.1. A pseudofunctor F is given by: • Two bicategories B and B . For every a, b ∈ B 0 , a functor F a,b : B(a, b) → B (F 0 (a), F 0 (b)).
• For every a, b, c ∈ B 0 , a natural isomorphism φ a,b,c : • For every a ∈ B 0 , a natural isomorphism ψ a : This data must satisfy the following axioms: • For every composable 1-cells f, g and h in B: • For every 1-cell f : a → b in B: • For every 1-cell f : a → b in B: • where C and D are sets, • where f is an application from C to D, • where Φ is a functor from PFonct[f ] to sCat such that, for every c ∈ C the following equality holds: Φ( c f (c) ) =  ) co → sOrd as extensions of the functors X C , X D , Y C and Y D from Proposition 3.1.5, and by setting for every a ∈ C: where is the terminal ordered set, and for every i ∈ N * : We now define an (X f , Y f , N)-derivation d f on PFonct[f ] * 2 as an extension of d C , by setting for every i, j, k ∈ N * : It remains to show that the inequalities required to apply Theorem 1.2.4 are satisfied. Since X f (resp. Y f ) extends X C and X D (resp. Y C and Y D ), the only inequalities that need to be checked are those corresponding to the 3-cells and . Indeed for every i, j ∈ N * , we have: Concerning d f , the 3-cells from BiCat[C] have already been checked in Proposition 3.1.5. For the other 3-cells, we have, for every i, j, k ∈ N * : Proof. We have shown that it is 3-terminating, so using Proposition  • For every a ∈ B 0 , a functor τ a : → B (F 0 (a), F 0 (a)), that is a 1-cell τ a : F 0 (a) → F 0 (a) in B .

Coherence for pseudonatural transformations
• For every a, b ∈ B 0 , a natural isomorphism σ a,b : This data must satisfy the following axioms: • For every (f, g) ∈ B(a, b) × B(b, c): • For every a ∈ B 0 :  • where C and D are sets, • where f , g : C → D are applications, • where Φ is a functor from PNTrans[f , g] to sCat, such that for every c ∈ C, d ∈ D and 1-cell  This result induces the following classification of the cells of the (4, 2)-polygraph PNTrans[f, g], depending on which structure they come from. We also distinguish two types of cells: product cells and unit cells. Moreover, in the following table, every line corresponds to a dimension.

Origin Dimension Product cells Unit cells
Source bicategory

2-cells 3-cells 4-cells ,
Target pseudofunctor 2 ) co → sOrd, we extend the functors X f , X g , Y f and Y g from Proposition 3.2.6, by setting:

2-cells 3-cells 4-cells
We now define an (X f ,g , Y f ,g , N)-derivation d f ,g of the 2-category PNTrans[f , g] * 2 as the extension of d f satisfying, for every i, j ∈ N * : It remains to show that the required inequalities are satisfied. Since X f ,g (resp. Y f ,g ) is an extension X f and X g (resp. Y f and Y g ), it only remains to treat the case of the 3-cell . For every i, j ∈ N * , we have: Concerning d f ,g , the 3-cells from PFonct[f ] were already treated in Proposition 3.1.5. For the others we have, for every i, j, k ∈ N * : Definition 3.3.7. We define a weight application w as the 1-functor from PNTrans[f , g] * 1 to N, defined as follows on PNTrans[f , g] 1 : • for all a, b ∈ C, w( a b ) = 1, • for all a, b ∈ D, w( a b ) = 1, • for all a ∈ C and b ∈ D, w( a b ) = 0. . This theorem will be proven in Section 4. Contrary to the case of bicategories and pseudofunctors, we cannot directly apply Theorem 1.4.4 to the (4, 2)-polygraph PNTrans[f , g], because the following critical pair is not confluent: Theorem 1.4.9 will be used in order to avoid this difficulty.

Proof of the coherence for pseudonatural transformations
In this section we prove Theorem 3.3.8. We fix for the rest of this section two sets C and D, together with two applications f , g : C → D. Let A, B ∈ PNTrans[f , g] * (2) be 3-cells whose 1-target is of weight 1. We want to build a 4-cell α : A 1 c B ∈ PNTrans[f , g] * (2) . The 1-cells of weight 1 are of one of the following forms, with a, a ∈ C and b, b ∈ D: In Section 4.1, we show that if the common 1-target of A and B is not of the last form, then they are generated by a sub-4-polygraph PFonct[f , g] of PNTrans[f , g]. We then show using Theorem 1.4.4 that this 4-polygraph is coherent.
There remains to treat the case where the 1-target of A and B is of the last form. We define two sub-(4, 2)-polygraphs of PNTrans PNTrans PNTrans In Section 4. We then define a sub-3-polygraph PNTrans u [f , g] of PNTrans[f , g]. The rewriting system induced by the 3-cells PNTrans u [f , g] corresponds to simplifying the units out.
Using the properties of this rewriting system, we extend the result of Section 4.2, first to 3-cells A and B in PNTrans + [f , g] in Section 4.3, and finally to general A and B whose 1-target is a f (a) b in Section 4.4, thereby concluding the proof.

A convergent sub-polygraph of PNTrans[f , g]
where i and j are non-zero integers, the a k are in C and the b k are in D.
Proof. Let us show first that the set of all 1-cells of the form (8) is stable when rewritten by PFonct[f , g] * 2 . To prove this, we examine the case of every cell of PFonct[f , g] * 2 of length 1: bj : a1 ai f (ai) b1 bj ⇒ a1 ai−1 f (ai−1) f (ai) b1 bj Let us now prove the lemma: we reason by induction on the length of h. If h is of length 0, it is an identity, so h is in PFonct[f , g] * .
If h is of length 1 and h is not in PFonct[f , g] * , then h has to be of the form . So its target is of the form: which is indeed of the form (8), with b 1 = g(a k ). Let now h be of length n > 1. We can write h = h 1 1 h 2 , where h 2 is of length 1, and h 1 is strictly shorter than h. Let us apply the induction hypothesis to h 2 . If the target of h 2 is of the form (8), then so is the target of h, since t(h 2 ) = t(h). Otherwise, then h 1 ∈ PFonct[f , g] * , and we can apply the induction hypothesis to h 2 . If h 2 also is in PFonct[f , g] * , then so is h.
It remains to treat the case where t(h 1 ) is of the form (8) , and h 2 is in PFonct[f , g] * . But we have shown that the 1-cells of the form (8)   , one of the following holds: • The 1-target of A is of the form (8).
Proof. Let us start by the case where A is a 3-cell of length 1 in PNTrans[f , g] * 3 . If the 1-target of A is not of the form (8) then, according to Lemma 4.
, and so is A. In the general case, A is a composite of 3-cells of one of the two previous forms, and all of them have the same 1-target as A. Thus if the 1-target of A is not of the form (8), all those 3-cells are in , and so is A.

The lexicographic order on
H h   Table 4.

Adjunction of the units 2-cells
is said unitary if it is generated by the sub-2-polygraph of PNTrans[f , g] whose only 2-cells are and .

Lemma 4.3.2.
Let h ∈ PNTrans[f , g] * 2 whose target is of the form a f (a) b , where a ∈ C and b ∈ D. If there is a decomposition h = h 1 1 h 2 , where h 1 ∈ PNTrans u [f , g] * and h 2 ∈ PNTrans[f , g] * are not identities, and h 1 is a unitary 2-cell, then there is a 3-cell A ∈ PNTrans u [f , g] * 3 of source h which is not an identity.
Proof. Let us start with the case where h 1 is of length 1. We reason by induction on the length of h 2 . If h 2 is of length 1, since the target of h 2 is of the form a f (a) b , h 2 is one of the following 2-cells: Hence h is one of the following 2-cells: And all of these 2-cells are indeed the sources of 3-cells in PNTrans u [f , g] * 3 . In the general case, let us write h 2 = h 0 1 h 2 , where h 0 is of length 1. Two cases can occur.
• Otherwise, h 1 1 h 0 is one of the following 2-cells, and all of them are sources of 3-cells in PNTrans u [f , g] * .
In the case general case where h 1 is of any length, let h 1 , h 1 ∈ PNTrans[f , g] * 2 with h 1 of length 1 such that h 1 = h 1 1 h 1 . Then there is a non-empty 3-cell A ∈ PNTrans u [f , g] * 3 of source h 1 1 h 2 , and one can take the 3-cell h 1 1 A . If h is a normal form for PNTrans u [f , g], then one of the following holds: • The 2-cell h equals the composite .
Proof. We reason by induction on the length of h. If h is of length 1, the cells of PNTrans[f , g] * 2 of length 1 and of target a f (a) b are: Otherwise, let us write h = h 1 1 h 2 , where h 1 is of length 1. We can apply the induction hypothesis to h 2 , which leads us to distinguish three cases: a f (a) . The only such cell is the identity, and h = h 2 = .
• If h 1 and h 2 are in PNTrans • Lastly, if h 2 is in PNTrans ++ [f , g] * and h 1 is in PNTrans u [f , g] * , then because of Lemma 4.3.2, h is the source of a 3-cell in PNTrans u [f , g] * of length 1, which is impossible since, by hypothesis, h is a normal form for PNTrans u [f , g].

Adjunction of the units 3-cells
In this section, we consider the rewriting system formed by the 3-cells of PNTrans u [f , g]. Since it is a sub-3-polygraph of PNTrans[f , g] (which 3-terminates by Proposition 3.3.6), PNTrans u [f , g] is 3-terminating. The fact that it is 3-confluent is a consequence of the following more general Lemma: and B ∈ PNTrans u [f , g] * and a 4-cell α A,B ∈ PNTrans[f , g] * (2) 4 of the following shape: Proof. Let us start by the case where (A, B) is a critical pair of PNTrans[f , g] 3 . If A and B are in Otherwise, the only critical pair left is the following one: Let us now study the case where (A, B) is a local branching of PNTrans[f , g] 3 . We distinguish three cases depending on the shape of the branching: • If (A, B) is an aspherical branching, then one can take identities for A and B , and α = 1 A .
• If (A, B) is a Peiffer branching, let A and B be the canonical fillers of the confluence diagram of (A, B), and α be an identity. • If A or B is an identity, then the result holds immediately.
• Otherwise, we write A = A 1 2 A 2 and B = B 1 2 B 2 , where A 1 and B 1 are of length 1. We now build the following diagram: In this diagram, α A1,B1 is obtained thanks to our study of the local branchings. The existence of α A2,B 1 and α A 1 ,B2 (followed by α A 2 ,B 2 ) then follows from the induction hypothesis. • If it is a Peiffer branching, then the required cell is provided by the canonical filling.
• If it is an overlapping branching, then it is enough to check the underlying critical pair.
It remains to examine those critical pairs: Proof. We reason by induction on the length of A: • If A is an identity, then it is in PNTrans + [f , g].
• Otherwise, let us write . There exist C 1 , C 2 ∈ PNTrans u [f , g] * 3 whose target is a normal form for PNTrans u [f , g], a 3-cell A ∈ PNTrans + [f , g] * (2) 3 and a 4-cell α ∈ PNTrans[f , g] * (2) 4 of the following shape: The following is a consequence of the target of D i being a normal form for PNTrans u [f , g]: • Since PNTrans u [f , g] is 3-convergent, for any i < n, the cells D i and D i+1 are parallel.
Since PNTrans u [f , g] is a sub-polygraph of PFonct[f , g] which is 3-coherent, there exists, for every i < n, a 4-cell γ i : . We can now conclude the proof of this Lemma by taking C 1 = D 1 , C 2 = D n and A = ( 2 B n , and by defining α as the following composite: We can now conclude the proof Theorem 3.3.8. . According to Lemma 4.4.4, there exist C 1 , C 2 , C 1 , C 2 ∈ PNTrans u [f , g] * whose targets are normal forms for PNTrans u [f , g], A , B ∈ PNTrans + [f , g] * (2) and α 1 , such that we have the diagrams: The 3-cells A and B are parallel, and the 3-cells C 1 and C 2 (resp. C 1 and C 2 ) have the same source and have a normal form for PNTrans u [f , g] as target. Since PNTrans u [f , g] is 3-convergent, this implies that the 3-cells C 1 and C 2 (resp. C 1 and C 2 ) are parallel. This has two consequences: • The critical pairs of PNTrans u [f , g] already appeared in PFonct[f , g], and we showed that they admit fillers. Hence there exist cells β 1 : C 1 1 c C 2 and β 2 : • To conclude, we define α as the following composite (where we omit the context of the 4-cells):

Transformation of a (4, 2)-polygraph into a (4, 3)-white-polygraph
The proof of Theorem 1.4.9 will occupy the rest of this article. We start with a (4, 2)-polygraph A satisfying the hypotheses of Theorem 1.4.9. Let S A be the set of all 2-cells in A * 2 whose target is a normal form. Then proving Theorem 1.4.9 consists in showing that A is S A -coherent.
In this section we successively transform A four times, leading to five pointed (4, 3)-white-categories, namely (A * (2) , S A ), (B w(2) , S B ), (C w(3) , S C ), (D w(3) , S D ) and (E w(3) , S E ), and we show each time that the new pointed (4, 3)-white-category is stronger than the previous one. A brief description of each pointed (4, 3)-white-category can be seen in Table 5. Finally in Section 5.5, we perform a number of Tietze-transformations on the 4-white-polygraph E, leading to a 4-white-polygraph F.
Thanks to Lemma 2.2.8 and Proposition 2.2.14, we know that in order to show that A * (2) is S Acoherent, it is enough to show that F w(3) is S E -coherent. This will be done in Section 6.

Name
Description  In what follows, we will use as a running example the polygraph A = Assoc which consists of one 0-cell, one 1-cell , one 2-cell : ⇒ , one 3-cell : , and one 4-cell : In particular, Assoc satisfies the 2-Squier condition of depth 2. The 2-category Assoc * 2 is 2convergent and its only normal form is the 1-cell .
The corresponding set S A is then the set of 2-cells in Assoc * 2 from any 1-cell to .

Weakening of the exchange law
We construct dimension by dimension a (4, 2)-white-polygraph B, together with a white-functor F : B w(2) → A * (2) . We then define a subset S B of B w(2) and show (Proposition 5.1.4) using F that (B w(2) , S B ) is stronger than (A * (2) , S A ).
In low dimensions, we set B i = A i , for every i ≤ 2, and the functor F is the identity on generators.
Proof. By construction, A * 2 is the quotient of B w 2 by the equivalence relation generated by: And F is the canonical projection induced by the quotient.
In what follows, we suppose chosen a section i : A * → B w of F , which is possible thanks to Lemma 5.1.1.
We extend B into a 3-white-polygraph and F : B w → A * into a 3-white-functor by setting B 3 := A 3 ∪ K: • The set K is the set of 3-cells A f v,ug , of shape: Proof. Let f, g ∈ B w 2 . The image of any cell in K w(2) 3 by F is an identity. So if there exists a 3-cell , necessarily F (f ) = F (g). Conversely, the set A * 2 is the quotient of B w 2 by the equivalence relation generated by: is a strict Peiffer branching, generate this relation, and they are in K. Hence the result. satisfying F (B) = A. Let us show that E = A * 3 . We already know that E contains the identities thanks to Lemma 5.1.2.
The 3-cells of length 1 in A * 3 are in E. Indeed, let A ∈ A * 3 be a 3-cell of length 1, and f, g ∈ B w 2 such that F (f ) = s(A) et F (g) = t(A). There exist u, v ∈ A * 1 , f , g ∈ A * 2 , and A ∈ A 3 such that Letũ,ṽ,f ,g be in the preimages respectively of u, v, f , g under F (they exist thanks to Lemma 5.1.1), and let B 1 :=f 1 (ũA ṽ) 1g ∈ B w(2) 3 . By construction, F (B 1 ) = A, which leads to the equalities: Thus, according to Lemma 5.1.2, there exist 3-cells . Let B := C 1 2 B 1 2 C 2 : by construction, B has the required source and target, and moreover: The set E is stable under composition. Indeed let A 1 , A 2 ∈ E such that t(A 1 ) = s(A 2 ), and f, g ∈ B w 2 satisfying F (f ) = s(A 1 ) and The set E is stable under 2-composition. Indeed let A ∈ E and f, g ∈ B w 2 such that F (f ) = s(A −1 ) and F (g) = t(A −1 ). There exists B ∈ B w(2) such that: Hence the cell B −1 satisfies the required property.
We now extend B into a (4, 2)-white-polygraph and F : B w(2) → A * (2) into a 4-white-functor by setting B 4 = A 4 ∪ L: • For every 3-cell A ∈ A 4 , the source and target of A in B • For every 3-fold strict Peiffer branching (f, g, h), the set L contains a 4-cell A f,g,h , whose shape depends on the form of the branching (f, g, h). If (f, g, h) = (f v, g v, uh ), with (f , g ) a critical pair, and h : v ⇒ v then A f,g,h is of the following shape: If (f, g, h) = (f v, ug , uh ), with (g, h) a critical pair, and f : u ⇒ u then A f,g,h is of the following shape: . And we define F (A f,g,h ) := 1 f 0Ag ,h .
If (f, g, h) = (f vw, ug w, uh w), then A f,g,h is of the following shape, where A and B are in K w(2) 3 : Let now S B be the set of all 2-cells in B w whose 1-target is a normal form.

Weakening of the invertibility of 3-cells
We construct dimension by dimension a 4-white-polygraph C, together with a 3-white-functor G : C w(3) → B w (2) . We then define a subset S C of C w(3) and show (Proposition 5.2.2) using G that (C w(3) , S C ) is stronger than (B w(2) , S B ).
In low dimensions, we set C i = B i for i ≤ 2, with the functor G being the identity. We extend C into a 3-white-polygraph by setting C 3 := B 3 ∪ B op 3 , where the set B op 3 contains, for every A ∈ B 3 , a cell denoted by A op , whose source and target are given by the equalities: And the functor G : C w → B w(2) is defined as follows for every A ∈ B 3 : Proof. By definition, B w(2) 3 is the quotient of C w 3 by the relations A op 2 A = 1 and A 2 A op = 1, and G is the corresponding canonical projection.
We extend C into a 4-white-polygraph by setting C 4 := B 4 ∪ {ρ A , λ A |A ∈ B 3 }, where the applications source and target s, t : C 4 → C w 3 are defined as follows: • For A ∈ B 4 , the cell s C (A) (resp. t C (A)) is any cell in the preimage of s B (A) under G, which is non-empty thanks to Lemma 5.2.1. And we set G(A) := A.
• For every A ∈ B 3 , the cells ρ A and λ A have the following shape: Let S C be the set of all 2-cells in C w whose 2-target is a normal form. Proof. The functor G restricts into a functor G S C : C w(3) S C → B w(2) S B , which is i-surjective for i < 2 thanks to Lemma 5.2.1. Hence we can conclude thanks to Lemma 2.2.8.

Adjunction of formal inverses to 2-cells
Let D be the 4-white-polygraph defined as follows: where for every f ∈ C 2 , the setC 2 contains a cellf with source t(f ) and with target s(f ). Let S D be the set of all 2-cells of the sub-2-white-category C w 2 of D w 2 whose target is a normal form.
Notation 5.3.1. The application C 2 →C 2 extends into an application C w 2 →C 2 w which exchanges the source and targets of the 2-cells.
We denote a 2-cell f by and by e y f 7 W if f is any cell in C w 2 . , and suppose that f is in D w(3) S D . In particular t 2 (f ) and s 2 (f ) are in C w 2 . Since ι also satisfies the hypotheses of Proposition 2.3.11, with k 0 = 2, it is 2-discriminating on D w(3) i . Thus f is in C w (3) , and in C w(3) S C since its 1-target is a normal form.

Adjunction of connections between 2-cells
Let E be the following 4-white-polygraph: The cells η f , f , τ f and σ f have the following shape: Let R := {σ f , τ f }, and R w (resp. R w(3) ) be the sub-4-white-category (resp. sub-(4, 3)-whitecategory) of E w(3) generated by the cells in R. A 4-cell of length 1 in R w is called an R-rewriting step.
Let S E be the set of all 2-cells of the sub-2-white-category C w 2 of E w 2 whose target is a normal form. Using properties of the rewriting system induced by R w , we are going to define a functor K : Lemma 5.4.2. Let α ∈ E w 4 and β ∈ R w of length 1 with the same source. There exist α ∈ E w 4 and β ∈ R w of maximum length 1, such that: Proof. The result holds whenever (α, β) is a Peiffer or aspherical branching.
If (α, β) is an overlapping branching, then the source of α must contain an η f or an f . The only cells of length 1 in E w 4 that satisfy this property are those in R w . Hence α is in R w . Thus the branching (α, β) is one of the following two, and both of them satisfy the required property: • Otherwise, we have B 1 = f 1 1 f 1 f 2 , with f 2 = f 1 f 2 . We then get the required factorisation of A by setting A 1 = 1 f 2 and Lemma 5.4.5. Let β ∈ R w , and α be a 4-cell E w 4 of same source. There exist α ∈ E w 4 and β ∈ R w of maximum length that of β such that we have the following square: Proof. We reason using a double induction on the lengths of β and α. If β (resp. α) is an identity, then the result holds by setting α = α (resp. β = β).
Lemma 5.4.6. The application A →Â extends into a 1-functor K : E w(3) S E → D w(3) S D , which is the identity on objects.
Proof. The application A →Â does not change the source or target. Moreover, given a 3-cell A ∈ E w(3) , if A is in E w(3) S E then in particular the source and target of A are in C w 2 . ThusÂ is in D w 3 S D (Lemma 5.4.4).
Let A, B be 3-cells in E w(3) which belong to E w(3) S E . We just showed thatÂ andB are in D w 3 S D , hence so isÂ 2B . SoÂ 2B is a normal form for R which is attainable from A 2 B. Since R is 4-convergent, this means that A 2 B =Â 2B . So A →Â does indeed define a functor. . Suppose that α lies in E w 4 . Let β ∈ R w be a cell from A toÂ. Applying Lemma 5.4.5 to α and β, we get cells α and β of sources respectivelyÂ and B. Let B be their common target. By hypothesisÂ is in D w 3 , and the only cells in E w 4 whose source is in D w 3 are the cells in D w 4 . Thus α is in D w 4 , and so is B . So B is a normal form for R w which is attainable from B. By unicity of the R w -normal-form, B =B, and so α is a cell in D w 4 of source K(A) and of target K(B), hence K(A) = K(B).
In general ifĀ =B, there exist A 1 , . . . , A n ∈ E w 3 with A 1 = A, B n = B and for every i there exist cells α i : A 2i 1 c A 2i−1 and β i : A 2i → A 2i+1 in E w 4 . Hence using the previous case K(A 1 ) = . . . = K(A n ), that is K(A 1 ) = K(A n ).
SoK : E w(3) S E → D w(3) S D is well defined, and it is 0 and 1-surjective because K is. Hence (E, S E ) is stronger than (D, S D ).
Example 5.4.8. In the case where A = Assoc, let A = . The set E 3 contains the following 3-cells: And the set E 4 the following 4-cells:

Reversing the presentation of a (4, 3)-white-category
We start by collecting some results on the cells of E.
Lemma 5.5.1. The set E 3 is composed exactly of the following cells: • For every f ∈ A 2 , 3-cells η f and f .
• For every non-aspherical minimal branching (f, g), a 3-cell A f,g of shape: And in particular for every non-aspherical minimal branching (f, g), we have A op f,g = A g,f . Proof. If (f, g) is a critical pair: if it was associated to a 3-cell in A then A f,g is this corresponding cell. Otherwise A f,g is in fact the cell A op g,f from Section 5.2. If (f, g) is a strict Peiffer branching, then A f,g is the cell defined in Section 5.1. Otherwise, (g, f ) if v e (f ) < v e (g), then there exists e > e such that v e (f ) > v e (g). Lemma 6.2.2. Let E be a set and a ∈ E. The set of all f ∈ N[E] such that f < a is equal to the set of all f ∈ N[E] satisfying the following implication for every b ∈ E: In particular, this set is a sub-monoid of N[E]. Proof.
Necessarily v a (f ) = 0, otherwise we would have a < a. Thus in particular f = a.
By definition of f this implies that b < a, and since 0 = v a (f ) < v a (a) = 1 we get that f < m a .
Conversely, let f < m a. Let us show by contradiction that v a (f ) = 0.
If v a (f ) = 0, we distinguish two cases: Hence necessarily v a (f ) = 0. Let b ∈ E such that v b (f ) > 0, and let us show that b < a. We just showed that b = a, and so v b (f ) > v b (a). Thus there exists c > b such that v c (a) > v c (f ). In particular this implies v c (a) > 0. So c = a and finally a > b. Lemma 6.2.3. Let (E, <) be a set equipped with a strict ordering. The relation > m is compatible with the monoidal structure on N(E), that is, for every f, f , g ∈ N(E), if f > m f , then f + g > m f + g.
Proof. Let f, f , g ∈ N(E), and suppose that f > m f . Let us show that f + g > m f + g. Firstly, f = f , hence f + g = f + g.
Let e ∈ E such that v e (f + g) < v e (f + g).
Since v e is a morphism of monoids, this implies that v e (f ) < v e (f ). Hence there exists e > e such that v e (f ) > v e (f ), and so v e (f + g) > v e (f + g) The proof of the following theorem can be found in [1]. Theorem 6.2.4. Let (E, >) be a set equipped with a strict ordering. Then > m is a well-founded ordering if and only if > is.
Since A is 2-terminating, the set A * 1 is equipped with a well-founded ordering ⇒. This induces a well founded ordering ⇒ m on N[A * 1 ]. We now define two applications p : . Using ⇒ m , those applications induce well-founded orderings on F w 2 and F w 3 . We then show a number of properties of these applications in preparation for Section 2.2. Definition 6.2.5. We define an application p : • for every composable f 1 , f 2 ∈ F w 2 , we set p(f 1 1 f 2 ) := p(f 1 ) + p(f 2 ). For every f, g ∈ F w 2 , we set f > g if p(f ) ⇒ m p(g). The relation > is a well-founded ordering of F w 2 .
• For every 3-cell A ∈ F 3 and u, v ∈ A * 1 , if A is not an η f then w η (uAv) = 0.
Definition 6.2.7. A product of the formf 1 g ∈ F w 2 , where f and g are nonempty cells in B w 2 is called a cavity. It is a local cavity if f and g are of length 1. Let C F be the set of all cavities. Lemma 6.2.8. Let f, g ∈ B w 2 . Suppose f is not an identity and t(f ) = s(g). The following inequality holds: s(f ) > p(g) Proof. We reason by induction on the length of g. If g is empty, then p(g) = 0 < s(f ).
In particular for every cell C f,g , we have s(C f,g ) > t(C f,g ).
Considering the second one, using Lemma 6.2.8, we have the inequalities u = s(f 1 ) > p(g 1 ) and u = s(f 2 ) > p(g 2 ). By 6.2.2, we then have u > p(g 1 ) + p(g 2 ) = p(t(A)). Definition 6.2.10. Let h ∈ F w 2 . A factorisation h = h 1 1f1 1 f 2 1 h 2 of h, with f 1 , f 2 ∈ B w 2 of length 1 and h 1 , h 2 ∈ F w 2 is called a cavity-factorisation of h. Thus a cavity-factorisation is represented as follows: Lemma 6.2.11. Let h ∈ F w 2 be a 2-cell which is not an identity, and whose source and target are a normal form for A 2 . Then there exists a cavity-factorisation of h.
Let us show that g 1 and g 2n are not identities: • If g 1 is an identity, then since h isn't, either n ≥ 2 or n = 1 and g 2n is not an identity. In both cases g 2 is of length at least 1, and has s(h) as target, which contradicts the fact that s(h) is a normal form for A 2 .
• The case where g 2n is an identity is symmetric.
Lemma 6.2.12. Let h ∈ F w 2 be a 2-cell of source and targetû, a normal form for A 2 . There exists a 3-cell A : h 1û such that w η (A) = 0.
Proof. We reason by induction on h using the ordering >. If h is minimal, then h = 1û and we can set A := 1 h . Otherwise by Lemma 6.2.11 there exists a cavity-factorisation h = h 1 1 f 1 1 f 2 1 h 2 of h. Let A 1 := C f1,f2 : we have w η (A 1 ) = 0 and by Lemma 6.2.9, s(A 1 ) > t(A 1 ). Since the ordering is compatible with composition, we get h > h 1 1 t(A 1 ) 1 h 2 . By induction hypothesis, there exists a 3-cell A 2 : h 1 1 t(A 1 ) 1 h 2 1û ∈ F w 3 such that w η (A 2 ) = 0. Let A := (h 1 1 A 1 1 h 2 ) 1 A 2 . We have w η (A) = w(h 1 1 A 1 1 h 2 ) + w(A 2 ) = w(A 1 ) + 0 = 0. Lemma 6.2.13. Let h ∈ F w 2 of source and targetû a normal form for A 2 , and A : h 1û ∈ F w 3 . For every cavity-factorisation h = h 1 1f1 1 f 2 1 h 2 , there exists a factorisation of A = (h 1 1 A 1 1 h 2 ) 2 A 2 , with A 1 , A 2 ∈ F w 3 , and either A 1 = C f1,f2 or A 1 =f 1 1 η f3 1 f 2 , with f 3 ∈ B w 2 of length 1. Proof. We reason by induction on the length of A. If A is of length 0, then there is no cavity-factorisation of h = 1û and the result holds.
If A is not of length 0, let h = h 1 1f1 1 f 2 1 h 2 be a cavity-factorisation of h. Let us write A = B 1 C, where B is of length 1. If B is not of the required form, then either B = B 1f1 1 f 2 1 h 2 , or B = h 1 1f1 1 f 2 1 B . Let us treat the first case, the second being symmetrical. The source of C admits a cavity-factorisation s(C) = t(B ) 1f1 1 f 2 1 h 2 . By induction hypothesis, we can factorise C as follows: A 1 C Lemma 6.2.14. Let h ∈ F w 2 and u ∈ A * 1 such that u > p(h), u > s(h) and u > t(h). For every 3-cell A ∈ F w 3 of source h, the inequality u > w η (A) holds. Proof. We reason by induction on the length of A. If A is of length 0, w η (A) = 0 and the result holds.
Otherwise, let us write A = A 1 2 A 2 , with A 1 of length 1. We distinguish two cases depending on the shape of A 1 .
• Otherwise, we have on the one hand that w η (A 1 ) = 0, and on the other hand that s(A 2 ) = t(A 1 ) < s(A 1 ) = h < u by Lemma 6.2.9. Thus w η (A) = w η (A 2 ) < u.
In order to construct the 4-cells α 1 and α 2 , let us show that we can apply the induction hypothesis to the couples (A 2 , D 1 2 D 3 ) and (D 2 2 D 3 , B 2 ). Let v be the common source of f 1 and f 2 .
Hence we can apply the induction hypothesis to the couples (A 2 , D 1 2 D 3 ) and (D 2 2 D 3 , B 2 ), which provides α 2 and α 3 . Proof. Let A, B : f h ∈ F w 3 whose 1-target is a normal formû, with f, g ∈ B w 2 . The 3-cells (h 1 A) 2 h and (h 1 B) 2 h are parallel, and their target is 1û. In particular they verify the hypothesis of Proposition 6.3.1. So there exists α : (h 1 A) 2 h 1 c (h 1 B) 2 h . Then the following composite is the required cell from A to B: We can now complete the proof of Theorem 1.4.9. Indeed we showed that F w(3) is S E -coherent. Using Proposition 2.2.14, that means that E w(3) is S E -coherent, and finally using Lemma 2.2.8 that A * (2) is S A -coherent, that is that for every 3-cells A, B ∈ A * (2) 3 , whose 1-target is a normal form, there exists a 4-cell α : A 1 c B ∈ A * (2) 4 .