Léonard Guetta

POLYGRAPHS AND DISCRETE CONDUCHÉ ω-FUNCTORS

We define a class of morphisms between strict ω-categories called discrete Conduché ω-functors that generalize discrete Conduché functors between 1-categories and we study their properties related to polygraphs. The main result we prove is that for every discrete Conduché ω-functor f : C → D, if D is a free strict ω-category on a polygraph then so is C.

Introduction

In [START_REF] Giraud | Méthode de la descente[END_REF]Theorem 4.4], Giraud introduced necessary and sufficient conditions for a functor f : C → D to be exponentiable in the category of (small) categories Cat, i.e. such that the pullback functor

f * : Cat/D → Cat/C
induced by f admits a right adjoint. A functor satisfying these conditions is usually called a Conduché functor or Conduché fibration (named after Conduché who rediscovered Giraud's theorem in [START_REF] Conduché | Au sujet de l'existence d'adjoints à droite aux foncteurs "image réciproque" dans la catégorie des catégories[END_REF]). In the present article, we will focus on a particular case of this notion. Recall that a category C is free on a graph G if

C ≃ L(G)
where G is a graph and L is the left adjoint to the forgetful functor from Cat to the category of graphs. It was remarked in [START_REF] Street | Categorical structures[END_REF] that discrete Conduché functors, called ulf functors there, have some properties related to free categories on graphs. For example, the following theorem follows immediatly from the first section of op. cit. In the setting of strict ω-categories, that we shall simply call ω-categories, the notion of free category on a graph can be generalized to the notion of free ω-category on a polygraph in the terminology of [START_REF] Burroni | Higher-dimensional word problems with applications to equational logic[END_REF] (or free ω-categories on a computad in the terminology of [START_REF] Street | Limits indexed by category-valued 2-functors[END_REF] or [START_REF] Makkai | The word problem for computads[END_REF]).

In the present paper we shall introduce a notion of discrete Conduché functor between ω-categories and prove the following generalization of the previous theorem.

Theorem 1. Let C and D be ω-categories and f : C → D be a discrete Conduché ω-functor. If D is free on a polygraph then C is free on a polygraph.

We will even be more precise and explicitly construct the polygraph generating C from the one generating D. As a by-product we will also prove the following theorem.

Theorem 2. Let C and D be ω-categories and f : C → D be a discrete Conduché ω-functor. If C is free on a polygraph and f n : C n → D n is surjective for every n ∈ N, then D is free on a polygraph. Note also that, as one would expect, discrete Conduché ω-functors are exponentiable morphisms in the category of ω-categories. However the proof of that fact goes beyond the scope of this paper.

The original motivation for the present paper comes from a seemingly unrelated topic. Let D(Z) be the localization of the category of chain complexes with respect to quasi-isomorphisms and Cat ω the category of ω-categories and ω-functors. In [START_REF] Métayer | Resolutions by polygraphs[END_REF], Métayer defines a functor H pol (-) : Cat ω → D(Z) called the polygraphic homology functor by means of a so-called polygraphic resolution. As it turns out, free ω-categories on polygraphs are the cofibrant objects of a "folk" model structure on Cat ω and the polygraphic homology functor can be understood as the left derived functor of a well-known abelianization functor (see [START_REF] Métayer | Resolutions by polygraphs[END_REF], [START_REF] Métayer | Cofibrant objects among higher-dimensional categories[END_REF] and [START_REF] Lafont | A folk model structure on omega-cat[END_REF]).

In [START_REF] Lafont | Polygraphic resolutions and homology of monoids[END_REF], Lafont and Métayer prove that when we restrict this functor to the category of monoids, considered as a subcategory of Cat and hence of Cat ω , then it is isomorphic to the "classical" homology functor of monoids (which can be defined as the singular homology of the classifying space of the monoid).

While extending the previous result from monoids to 1-categories [Gue], I encountered the following question:

Let f : P → C be an ω-functor with P a free ω-category on a polygraph, C a 1-category and let c be an object of C. Consider the ω-category P/c defined as the following fibred product in Cat ω :

P/c P C/c C f
where the anonymous arrow from the slice category C/c to C is the obvious forgetful functor. Question: Is P/c free on a polygraph? Now, it is straightforward to check that the arrow C/c → C is a discrete Conduché functor. Moreover, as we shall see, discrete Conduché functors are stable by pullback. Hence, the arrow P/c → P is a discrete Conduché ω-functor. Then Theorem 1 provides a positive answer to the previous question.

The same strategy also yields an alternative proof of Proposition 6 of [START_REF] Lafont | Polygraphic resolutions and homology of monoids[END_REF]. It suffices to notice that the so-called "unfolding" of an ω-functor

f : P → M,
where M is a monoid (definition 13 of op. cit.) is just the category P/⋆, with ⋆ the only object of M when seen as a category.

1.1. An ω-graph consists of -a sequence (C n ) n∈N of sets, -maps s k n , t k n : C n → C k for all k < n ∈ N, subject to the globular identities s l n = s l k • s k n = s l k • t k n , t l n = t l k • t k n = t l k • s k n , whenever l < k < n ∈ N. When the context is clear, we often write s k (resp. t k ) instead of s k n (resp. t k n ). Elements of C n are called n-cells. For an n-cell x and k < n, s k (x) is its k-source and t k (x) its k-target. When n > 0, we use s(x) (resp. t(x)) as a synonym for s n-1 n (x) (resp. t n-1 n (x)
). Two n-cells x and y are parallel if n = 0 or n > 0 and s(x) = s(y) and t(x) = t(y).

We define the set C n × C k C n as the following fibred product

C n × C k C n C n C n C k . t k s k That is, elements of C n × C k C n are pairs (x, y) of n-cells such that s k (x) = t k (y).
We say that two n-cells x and y are k-composable if the pair (x, y) belongs to

C n × C k C n .
1.2. Given two ω-graphs C and D, a morphism of ω-graphs

f : C → D is a sequence of maps (f n : C n → D n ) n∈N such that, for all k < n ∈ N, both diagrams C n D n C k D k fn s k s k f k C n D n C k D k fn t k t k f k are commutative.

An ω-category consists of an ω-graph C together with maps

* n k : C n × C k C n → C n 1 n k : C k → C n for each pair k < n subject to the following axioms.
Source and target axioms:

(1) For all l ≤ k < n ∈ N, for all k-composable n-cells x and y, we have

s l (x * n k y) = s l (x), t l (x * n k y) = t l (x).
(2) For all k < l < n ∈ N, for all k-composable n-cells x and y, we have

s l (x * n k y) = s l (x) * l k s l (y), t l (x * n k y) = t l (x) * l k t l (y). (3) For all k < n ∈ N, for every k-cell x, we have s k (1 n k (x)) = x = t k (1 n k (x)).

Unit axioms:

(4) For all l < k < n ∈ N,

1 n k • 1 k l = 1 n l . (5) For all k < n ∈ N, for every n-cell x, we have x * n k 1 n k (s k (x)) = x = 1 n k (t k (x)) * n k x. ( 6
) For all k < l < n ∈ N, for all k-composable n-cells x and y, we have

1 n l (x * l k y) = 1 n l (x) * n k 1 n l (y).

Associativity axiom:

(7) For all k < n ∈ N, for all n-cells x, y and z such that x and y are k-composable, and y and z are k-composable, we have

(x * n k y) * n k z = x * n k (y * n k z).

Exchange law:

(8) For all k < l < n ∈ N, for all n-cells x, x ′ , y and y ′ such that -x and y are l-composable, x ′ and y ′ are l-composable, -x and x ′ are k-composable, y and y ′ are k-composable, we have

((x * n k x ′ ) * n l (y * n k y ′ )) = ((x * n l y) * n k (x ′ * n l y ′ )).
The same letter will refer to an ω-category and its underlying ω-graph. We will almost always write * k instead of * n k , and, for an n-cell x, 1 x will sometimes be used as a synonym for 1 n+1 n (x). Moreover, for consistency, we set 1 n n (x) := x for any n-cell x. 1.4. Let C and D be ω-categories. An ω-functor is a morphism of ω-graphs f : C → D that satisfies the following axioms:

(1) For all k < n ∈ N, for all k-composable n-cells x and y, we have

f n (x * k y) = f n (x) * k f n (y).
(2) For all k < n ∈ N, for every k-cell x, we have

f n (1 n k (x)) = 1 n k (f (x))
. For an n-cell x, we will often write f (x) instead of f n (x). The category of ω-categories and ω-functors is denoted by Cat ω .

1.5. Let x be a k-cell in an ω-category. We say that x is degenerate if there exists

x ′ ∈ C k ′ with k ′ < k such that x = 1 k k ′ (x ′
). Note that 0-cells are never degenerate.

1.6. For n ∈ N, an n-category is an ω-category such that every k-cell with k > n is degenerate. An n-functor is an ω-functor between two n-categories. The category of n-categories and n-functors is denoted by Cat n .

There is an obvious inclusion functor

Cat n → Cat ω .
This functor has a left and a right adjoint. In the sequel, we shall only use the right adjoint, which will be denoted by

τ ≤n : Cat ω → Cat n .
For an ω-category C, the ω-category τ ≤n (C) is obtained by removing all non-degenerate k-cells of C such that k > n.

Remark 1.7. It follows from the axioms of ω-categories and ω-functors that, for n-categories and n-functors, everything involving k-cells such that k > n can be recovered from the rest. For example, we will often consider that the data of an n-category C only consists of

-(C k ) 0≤k≤n , -(s k l ) 0≤k<l≤n , -(t k l ) 0≤k<l≤n , -( * l k ) 0≤k<l≤n , -(1 l k ) 0≤k<l≤n .
1.8. Let n ∈ N. It follows from the definition of ω-categories and ω-functors that we have a functor Cell n : Cat ω → Set that associates to each ω-category C, the set C n of its n-cells and to an ω-functor f : C → D, the map

f n : C n → D n .
This functor is representable and we define the n-globe D n to be the ωcategory representing this functor. (D n is in fact an n-category.) Here are some pictures in low dimension:

D 0 = • , D 1 = • • , D 2 = • • , D 3 = • • ⇛ .
We will make no distinction between an n-cell x and the ω-functor

x : D n → C associated to it. For k < n ∈ N,
f n × f k f n : C n × C k C n → D n × D k D n .
This functor is represented by the ω-category

D n D k D n
(which is also an n-category) defined as the following amalgamated sum

D k D n D n D n D k D n . σ k n τ k n
The arrow * n k induces a natural transformation ∇ n k : Comp n k → Cell n which in turn is represented by an ω-functor

∇ n k : D n → D n D k D n .
For example, having a commutative triangle

D n C D n D k D n x ∇ n k (y,y ′ )
means exactly that we have n-cells x, y, y ′ of C such that y and y ′ are kcomposable and x = y * k y ′ .

Polygraphs

Definition 2.1. Let C be an ω-category and Σ ⊆ C k with k ∈ N. We say that Σ is a k-basis if either k = 0 and Σ = C 0 or k > 0 and the following universal property is satisfied: for every kcategory D, every (k -1)-functor

f : τ ≤k-1 (C) → τ ≤k-1 (D)
and every map ϕ : Σ → D k such that for every k-cell x of C, we have

s(ϕ(x)) = f k-1 (s(x)) and t(ϕ(x)) = f k-1 (t(x)), there exists a unique k-functor f ′ : τ ≤k (C) → D such that τ ≤k-1 (f ′ ) = f and f ′ k (x) = ϕ(x) for every x ∈ Σ.
Remark 2.2. Note that an n-category has a k-basis for any k > n, namely the empty set. Definition 2.3. An ω-category C is free if it has a k-basis for every k ∈ N. Definition 2.4. Let C be an ω-category and n ∈ N. An n-cell x of C is indecomposable if:

(1) x is not degenerate, (2) for any 0

≤ k < n, if x = x 1 * k x 2 with x 1 , x 2 ∈ C n then x 1 = 1 n k (t k (x)) or x 2 = 1 n k (s k (x)
). In particular, any 0-cell is indecomposable.

Proposition 2.5. Let C be a free ω-category. For each k ∈ N, there is a unique k-basis of C, namely the set of indecomposable k-cells.

Proof. See [Mak05, Section 4, Proposition 8.3].

Proposition 2.5 allows us to talk about the k-basis of a free category

C. The sequence (Σ k ⊆ C k ) k∈N where each Σ k is the k-basis of C is simply called the basis of C. Definition 2.7. An ω-functor f : C → D between two free ω-categories is rigid if for every k ∈ N, we have f k (Σ C k ) ⊆ Σ D k where Σ C k (resp. Σ D k ) is the k-basis of C (resp. D).
Free ω-categories and rigid ω-functors form a category denoted by Pol.

Remark 2.8. Objects of Pol are commonly called polygraphs and morphisms of Pol are commonly called morphisms of polygraphs. Although the terms "polygraph" and "free ω-category" are synonyms, we prefer to use the former one when we think of them as objects of the category Pol and the latter one when we think of them as objects of the category Cat ω .

Discrete Conduché ω-functors

Recall that given a category C and M a class of arrows of C, an arrow

f : X → Y of C is said to be right orthogonal to M if for every m : A → B ∈ M and every commutative square A X B Y m f
there exists a unique l : B → X (referred to as a lifting) such that the

diagram A X B Y. m f l is commutative. Definition 3.2. Let f : C → D be an ω-functor. We say that f is a discrete Conduché ω-functor if it is right orthogonal to the arrows D n D k κ n k and D n D n D k D n ∇ n k for any k, n ∈ N such that 0 ≤ k < n.
Remark 3.3. Since the class of discrete Conduché ω-functors is a right orthogonal class, it has many good properties. One of them is that discrete Conduché ω-functors are stable by pullback.

3.4.

Unfolding the Definition 3.2, the right orthogonality to κ n k means that for any x ∈ C n , for any y ∈ D k such that

f (x) = 1 n k (y) there exists a unique 1 z ∈ C k such that x = 1 n k (z) and f (z) = y. Similarly, the right orthogonality to ∇ n k means that for any x ∈ C n , if f (x) = y 1 * k y 2 with y 1 , y 2 ∈ D n that are k-composable, then there exists a unique pair (x 1 , x 2 ) of elements of C n such that (1) s k (x 1 ) = t k (x 2 ) and x = x 1 * k x 2 , (2) f (x 1 ) = y 1 and f (x 2 ) = y 2 .
As it turns out, there are redundancies in the definition of discrete Conduché ω-functor.

Lemma 3.5. Let k < n ∈ N and f : C → D an ω-functor. If f is right orthogonal to D n D n D k D n ∇ n k then it is right orthogonal to D n D k . κ n k 1
Note that since the map z → 1 n k (z) is injective, the uniqueness comes for free.

Proof. Let x ∈ C n and suppose that f

(x) = 1 n k (y) with y ∈ D k . Notice that f (x) = 1 n k (y) * k 1 n k (y) and x = x * k 1 n k (s k (x)) = 1 n k (t k (x)) * k x and f (1 n k (s k (x))) = 1 n k (s k (f (x))) = 1 n k (y) = 1 n k (t k (f (x))) = f (1 n k (t k (x))
). Using the uniqueness part of the right orthogonality to ∇ n k , we deduce that

x = 1 n k (s k (x)) = 1 n k (t k (x)). Thus, if we set z = s k (x) = t k (x), we have x = 1 n k (z) and f (z) = y,
which is what we needed to prove. Remark 3.6. In light of the previous lemma, the reader may wonder why we included the right orthogonality to κ n k in the definition of discrete Conduché ω-functor. The motivation for such a choice is that it should be possible to apply this definition mutatis mutandis for non-strict ω-categories where Lemma 3.5 might not hold anymore.

Lemma 3.7. Let k < m < n ∈ N and f : C → D be an ω-functor. If f is right orthogonal to ∇ n k and κ n m , then it is right orthogonal to ∇ m k . Proof. Let x ∈ C m and suppose that f (x) = y 1 * k y 2 with y 1 , y 2 ∈ D m that are k-composable. Then 1 n m (x) ∈ C n and f (1 n m (x)) = 1 n m (y 1 ) * k 1 n m (y 2 ).
From the right orthogonality to ∇ n k , we know that there exist

z 1 , z 2 ∈ C n that are k-composable such that f (z 1 ) = 1 n m (y 1 ), f (z 2 ) = 1 n m (y 2 ) and 1 n m (x) = z 1 * k z 2 .
From the right orthogonality to κ n m , we know that there exist

x 1 , x 2 ∈ C m such that z 1 = 1 n m (x 1 ), z 2 = 1 n m (x 2 ), f (x 1 ) = y 1 and f (x 2 ) = y 2 . It follows that s k (x 1 ) = t k (x 2 ) and 1 n m (x) = 1 n m (x 1 ) * k 1 n m (x 2 ) = 1 n m (x 1 * k x 2 ), hence x = x 1 * k x 2 .
This proves the existence part of the right orthogonality to ∇ m k . Now suppose that there are two pairs (x 1 , x 2 ) and (x ′ 1 , x ′ 2 ) of m-cells of C that lift the pair (y 1 , y 2 ) in the usual way. It follows that (

1 n m (x 1 ), 1 n m (x 2 )) and (1 n m (x ′ 1 ), 1 n m (x ′ 2 )) lift the pair (1 n m (y 1 ), 1 n m (y 2
)) in the usual way. From the uniqueness part of the right orthogonality to ∇ n k , we deduce that

1 n m (x 1 ) = 1 n m (x ′ 1 ) and 1 n m (x 2 ) = 1 n m (x ′ 2 ), hence x 1 = x ′ 1 and x 2 = x ′ 2 .

3.8.

Recall that with the definition we chose (paragraph 1.6), an n-functor is a particular type of ω-functor. Hence, it makes sense to call an n-functor a discrete Conduché n-functor when it is a discrete Conduché ω-functor. Proposition 3.9. Let f : C → D be an n-functor. It is a discrete Conduché n-functor if and only if it is right orthogonal to ∇ n k for any k ∈ N such that k < n.

Proof. From Lemma 3.5, we know that we only need to show that f is right orthogonal to ∇ m k for all k < m ∈ N. When m ≤ n, this follows from Lemma 3.5 and Lemma 3.7. As for the case m > n, this follows from the fact that for any m-cell x in an n-category with m > n, there exists a unique n-cell x ′ such that x = 1 m n (x ′ ). Details are left to the reader. Corollary 3.10. Let f : C → D be an ω-functor and n ∈ N. The n-functor τ ≤n (f ) :

τ ≤n (C) → τ ≤n (D) is a discrete Conduché n-functor if and only if f is right orthogonal to ∇ n k for any k ∈ N such that k < n. Lemma 3.11. Let f : C → D be a discrete Conduché ω-functor and x a cell of C. Then x is an indecomposable cell if and only if f (x) is an indecomposable cell.
Proof. If x is a 0-cell, there is nothing to show since every 0-cell is indecomposable. We suppose now that x is an n-cell with n > 0.

Suppose that x is indecomposable. The right orthogonality to κ n k for any 0 ≤ k < n implies that f (x) is non-degenerate since, if it were degenerate, x would be too. Suppose that

f (x) = y 1 * k y 2 with y 1 , y 2 ∈ D n that are k-composable. The right orthogonality to ∇ n k implies that x = x 1 * k x 2 with f (x 1 ) = y 1 and f (x 2 ) = y 2 . Since x is indecomposable, x 1 or x 2 has to be of the form 1 n k (z) with z ∈ C k .
Thus, y 1 or y 2 has to be of the form 1 n k (z ′ ) with z ′ ∈ D k . This proves that f (x) is indecomposable. Suppose that f (x) is indecomposable. Then x is non-degenerate because otherwise f (x) would be degenerate. Suppose that

x = x 1 * k x 2 with x 1 , x 2 ∈ C n that are k-composable. Thus, f (x) = f (x 1 ) * k f (x 2 ). Since f (x) is indecomposable, either f (x 1 ) or f (x 2 ) has to be of the form 1 n k (z) with z ∈ D k .
From the right orthogonality to κ n k , it follows that either x 1 or x 2 has to be of the form 1 n k (z ′ ) with z ′ ∈ C k . This proves that x is indecomposable.

From the previous lemma and Proposition 2.5, we deduce the following proposition.

Proposition 3.12. Let f : C → D be an ω-functor with C and D free ω-categories. If f is a discrete Conduché ω-functor then f is rigid. -C is an n-category, -Σ is a set, -σ and τ are maps Σ → C n such that for every element x of Σ, the n-cells σ(x) and τ (x) are parallel.

Cellular extensions and technicalities on words

When the natural number n is understood, a cellular extension means a cellular extension of some n-category. Definition 4.2. Let E = (C, Σ, σ, τ ) and E ′ = (C ′ , Σ ′ , σ ′ , τ ′ ) be two cellular extensions of n-categories. A morphism of cellular extensions from E to E ′ is a pair (f, ϕ) where:

-

f is n-functor from C to C ′ , -ϕ is a map Σ → Σ ′ , -the following squares are commutative Σ Σ ′ C n C ′ n ϕ σ σ ′ fn Σ Σ ′ C n C ′ n . ϕ τ τ ′ fn
Cellular extensions of n-categories and morphisms between them form a category Cat + n . There is an obvious functor U n : Cat n+1 → Cat + n that sends an (n + 1)-category C to the cellular extension (τ ≤n (C), C n+1 , s, t). We shall see later that this functor has a left adjoint.

4.3.

Let E = (C, Σ, σ, τ ) be a cellular extension of an n-category. We consider the alphabet that has:

-a symbol c α for each α ∈ Σ, -a symbol i x for each x ∈ C n , -a symbol * k for each 0 ≤ k ≤ n, -a symbol of opening parenthesis (, -a symbol of closing parenthesis ). We write W[E] for the set of finite words on this alphabet. If w and w ′ are elements of W[E], we write ww ′ for their concatenation.

The length of a word w, denoted by L(w), is the number of symbols that appear in w.

We now recursively define the set T [E] ⊆ W[E]

of well formed words (or terms) on this alphabet together with maps s, t : T [E] → C n that satisfy the globular conditions:

-

(c α ) ∈ T [E] with s((c α )) = σ(α) and t((c α )) = τ (α) for each α ∈ Σ, -(i x ) ∈ T [E] with s((i x )) = t((i x )) = x for each x ∈ C n , -(v * n w) ∈ T [E] with s((v * n w)) = s(w) and t((v * n w)) = t(v) for v, w ∈ T [E] such that s(v) = t(w), -(v * k w) ∈ T [E] with s((v * k w)) = s(v) * k s(w) and t((v * k w)) = t(v) * k t(w) for v, w ∈ T [E] and 0 ≤ k < n, such that s k (s(v)) = t k (t(w)).
We define s k , t k : T [E] → C k as iterated source and target (with s n = s and t n = t for consistency). We say that two well formed words v and w are parallel if s(v) = s(w) and t(v) = t(w)

and we say that they are k-composable for a k ≤ n if

s k (v) = t k (w).
For the rest of the section, we fix some cellular extension of an n-category E = (C, Σ, σ, τ ). All the words considered are elements of W[E].

Definition 4.5. The size of a well formed word w, denoted by |w|, is the number of symbols * k for any 0 ≤ k ≤ n that appear in the well formed word w. Definition 4.6. A word v is a subword of a word w if there exist words a and b such that w can be written as

w = avb.
Remark 4.7. Beware that in the previous definition, none of the words were supposed to be well formed. In particular, a subword of a well formed word is not necessarily well formed.

4.8.

Since a word w is a finite sequence of symbols, it makes sense to write w(i) for the symbol at position i of w, with 0 ≤ i ≤ L(w) -1.

For any 0 ≤ i ≤ L(w) -1, define P w (i) to be the number of opening parenthesis in w with position ≤ i minus the number of closing parenthesis in w with position ≤ i. This defines a function (1) it is not empty, (2) P w (i) ≥ 0 for any 0 ≤ i ≤ L(w) -1, (3) P w (i) = 0 if and only if i = L(w) -1.

P w : {0, . . . , L(w) -1} → Z.

4.11.

It follows from the previous definition that the first letter of a well parenthesized word is necessarily an opening parenthesis and that the last letter is necessarily a closing parenthesis. Thus, the length of a well parenthesized word is not less than 2.

Moreover, it is immediate that if w 1 and w 2 are well parenthesized words then, for any 0 ≤ k ≤ n, (w 1 * k w 2 ) is well parenthesized. Lemma 4.12. A well formed word is well parenthesized.

Proof. Let w be a well formed word. We proceed by induction on |w|. If |w| = 0, then w is either of the form (c α ) or of the form (i x ). In either case, the assertion is trivial. Now suppose that |w| > 0, we know by definition that w = (w 1 * k w 2 )

with w 1 , w 2 well formed words such that |w 1 |, |w 2 | < |w|. The desired properties follow easily from the induction hypothesis. Details are left to the reader.

The converse of the previous lemma is obviously not true. However, Corollary 4.14 below is a partial converse. Lemma 4.13. Let w be a well parenthesized word of the form

w = (w 1 * k w 2 )
with w 1 and w 2 well parenthesized words, and 0 ≤ k ≤ n and let v be a subword of w. If v is well parenthesized then one the following holds:

(1

) v = w, (2) v is a subword of w 1 , (3) v is a subword of w 2 .
Proof. Let a and b be words such that

avb = w = (w 1 * k w 2 ).
Let l 1 , l 2 , l, l a , l b , l v respectively be the length of w 1 , w 2 , w, a, b, v. Notice that

l a + l v + l b = l = l 1 + l 2 + 3.
Notice that since v is well parenthesized, the following cases are forbidden:

(1)

l 1 ≤ l a ≤ l 1 + 1, (2) l 2 ≤ l b ≤ l 2 + 1, (3) l a ≥ l -1, ( 4
) l b ≥ l -1. Indeed, the first case would imply that the first letter of v is a closing parenthesis or the symbol * k . Similarly, the second case would imply that the last letter of v is an opening parenthesis or the symbol * k . The third and fourth cased would imply that l v < 2 which is also impossible.

That leaves us with the following cases:

(1) l a = 0, (2) l b = 0, (3) 0 < l a < l 1 and 0 < l b < l 2 , (4) 0 < l a < l 1 and l b > l 2 + 1, (5) l 1 + 1 < l a and 0 < l b < l 2 . If we are in the first case, then

P w (j) = P v (j) for 0 ≤ j ≤ l v -1. That implies that P w (l v -1) = 0 which means that l = l v , hence w = v.
By a similar argument that we leave to the reader, we can show that the second case implies that w = v.

If we are in the fourth (resp. fifth) case, then it is clear that v is a subword of w 1 (resp. w 2 ).

Suppose now that we are in the third case. Intuitively, it means that the first letter of v is inside w 1 and the last letter of v is inside w 2 . Notice first that (⋆)

l a < l 1 < l a + l v -3,
where the inequality on the right comes from the fact that l v ≥ 2 because v is well formed. Besides, by definition of P w , P w (j) = P v (j -l a ) + P w (l a )

for l a ≤ j < l v + l a . In particular, we have 1 = P w 1 (l 1 -1) + 1 = P w (l 1 -1) = P v (l 1 -1) + P w (l a ).

From (⋆) and since v is well parenthesized, we deduce that

P v (l 1 -1) > 0.
Hence, P w (l a ) ≤ 0 which is impossible because w is well formed and l a < l -1.

Corollary 4.14. Let w be a well parenthesized word. If w is a subword of a well formed word, then it is also well formed.

Proof. Let u be a well formed word such that w is a subword of u. We proceed by induction on |u|. If |u| = 0, then u is either of the form (c α ) or of the form (i x ). In both cases, w = u since the only well parenthesized subword of u is u itself.

Suppose now that |u| > 0. By definition,

u = (u 1 * k u 2 )
with |u 1 |, |u 2 | < |u|. By Lemmas 4.12 and 4.13, we have that either: -w = u in which case w is well formed by hypothesis, -w is a subword of u 1 and from the induction hypothesis we deduce that w is well formed, -w is a subword of u 2 which is similar to previous case. Lemma 4.15. Let w be a well formed word of the form

w = (w 1 * k w 2 )
with w 1 and w 2 well formed words, and 0 ≤ k ≤ n, and let v be a subword of w. If v is well formed, then we are in one of the following cases:

(1

) v = w, (2) v is a subword of w 1 , (3) v is a subword of w 2 .
Proof. This follows immediately from Lemma 4.12 and Lemma 4.13.

Corollary 4.16. Let u be a well formed word of the form vew with v, w and e words and such that e is well formed. If e ′ is a well formed word that is parallel to e, then the word ve ′ w is also well formed.

Corollary 4.19. Let w be a well formed word and suppose that it can be written as

w = (w 1 * k w 2 )
with w 1 and w 2 well formed words and 0 ≤ k ≤ n. Then s k (w 1 ) = t k (w 2 ).

Proof. By hypothesis, |w| ≥ 1. From the definition of well formed words, we know that w is of the form

(w ′ 1 * k ′ w ′ 2 ) with w ′
1 and w ′ 2 well formed words and 0 ≤ k ′ ≤ n such that (1) e is of the form ((x

s k ′ (w ′ 1 ) = t k ′ (w ′ 2 ). From Lemma 4.18, we have that w ′ 1 = w 1 , w ′ 2 = w 2 and k = k ′ .
* k y) * k z)
and e ′ is of the form

(x * k (y * k z)) with x, y, z ∈ T [E] and 0 ≤ k ≤ n, (2) e is of the form ((i c ) * k x)
and e ′ is of the form x

with x ∈ T [E], 0 ≤ k ≤ n and c = 1 n k (t k (x)), (3) e is of the form (x * k (i c ))
and e ′ is of the form x

with x ∈ T [E], 0 ≤ k ≤ n and c = 1 n k (s k (x)), (4) e is of the form ((i c ) * k (i d ))
and e ′ is of the form

(i c * k d ) with c, d ∈ C n and 0 ≤ k < n,
(5) e is of the form ((x * k y) * l (z * k t)) and e ′ is of the form

((x * l z) * k (y * l t)) with x, y, z, t ∈ T [E], 0 ≤ l < k ≤ n.

We will use the notation

µ : u → u ′
to say that µ is an elementary move from u to u ′ .

We now define an oriented graph2 G[E] with:

-T [E] as its set of objects, -for all u, u ′ in T [E], the set of elementary moves from u to u ′ as its set of arrows from u to u ′ . We will use the categorical notation

G[E](u, u ′ )
to denote the set of arrows from u to u ′ .

We will also sometimes write

u ↔ u ′
to say that there exists an elementary move from u to u ′ or from u ′ to u. Proof. Let µ = (v, w, e, e ′ ) : u → u ′ be an elementary move from w to w ′ . We are going to prove that s(u) = s(u ′ ) and t(u) = t(u ′ ) with an induction on L(v)+ L(w)(cf. 4.3). Notice first that, by definition of elementary moves, |u| ≥ 1 and thus

u = (u 1 * k u 2 ) with u 1 , u 2 ∈ T [E].
Base case: If L(v) + L(w) = 0, it means that both v and w are both the empty word. It is then straightforward to check the desired property using Definition 5.1. Inductive step: Suppose now that L(v) + L(w) ≥ 0. Since e is a subword of u that is well formed, from Lemma 4.15 we are in one of the following cases:

Lemma 5.7. The map ϕ commutes with source and target, i.e. for a well formed word w, we have

s( ϕ(w)) = f (s(w)) and t( ϕ(w)) = f (t(w)).
Proof. The lemma is proven with an induction left to the reader.

Lemma 5.8. Let v and w be two well formed words. If v ∼ w then

ϕ(v) = ϕ(w).
Proof. It suffices to prove that for any elementary move µ : v → w, we have ϕ(v) = ϕ(w), which is immediate from the axioms for ω-category (see paragraph 1.3).

5.9. Let E = (C, Σ, σ, τ ) be a cellular extension of an n-category.

From Lemma 5.4, we deduce that s, t :

T [E] → C n induce maps s, t : T [E]/∼ → C n . Let [v] and [w] be two elements of T [E]/∼ such that s k ([v]) = t k ([w]
) for some 0 ≤ k ≤ n. From Lemma 5.5, we can define without ambiguity:

[v] * k [w] := [v * k w].
We leave it to the reader to show that these data add up to an (n + 1)category

E * with τ ≤n (E * ) = C and E * n+1 = T [E]/∼. Note that we have a canonical map j E : Σ → E * n+1 α → [(c α )]
and the following two triangles are commutative

Σ E * n+1 E n j E σ s n Σ E * n+1 E n . j E τ t n
Lemma 5.10. The map j E : Σ → E * n+1 is injective. Proof. For any α ∈ Σ, it is straightforward to check that the number of occurences of c α in a well formed word w depends only on its equivalence class

[w]. In particular, for α = β in Σ, [(c α )] = [(c β )].
As a consequence of the previous lemma, we will always consider Σ as a subset of E * n+1 and j E as the canonical inclusion. Proposition 5.11. Let E = (C, Σ, σ, τ ) be a cellular extension of an ncategory. Then Σ is an (n + 1)-basis of the (n + 1)-category E * .

Proof. Let D be an (n + 1)-category, f : C → τ ≤n (D) an n-functor and a map ϕ : Σ → D n+1 compatible with source and target. In other words, we have a morphism of cellular extension (ϕ, f ) : E → U n (D). From paragraph 5.6, we have a map ϕ : T [E] → D n+1 , whose restriction to Σ is ϕ. From all the results from 5.4 to 5.8, ϕ induces a map

T [E]/∼ → D n+1 ,
which is compatible with source, target, units, and composition. This proves the existence of an (n + 1)-functor

f ′ : E * → D
such that τ ≤n (f ′ ) = f and the restriction of f ′ n+1 to Σ is ϕ. Let f ′′ : E * → D be another (n + 1)-functor with the same properties and let x be an (n + 1)-cell of E * . By definition, there exists a well formed word w such that x = [w]. By a quick induction on |w| that we leave to the reader, we easily prove that

f ′′ n+1 = f ′ n+1 .
Since by definition τ ≤n (f ′′ ) = f , we have f ′ = f ′′ . We leave it to the reader to extend the correspondance

E = (C, Σ, σ, τ ) → E * to a functor Cat + n → Cat n+1 . Corollary 5.12. Let n ∈ N. The functor Cat + n → Cat n+1 E → E *
is left adjoint to the functor

U n : Cat n+1 → Cat + n .
Proof. This is a reformulation of the universal property from Definition 2.1. Remark 5.13. Note that this left adjoint has already been explicitly constructed in the literature, for example in [START_REF] Makkai | The word problem for computads[END_REF] or in [START_REF] Métayer | Cofibrant objects among higher-dimensional categories[END_REF]. Our construction is greatly inspired from the latter reference but it differs in one subtle point. Métayer defines an elementary relation on parallel well formed words and then takes the congruence generated by it, whereas we directly defined an explicit equivalence relation (Definition 5.3) and then showed that that two equivalent well formed words are necessarily parallel (Lemma 5.4) and that it is in fact a congruence (Lemma 5.5).

5.14.

Let C be an ω-category and Σ a subset of C n+1 . We define the cellular extension E Σ = (τ ≤n (C), Σ, s, t) where s and t mean the restrictions of s, t :

C n+1 → C n to Σ ⊆ C n+1 .
In order to simplify the notations, we will allow ourselves to write T [Σ] instead of T [E Σ ] when there is no ambiguity on the rest of the data.

The canonical inclusion ι : Σ ֒→ C n+1 induces a canonical morphism (ι, id τ ≤n (C) ) :

E Σ → U n (C) = (τ ≤n (C), C n+1 , s n , t n ).
From paragraph 5.6, we obtain a map, that we denote ρ Σ instead of ι,

ρ Σ : T [Σ] → C n+1 such that -ρ Σ ((c α )) = α for α ∈ Σ, -ρ Σ ((i x )) = 1 x for x ∈ C n , -ρ Σ ((v * k w)) = ρ Σ (v) * k ρ Σ (w) for 0 ≤ k ≤ n, v
and w two well formed words that are k-composable.

5.15.

Let a be an element of C n+1 , we define T [Σ] a to be

T [Σ] a = {w ∈ T [Σ] | ρ Σ (w) = a}. Lemma 5.8 implies that if v ∈ T [Σ] a and v ∼ w, then w ∈ T [Σ] a . We define G[Σ] a to be the full subgraph of G[Σ] whose set of objects is T [Σ] a .
Proposition 5.16. Let C be an ω-category and Σ ⊆ C n+1 . Then Σ is an

(n + 1)-basis of C if and only if for every a ∈ C n+1 , the graph G[Σ] a is 0-connected (i.e.

non-empty and connected).

More precisely this means that for every a ∈ C n+1 :

-there exists

w ∈ T [Σ] such that ρ Σ (w) = a, -for every v, w ∈ T [Σ], if ρ Σ (v) = a = ρ Σ (w) then v ∼ w.
Proof. From Proposition 5.11, we know that E * Σ has Σ as an (n + 1)-basis. Hence, the canonical morphism

E Σ → U n (C) from paragraph 5.14 induces a map T [Σ]/∼ → C n+1 ,
which is nothing but the map obtained from ρ Σ by applying Lemma 5.8. This maps sends Σ (as a subset of T [Σ]/∼) to Σ (as a subset of C n+1 ). Since Σ is an (n + 1)-basis of E * Σ , we easily deduce that Σ is an (n + 1)-basis of C if and only if the previous map is an isomorphism, which is exactly what we wanted to prove. Details are left to the reader. 6. Discrete Conduché ω-functors and polygraphs 6.1. Let f : C → D be an ω-functor, n > 0, Σ C ⊆ C n and Σ D ⊆ D n such that f n (Σ C ) ⊆ Σ D . We recursively define a map (1) if u is well formed then f (u) is well formed, (2) if f (u) is well formed and if u is a subword (4.6) of a well formed word then it is also well formed.

f : W[Σ C ] → W[Σ D ] with -f (c α ) = c f (α) for α ∈ Σ C , -f (i x ) = i f (x) for x ∈ C n , -f ( * k ) = * k for 0 ≤ k < n, -f ( ( ) = (, - f
Proof. The first part of the previous lemma is proved with a short induction left to the reader. For the second part, first notice that the map

f : W[Σ C ] → W[Σ D ]
satisfies the following property:

For any w ∈ W[Σ C ], w is well parenthesized if and only if f (w) is well parenthesized.

It suffices then to apply Lemma 4.12 and then Corollary 4.14.

6.3.

The first part of Lemma 6.2 shows that f induces a map

f : T [Σ C ] → T [Σ D ].
Moreover, we have a commutative square

T [Σ C ] C n T [Σ D ] D n ρ C f fn ρ D
where ρ C and ρ D respectively stand for ρ Σ C and ρ Σ D . Thus, for every a ∈ C n we can define a map:

f a [Σ C ] a → T [Σ D ] f (a)
w → f (w).

Recall from Corollary 3.10 that for an ω-functor f : C → D and n > 0, τ ≤n (f ) is a discrete Conduché n-functor if and only if f is right orthogonal to ∇ n k for any k ∈ N such that k < n. Proposition 6.4. Let f : C → D be an ω-functor and n > 0. Then the following conditions are equivalent:

(1) τ ≤n (f ) : τ ≤n (C) → τ ≤n (D) is a discrete Conduché n-functor, (2) for every Σ D ⊆ D n and Σ C := f -1 (Σ D ) and for every a ∈ C n the map

f a : T [Σ C ] a → T [Σ D ] f (a)
defined above is bijective.

Proof. We begin with 1 ⇒ 2. Surjectivity: We are going to prove the following assertion:

∀l ∈ N, ∀a ∈ C n ,∀w ∈ T [Σ D ] f (a) such that |w| ≤ l ∃v ∈ T [Σ C ] a such that f a (v) = w.
We proceed by induction on l. Suppose first that l = 0. We are necessarily in one of the two cases:

(

ρ D (w 1 ) * k ρ D (w 2 ) = f (a).
Since by hypothesis f is right orthogonal to ∇ n k , we know that there exist a 1 ∈ C n and a 2 ∈ C n that are k-composable and such that a = a 1 * k a 2 , f (a 1 ) = ρ D (w 1 ) and f (a 2 ) = ρ D (w 2 ).

Since |w 1 | ≤ l and |w 2 | ≤ l, we can apply the induction hypothesis. Hence, there exist

v 1 ∈ T [Σ C ] a 1 and v 2 ∈ T [Σ C ] a 2 such that f a 1 (v 1 ) = f (v 1 ) = w 1 and f a 2 (v 2 ) = f (v 2 ) = w 2 .
Since ρ C commutes with source and target by Lemma 5.7, v 1 and v 2 are k-composable and the word (v 1 * k v 2 ) is a well formed. By definition of ρ C , we have

ρ C ((v 1 * k v 2 )) = ρ C (v 1 ) * k ρ C (v 2 ) = a 1 * k a 2 = a. Thus, (v 1 * k v 2 ) ∈ T [Σ C ] a and f a ((v 1 * k v 2 ) = f ((v 1 * k v 2 )) = ( f (v 1 ) * k f (v 2 )) = (w 1 * k w 2 ) = w.
Injectivity: We are going to prove the following assertion:

∀l ∈ N, ∀v ∈ T [Σ C ] a , w ∈ T [Σ C ] a such that |v| = |w| ≤ l f a (v) = f a (w) ⇒ v = w
We proceed by induction on l.

Suppose first that l = 0. We are necessarily in one of the four cases:

(1) v = (c α ) and w = (c β ) with α and β in Σ C . By definition of ρ

C , α = ρ C (v) = a = ρ C (w) = β. Hence, v = w. (2) v = (i x ) and w = (i y ) with x and y in C n-1 . By hypothesis ρ C (v) = a = ρ C (w) and by definition of ρ C , 1 x = ρ C (v) = a = ρ C (w) = 1 y , thus x = y and v = w. (3) v = (c α ) and w = (i x ) with α ∈ Σ C and x ∈ C n-1 . By hypoth- esis, (c f (α) ) = f (v) = f (w) = (i f (x) ) which is impossible. (4) v = (i x ) and w = (c α ) with α ∈ Σ C and x ∈ C n-1 , which is symmetric to the previous case. such that f (u ′ ) = v ′ and f (λ) = µ.
Proof. The proof is long and tedious as we have to check all the different cases of elementary moves. For the sake of clarity, we first outline a sketch of the proof that is common to all the cases of elementary moves and then we proceed to fill in the blanks successively for each case. Let µ = (v 1 , v 2 , e, e ′ ) : v → v ′ be an elementary move. Since, by definition,

f (u) = v = v 1 ev 2 u is necessarily of the form u = u 1 eu 2 with e, u 1 , u 2 ∈ W[E] such that f (e) = e
and f (u j ) = v j for j ∈ {1, 2}. From the second part of Lemma 6.2, we deduce that e is well formed. In each different case, we will prove the existence of a well formed word e ′ parallel to e and such that f (e ′ ) = e ′ .

From Corollary 4.16, we deduce that the word u ′ := u 1 e ′ u 2 is well formed. By definition, we have

f (u ′ ) = v ′ .
Moreover, in each case, it will be immediate that the pair (e, e ′ ) is such that the quadruple λ := (u 1 , u 2 , e, e ′ ) is an elementary move and that

f (λ) = µ.
All that is left now is to prove the existence of e ′ with the desired properties. Since f (e) = e, we deduce from Lemma 6.2 that x, y, z and (x * k y) are well formed. From Corollary 4.19, we deduce that s k (x) = t k (y) and s k (y) = t k (z). Thus, the word e ′ := (x * k (y * k z)) is well formed and it satisfies the desired properties. Second case: e is of the form

(x * k (i 1 n-1 k (z) ))
and e ′ is of the form

x with x ∈ T [Σ C ], 0 ≤ k < n and z ∈ D k . 3
Necessarily e is of the form

((i x ) * k (i y ))
and e ′ is of the form

(i x * k y ) with x, y ∈ D n-1 such that s k (x) = t k (y). Necessarily, e is of the form ((i x ) * k (i y )) with x, y ∈ C n such that f (x) = x and f (y) = y.
Using Corollary 4.19 and the fact that e is well formed, we deduce that s k (x) = t k (y). Thus, the word f (t) = t. From Lemma 6.2 and the fact that e is well formed, we deduce that x, y, z, t, (x * k y), (z * k t) are well formed and from Corollary 4.19, we deduce that s k (x) = t k (y), s k (z) = t k (t) and s l ((x * k y)) = t l ((z * k t)). Since l < k, we deduce from this last equality that s l (x) = s l (y) = t l (z) = t l (t).

Thus, the word e ′ := ((x * l z) * k (y * l t)) is well formed. It satisfies all the desired properties. Remark 6.8. In the proof of the previous theorem, we only used the hypothesis that f is right orthogonal to κ n k for any k such that 0 ≤ k < n -1. Corollary 6.9. Let f : C → D be an ω-functor, n > 0, Σ D ⊆ D n and

Σ C = f -1 (Σ D ). If τ ≤n (f ) is a discrete Conduché n-functor, then for every a ∈ C n f a : G[Σ C ] a → G[Σ D ] f (a)
is an isomorphism of graphs.

Proof. Proposition 6.4 exactly says that the map

f a : G[Σ C ] a → G[Σ D ] f (a)
is an isomorphism on objects and we know from Lemma 6.6 that it is a faithful morphism of graphs (same definition as for functors). All that is left to show is that it is also full.

In A.6. The terminal object of Pol is a rather complicated object (see [START_REF] Street | Categorical structures[END_REF] section 4] for an explicit description of the 2-cells of that polygraph) and the previous criterion seems hard to use in practise.

However, it can be checked that every free 1-category is tight and the previous proposition implies that a 1-functor f : C → D between free 1categories is rigid if and only if it is a Conduché functor. This fact can also be directly proved "by hand". We leave the details to the reader.

  Definition. A functor f : C → D is a discrete Conduché functor (or discrete Conduché fibration) if for every arrow γ : x → y in C and every factorization f (γ) = f (x) z f (y), α β there exists a unique factorization γ = x z y α β such that f (α) = α and f (β) = β.

Date:

  January 8, 2020. 2010 Mathematics Subject Classification. 18D05. 1 Theorem. Let f : C → D be a discrete Conduché functor. If D is free on a graph then C is free on a graph.

  we have an n-cell x of C and a k-cell y of C such that x = 1 n k (y). 1.9. Similarly, for k < n ∈ N, we have a functor Comp n k : Cat ω → Set that associates to each ω-category C the set C n × C k C n and to an ω-functor f : C → D the canonically induced map

  Definition 4.1. A cellular extension of an n-category C is a quadruple (C, Σ, σ, τ ) where:

Remark 4. 9 .

 9 Such a counting function is standard in the literature about formal languages. For example see [HU79, chapter 1, exercice 1.4]. Definition 4.10. A word w is well parenthesized if:

5.

  From cellular extensions to free ω-categories Definition 5.1. Let E = (C, Σ, σ, τ ) be a cellular extension of an n-category and let u, u ′ ∈ T [E]. An elementary move from u to u ′ is a quadruple µ = (v, w, e, e ′ ) with v, w ∈ W[E] and e, e ′ ∈ T [E] such that u = vew, u ′ = ve ′ w, and one of the following holds:

Definition 5. 3 .

 3 Let E = (C, Σ, σ, τ ) be a cellular extension of an n-category and u, u ′ ∈ T [E]. We say that the well formed words u and u ′ are equivalent and write u ∼ u ′ if they are in the same connected component of G[E]. More precisely, this means that there exists a finite sequence (u j ) 0≤j≤N of well formed words with u 0 = u, u N = u ′ and u j ↔ u j+1 for 0 ≤ j < N . The equivalence class of a well formed word u will be denoted by [u]. Lemma 5.4. Let u, u ′ ∈ T [E]. If u ∼ u ′ then u and u ′ are parallel.

  ( ) ) =).Notice that for any wordw ∈ W[Σ C ], | f (w)| = |w| and L( f (w)) = L(w). Lemma 6.2. Let f : C → D be an ω-functor, Σ C ⊆ C n and Σ D ⊆ D n such that f n (Σ C ) ⊆ Σ D . For every u ∈ W[Σ C ]:

  1) w = (c β ) with β ∈ Σ D . By hypothesis, ρ D (w) = f (a) and by definition of ρ D , ρ D (w) = β thus f (a) = β. By definition of Σ C , a ∈ Σ C and we can choose v = (c a ).(2) w = (i y ) with y ∈ D n-1 . By hypothesis, ρ D (w) = f (a) and by definition of ρ D , ρ D (w) = 1 y thus f (a) = 1 y . Since τ ≤n (f ) is a discrete Conduché n-functor, f is right orthogonal to κ n n-1 . Hence, there exists x ∈ C n-1 such that a = 1 x and f (x) = y. We can then choose v = (i x ) ∈ T [Σ C ] a . Now suppose that the assertion is true for a fixed l ∈ N and let w ∈ T [Σ D ] f (a) be such that |w| = l + 1. By definition of well formed words, we have w = (w 1 * k w 2 ) with 0 ≤ k < n and w 1 , w 2 ∈ T [Σ D ] such that |w 1 | ≤ l and |w 2 | ≤ l. By hypothesis, ρ D (w) = f (a) and by definition of ρ D , ρ D (w) = ρ D (w 1 ) * k ρ D (w 2 ) thus,

  First case: e is of the form ((x * k y) * k z) and e ′ is of the form (x * k (y * k z)) with x, y, z ∈ T [Σ D ]. The word e is then necessarily of the form ((x * k y) * k z).

  (x * k (i y ))with x ∈ T [E] (from Lemma 6.2 again) and y ∈ C n-1 such thatf (x) = x and f (y) = 1 n-1 k (z). Then we set e ′ := x. The only thing left to show is that y = 1 n-1 k (s k (x)). If k = n -1,this follows from Corollary 4.19 and the fact that e is well formed. If k < n -1, we need first to use the fact that f is right orthogonal to κ some z ∈ C k such that f (z) = z and then use Corollary 4.19 and the fact that e is well formed. Third case: Similar to the second one with unit on the left. Fourth case: e is of the form

e

  ′ := (i x * k y )is well formed. It satisfies all the desired properties.

Fifth

  case: e is of the form((x * k y) * l (z * k t))and e ′ is of the form((x * l z) * k (y * l t))with x, y, z, t ∈ T [Σ D ] and 0 ≤ l < k < n such that all the compatibilities of sources and targets needed are satisfied.Necessarily, e is of the form((x * k y) * l (z * k t)) with x, y, z, t ∈ W[Σ C ] such that f (x) = x, f (y) = y, f (z) = z,

  other words, we have to show that for any u, u ′ ∈ T [Σ C ] a the mapG[Σ C ](u, u ′ ) → G[Σ D ]( f (u), f (u ′ ))Counter-Example A.2. Let e be the terminal 1-category and let ⋆ be its unique object. Let E = (e, Σ, σ, τ ) be the cellular extension of e such that Σ has two elements a, b : ⋆ → ⋆ and let C := E * . By the Eckmann-Hilton argument, we have thata * 0 b = a * 1 b = b * 0 a,and C is (isomorphic to) the free commutative monoid generated by a and b, seen as a 2-category.Let E ′ = (e, Σ ′ , σ, τ ) be the cellular extension of e such that Σ ′ has one element c : ⋆ → ⋆ and letC ′ := E ′ * . Then C ′ is (isomorphic to)the free commutative monoid generated by c, seen as a 2-category. Let f : C → C ′ be the unique rigid functor such that f (a) = f (b) = c. Now set x := a * 0 b and consider the decomposition f (x) = c * 0 c. The fact that a * 0 b = b * 0 a but that a = b shows that the uniqueness of the lifting of the previous decomposition of f (x) fails. Remark A.3. While, in the previous counter-example, the uniqueness part of the definition of discrete Conduché 2-functor fails, the existence still holds. I believe that there should be examples where the existence part fails as well. A.4. The category Pol admits a terminal object ⊤. Hence, a rigid ω-functor f : C → D between two free ω-categories always fits in a commutative triangle C D ⊤ f where the anonymous arrows are the canonical rigid functors to the terminal polygraph. Since the class of discrete Conduché ω-functors is a right orthogonal class, it has the following cancellation property: for f : C → D and g : D → E two ω-functors, if g and g • f are Conduché ω-functors then so is f . Following the terminology of [Str96, section 5], we say that a free ωcategory C is tight if the canonical rigid functor C → ⊤ is a Conduché ω-functor. Putting all the pieces together, we obtain the following partial converse of Proposition 3.12. Proposition A.5. Let C and D be two free ω-categories and f : C → D a rigid ω-functor. If C and D are tight then f is a discrete Conduché ωfunctor.

Here, oriented graph is to be understood in the same way as the underlying (oriented) graph of a category.

Acknowledgements

I am deeply grateful to Georges Maltsiniotis for his help and support. Any attempt to sum it up in a few words would be reductive. I am also grateful to François Métayer and Chaitanya Leena-Subramaniam for their careful reading of this article that helped improve it a lot. Finally, I would like to thank the anonymous referee for their very useful remarks and suggestions.

ω-categories

This section is mainly devoted to fixing notations. Some facts are asserted and proofs are left to the reader.

Proof. We proceed by induction on |u|. Base case: If |u| = 0, then necessarily v and w are both the empty word and the assertion is trivial.

with u 1 and u 2 well formed words such that |u 1 |, |u 2 | < |u|. By hypothesis, e is a subword of u and from Lemma 4.15, we are in one of the following cases.

-u = e in which case the assertion is trivial.

-e is a subword of u 1 , which means that there exist words ṽ, w such that u 1 = ṽe w. Moreover, we have v = (ṽ and w = w * k u 2 ). By induction hypothesis, the word ṽe ′ w is well formed and thus (ṽe ′ w * k u 2 ) = vew is well formed.

-e is a subword of u 2 , which is symmetric to the previous case.

Lemma 4.17. Let w 1 , w 2 , w ′ 1 , w ′ 2 be well parenthesized words, and 0 ≤ k ≤ n and 0 ≤ k ′ ≤ n be such that

Let us define l := min(L(w 1 ), L(w ′ 1 )). Notice that P w (j) = P w 1 (j -1) + 1 = P w ′ 1 (j -1) + 1 for 0 < j ≤ l hence P w 1 (l -1) = P w ′ 1 (l -1). Since w 1 and w ′ 1 are well parenthesized, one of the members of the last equality (and thus both) is equal to 0. That implies that L(w 1 ) = L(w ′ 1 ) and the desired properties follow immediately from that. Lemma 4.18. Let w 1 , w ′ 1 , w 2 , w ′ 2 be well formed words, and

Proof. This follows from Lemma 4.12 and Lemma 4.17.

-e = u, which is exactly the base case.

-e is a subword of u 1 , which means that there exist ṽ, w ∈ T [E] such that u 1 = ṽe w. Moreover, we have v = (ṽ and w = w * k u 2 ). From Corollary 4.16, the word u ′ 1 := ṽe ′ w is well formed. Therefore we can use the induction hypothesis on μ := (ṽ, w, e, e ′ ) :

-e is a subword of u 2 , which is symmetric to the previous case. By definition of ∼, this suffices to show the desired properties.

be an elementary move. Set ṽ := (v and w := w * k v 2 ). Then, by definition, (ṽ, w, e, e ′ ) is an elementary move from (v

Similarly, if we have an elementary move from v 2 to v ′ 2 , we obtain an elementary move from (v

. By definition of ∼, this suffices to show the desired property.

Let E = (C, Σ, σ, τ ) be a cellular extension of an n-category, D an (n + 1)-category and

a morphism of cellular extensions. We recursively define a map

By hypothesis, we have

From Lemma 4.18, we deduce that

In order to apply the induction hypothesis, we need to show that ρ C (v j ) = ρ C (w j ) for j ∈ {1, 2}.

By hypothesis,

Hence,

From the induction hypothesis we have v j = w j for j ∈ {1, 2}, hence v = w. Now we prove 2 ⇒ 1.

Let a ∈ C n and suppose that f

and by hypothesis there exists a unique

Using Lemma 4.18, we deduce that k = k ′ and f (v j ) = (c b j ) for j ∈ {1, 2}. We set

, which proves the existence part of the right orthogonality to

1 and a 2 = a ′ 2 which proves the uniqueness part of the right orthogonality to ∇ n k .

Let

and the definition of elementary move (5.1) that for an elementary move

is an elementary move from f (u) to f (u ′ ). Thus, we have defined a map

Together with the map f :

] and, by restriction, a morphism of graphs

for any a ∈ C n . Lemma 6.6. With the notations of the above paragraph, the map

) and (v 2 , w 2 , e 2 , e ′ 2 ) be two elementary moves from u to u ′ such that

). In particular, we have Let µ : f (u) → f (u ′ ) be an element of the codomain. From Lemma 6.7 we know that there exists

In particular, we have

Since we have an elementary move from u to v and by hypothesis u ∈ T [Σ C ] a , we also have v ∈ T [Σ C ] a (see 5.15). Using the injectivity of the map

we conclude that v = u ′ . Proposition 6.10.

Proof. The case n = 0 is trivial. We know suppose that n > 0. From Corollary 6.9 we have that for every a ∈ C n , the map

is an isomorphism of graphs.


Arnaud Boutin
email: arnaud.boutin@u-psud.fr

Julien Doyon
email: julien.doyon@mcgill.ca

A Sleep Spindle Framework for Motor Memory Consolidation

Sleep spindle activity has repeatedly been found to contribute to brain plasticity and consolidation of both declarative and procedural memories. Here we propose a framework for motor memory consolidation that outlines the essential contribution of the hierarchical and multi-scale periodicity of spindle activity, as well as of the synchronization and interaction of brain oscillations during this sleep-dependent process. We posit that the clustering of sleep spindles in "trains", together with the temporally organized alternation between spindles and associated refractory periods, are critical for efficient reprocessing and consolidation of motor memories. We further argue that the long-term retention of procedural memories relies on the synchronized (functional connectivity) local reprocessing of new information across segregated, but inter-connected brain regions that are involved in the initial learning process.

Finally, we propose that oscillatory synchrony in the spindle frequency band may reflect the cross-structural reactivation, reorganization and consolidation of motor, and potentially declarative, memory traces within broader subcortical-cortical networks during sleep.

Introduction

A plethora of studies using a large variety of methodological approaches in animal models and humans have clearly demonstrated that memory is not a unitary process. On the one hand, declarative memory is defined as our capacity to acquire and recollect facts and events.

It is known to depend primarily upon integrity of the medial temporal lobe, and the hippocampus in particular (see 1,2, for reviews). On the other hand, procedural memory is described as our ability to acquire a variety of skills, including simple and complex motor skill behaviors such as shoe lacing or playing a musical instrument, respectively. To study this type of procedural memory, researchers have used a variety of motor sequence learning (MSL) paradigms that test the process by which discrete, independent actions become unified and performed effortlessly as a unitary sequence through repeated practice (see 3,4, for reviews).

Changes in performance seen during the acquisition of such skills are known to be incremental, implicit in nature, long lasting and dependent upon anatomical and functional plasticity within motor-related brain structures, including the striatum, cerebellum, motor-related cortical regions, as well as the hippocampus (e.g., [START_REF] Boutin | Transient synchronization of hippocampo-striato-thalamo-cortical networks during sleep spindle oscillations induces motor memory consolidation[END_REF][START_REF] Doyon | Distinct contribution of the cortico-striatal and cortico-cerebellar systems to motor skill learning[END_REF][START_REF] Doyon | Contributions of the basal ganglia and functionally related brain structures to motor learning[END_REF][START_REF] Doyon | Reorganization and plasticity in the adult brain during learning of motor skills[END_REF]. Yet, following the initial acquisition phase, the newly learned information is then thought to be processed and reactivated offline, such that initially labile memory traces become fixed into the physical structure of the brain through a cascade of events occurring at both the synaptic/cellular and the brain systems levels; a process called "memory consolidation" [START_REF] Dudai | The restless engram: consolidations never end[END_REF][START_REF] Dudai | The Consolidation and Transformation of Memory[END_REF]. In fact, studies using MSL paradigms have repeatedly shown that the procedural memory consolidation process of motor skill representations is primarily observed hours or days after the first training session, although Cohen and colleagues [START_REF] Bönstrup | A Rapid Form of Offline Consolidation in Skill Learning[END_REF] have recently reported that the same offline phenomena may also occur rapidly (i.e., on a time scale of seconds) between blocks of practice executed during the initial training session. Some studies have further revealed that such a consolidation process may depend on the type of MSL tasks being acquired, as the offline consolidation of skills acquired unintentionally (implicit learning) has been shown to be time-dependent, while skills acquired intentionally (explicit learning) are greatly dependent upon sleep for this process to occur [START_REF] Debas | Brain plasticity related to the consolidation of motor sequence learning and motor adaptation[END_REF][START_REF] Doyon | Contribution of night and day sleep vs. simple passage of time to the consolidation of motor sequence and visuomotor adaptation learning[END_REF][START_REF] Korman | Daytime sleep condenses the time course of motor memory consolidation[END_REF][START_REF] Robertson | Current concepts in procedural consolidation[END_REF]. Thus, although the conditions that trigger this latent mnemonic process are still conjectural, most of the work in this field of research has been done using explicit-related MSL paradigms to study its behavioral determinants, neural networks and sleep neurophysiological correlates.

Accordingly, this review will focus primarily on findings from sleep studies using explicitly learned materials, spanning declarative and procedural memory domains, to outline the role of sleep in the consolidation of motor skills, and to develop a framework dedicated to MSL consolidation, in particular.

Hierarchical multi-scale periodicity in sleep spindle activity and memory consolidation

There is now ample evidence indicating that sleep plays an active role in the consolidation of both explicit procedural and declarative memories (e.g., 10,16,17, for reviews).

Contemporary theories on memory consolidation postulate that the synchronization and interaction between specific brain rhythms during sleep may be responsible for the reactivationinduced strengthening and transformation of newly formed memory representations [START_REF] Vahdat | Network-wide reorganization of procedural memory during NREM sleep revealed by fMRI[END_REF][START_REF] Fogel | Reactivation or transformation? Motor memory consolidation associated with cerebral activation timelocked to sleep spindles[END_REF][START_REF] Genzel | Light sleep versus slow wave sleep in memory consolidation: a question of global versus local processes?[END_REF][START_REF] Rasch | About sleep's role in memory[END_REF].

Indeed, it has been suggested that low-frequency ~0.5-1 Hz slow oscillations (SO) during nonrapid eye movement (NREM) sleep drive and synchronize the repeated reactivation of newly encoded memories through spindle-shaped ~0.3-2s bursts of ~11-16Hz oscillatory activity (1,3,22, for reviews). As an electrophysiological hallmark of thalamo-cortical activity during NREM sleep, spindles are thought to be involved in sleep-dependent brain plasticity and memory consolidation through periodic and synchronized rhythmic activity over the brain [START_REF] Boutin | Transient synchronization of hippocampo-striato-thalamo-cortical networks during sleep spindle oscillations induces motor memory consolidation[END_REF][START_REF] Antony | Sleep Spindle Refractoriness Segregates Periods of Memory Reactivation[END_REF][START_REF] Boutin | Hierarchical organization of sleep spindles mediates motor memory consolidation[END_REF]. However, recent data from our own laboratory, and from other groups, suggest that the hierarchical organization and spontaneous rhythmic occurrence of sleep spindles are critical mechanisms necessary for effective covert memory reprocessing, reorganization and consolidation of memories. We thus propose here a sleep spindle framework for motor memory consolidation that integrates the rhythmic local reactivation of newly encoded information and the oscillatory interactions between segregated brain regions and nested frequencies for efficient strengthening and integration of new representations into long-term memory networks.

Using explicit MSL paradigms, numerous studies have revealed that both nocturnal and diurnal sleep enhance consolidation of newly acquired motor sequence knowledge, as expressed by post-sleep maintenance or additional performance gains in the absence of further task engagement [START_REF] Doyon | Current issues related to motor sequence learning in humans[END_REF][START_REF] King | Sleeping on the motor engram: The multifaceted nature of sleep-related motor memory consolidation[END_REF][START_REF] Albouy | Hippocampus and striatum: Dynamics and interaction during acquisition and sleep-related motor sequence memory consolidation[END_REF][START_REF] Walker | Sleep-Dependent Learning and Memory Consolidation[END_REF], for reviews). There is also ample evidence that motor memory consolidation is associated with state-like changes in sleep spindle characteristics (i.e., increases in spindle frequency, amplitude and density) following learning. However, most of the findings supporting the role of NREM sleep spindles in memory consolidation have been indirect in nature, with studies focusing on correlations between such spindle characteristics and memory improvements following a night of sleep or a mere daytime nap [START_REF] Boutin | Transient synchronization of hippocampo-striato-thalamo-cortical networks during sleep spindle oscillations induces motor memory consolidation[END_REF][START_REF] Fogel | The function of the sleep spindle: A physiological index of intelligence and a mechanism for sleep-dependent memory consolidation[END_REF][START_REF] Nishida | Daytime naps, motor memory consolidation and regionally specific sleep spindles[END_REF][START_REF] Morin | Motor Sequence Learning Increases Sleep Spindles and Fast Frequencies in Post-Training Sleep[END_REF]. To address this issue, researchers have more recently used a different experimental approach, and were able to confer a more causal role of spindles in declarative and procedural memory consolidation by modulating experimentally this type of brain oscillations using transcranial alternating current stimulation [START_REF] Lustenberger | Feedback-Controlled Transcranial Alternating Current Stimulation Reveals a Functional Role of Sleep Spindles in Motor Memory Consolidation[END_REF], or by inducing the reactivation of previously learned materials through olfactory/auditory stimulations using targeted memory reactivation (TMR) paradigms [START_REF] Oudiette | Upgrading the sleeping brain with targeted memory reactivation[END_REF][START_REF] Rudoy | Strengthening individual memories by reactivating them during sleep[END_REF][START_REF] Rasch | Odor Cues During Slow-Wave Sleep Prompt Declarative Memory Consolidation[END_REF][START_REF] Rasch | Reactivation and Consolidation of Memory During Sleep[END_REF][START_REF] Schönauer | Strengthening procedural memories by reactivation in sleep[END_REF].

Interestingly, TMR work has revealed that cued-memory reactivation during NREM sleep increases spindle activity and related consolidation of both declarative and procedural memories [START_REF] Rasch | Odor Cues During Slow-Wave Sleep Prompt Declarative Memory Consolidation[END_REF][START_REF] Laventure | NREM2 and Sleep Spindles Are Instrumental to the Consolidation of Motor Sequence Memories[END_REF][START_REF] Rihm | Reactivating Memories during Sleep by Odors: Odor Specificity and Associated Changes in Sleep Oscillations[END_REF][START_REF] Cousins | Cued Reactivation of Motor Learning during Sleep Leads to Overnight Changes in Functional Brain Activity and Connectivity[END_REF][START_REF] Antony | Cued memory reactivation during sleep influences skill learning[END_REF]. While previous declarative-memory studies have revealed that sensory stimulation delivered in synchrony with the SO up-states led to enhanced SO amplitudes, phaselocked spindle activity, and overnight memory consolidation [START_REF] Ngo | Auditory Closed-Loop Stimulation of the Sleep Slow Oscillation Enhances Memory[END_REF][START_REF] Shimizu | Closed-Loop Targeted Memory Reactivation during Sleep Improves Spatial Navigation[END_REF][START_REF] Ngo | Driving sleep slow oscillations by auditory closed-loop stimulation-a self-limiting process[END_REF][START_REF] Cairney | Memory Consolidation Is Linked to Spindle-Mediated Information Processing during Sleep[END_REF][START_REF] Göldi | Increased neuronal signatures of targeted memory reactivation during slow-wave up states[END_REF], recent evidence suggests that the effective reactivation of learned cue-associated memory content is dependent upon a precisely-timed cue stimulation that follows an innate, endogenous periodicity of spindle occurrence in the sleeping brain [START_REF] Antony | Sleep Spindles and Memory Reprocessing[END_REF][START_REF] Antony | Sleep Spindle Refractoriness Segregates Periods of Memory Reactivation[END_REF]. Indeed, by using real-time tracking of sleep spindles and systematic delivery of sensory cues at different time points following the offset of the preceding detected spindles, Antony and colleagues (23) found a precise temporal link between sleep spindles and cue-induced memory reactivation, with optimal memory reactivation being strongly linked to sleep spindle rhythmicity.

Additional work has also shown that the distribution of sleep spindles involved in memory consolidation is more localized and spatially restricted to learning-related brain regions than SOs, hence enabling the reactivation of specific memories [START_REF] Andrillon | Sleep Spindles in Humans: Insights from Intracranial EEG and Unit Recordings[END_REF][START_REF] Nir | Regional Slow Waves and Spindles in Human Sleep[END_REF]. Specifically, the widespread and in-phase distribution of SOs across brain regions has been thought to temporally coordinate the local spatial expression of spindle activity [START_REF] Cox | Large-scale structure and individual fingerprints of locally coupled sleep oscillations[END_REF][START_REF] Cox | Slow Oscillations during Sleep Coordinate Interregional Communication in Cortical Networks[END_REF]. According to this view, SOs would not only provide fine-tuned temporal frames to spindles such that they occur in their excitable up-states, but would also support their brain-wide spatial synchronization, hence facilitating the putative reactivation and consolidation of memories through systematic SO-spindle interactions [START_REF] Staresina | Hierarchical nesting of slow oscillations, spindles and ripples in the human hippocampus during sleep[END_REF]. Indeed, evidence suggests that the coupling between SO and spindle might play a key role in the process of memory consolidation [START_REF] Yordanova | Dynamic coupling between slow waves and sleep spindles during slow wave sleep in humans is modulated by functional presleep activation[END_REF][START_REF] Bergmann | Phase-Amplitude Coupling: A General Mechanism for Memory Processing and Synaptic Plasticity?[END_REF][START_REF] Muehlroth | Precise Slow Oscillation-Spindle Coupling Promotes Memory Consolidation in Younger and Older Adults[END_REF][START_REF] Helfrich | Old Brains Come Uncoupled in Sleep: Slow Wave-Spindle Synchrony, Brain Atrophy, and Forgetting[END_REF]. Recently, however, it has been proposed that additional oscillatory mechanisms, other than SOs and its coupling with spindles, relating to local spindle rhythmicity may also be of critical importance for the consolidation of both declarative and procedural memories [START_REF] Antony | Sleep Spindle Refractoriness Segregates Periods of Memory Reactivation[END_REF][START_REF] Boutin | Hierarchical organization of sleep spindles mediates motor memory consolidation[END_REF][START_REF] Lecci | Coordinated infraslow neural and cardiac oscillations mark fragility and offline periods in mammalian sleep[END_REF][START_REF] Lázár | Infraslow oscillations in human sleep spindle activity[END_REF][START_REF] Boutin | Clustered and temporally organised occurrence of NREM-stage2 sleep spindles mediates motor memory consolidation[END_REF]. In line with the latter viewpoint, studies have revealed that locally expressed spindles follow two types of rhythms during NREM sleep: the infraslow and mesoscale rhythms. In fact, results from our laboratory [START_REF] Boutin | Hierarchical organization of sleep spindles mediates motor memory consolidation[END_REF][START_REF] Boutin | Clustered and temporally organised occurrence of NREM-stage2 sleep spindles mediates motor memory consolidation[END_REF] and that of other researchers have shown that groups of periodically recurring spindles (called "trains") appear to cycle at an infraslow frequency scale (~0.02 Hz) during both light NREM stage 2 sleep (54,55) and deep NREM stage 3 slow-wave sleep (SWS) [START_REF] Antony | Sleep Spindle Refractoriness Segregates Periods of Memory Reactivation[END_REF]. Such "trains" of sleep spindles closely spaced in time are operationalized in our framework (see Figure 1A) as groups of two or more consecutive and region-specific spindles interspaced by less than or equal to 6 seconds [START_REF] Boutin | Hierarchical organization of sleep spindles mediates motor memory consolidation[END_REF], in comparison to those occurring in isolation (i.e., more than 6 seconds from region-specific neighboring spindles). Importantly, we have demonstrated that such groups of spindles play a more critical role than isolated ones in motor memory consolidation, as several metrics related to spindle trains (i.e., length of spindle trains, number of trains, ratio between clustered vs. isolated spindles) were positively correlated with the magnitude of overnight memory improvements. Finally, our findings [START_REF] Boutin | Hierarchical organization of sleep spindles mediates motor memory consolidation[END_REF][START_REF] Boutin | Clustered and temporally organised occurrence of NREM-stage2 sleep spindles mediates motor memory consolidation[END_REF] This finely-tuned alternance between region-specific spindle events and refractory periods has recently been proposed to be of paramount importance for optimal segregation of spindle events and interference-free reprocessing of memory traces. In fact, such temporally-coordinated spindles and periods of refractoriness are thought to be critical for efficient reprocessing and offline consolidation of both declarative and non-declarative procedural memories, respectively, during SWS-rich deep sleep [START_REF] Antony | Sleep Spindle Refractoriness Segregates Periods of Memory Reactivation[END_REF][START_REF] Schönauer | Sleep Spindles: Timed for Memory Consolidation[END_REF] and light NREM stage 2 sleep [START_REF] Boutin | Transient synchronization of hippocampo-striato-thalamo-cortical networks during sleep spindle oscillations induces motor memory consolidation[END_REF][START_REF] Boutin | Clustered and temporally organised occurrence of NREM-stage2 sleep spindles mediates motor memory consolidation[END_REF].

However, the question as to whether spindles belonging to the same "trains" mediate the repeated reactivation of a unique memory content, or whether they trigger reactivation of several previously learned memories, remains unanswered. Thus, in future studies it would be of interest to apply EEG/fMRI decoding-classification approaches (e.g., [START_REF] Wang | Targeted memory reactivation during sleep elicits neural signals related to learning content[END_REF][START_REF] Kouider | Inducing Task-Relevant Responses to Speech in the Sleeping Brain[END_REF][START_REF] Schönauer | Decoding material-specific memory reprocessing during sleep in humans[END_REF][START_REF] Pinsard | Consolidation alters motor sequence-specific distributed representations[END_REF][START_REF] Deuker | Memory consolidation by replay of stimulus-specific neural activity[END_REF], in order to compare the neural correlates of memory reactivations within and between spindle trains.

Neural substrates associated with spindle-related memory consolidation

Growing evidence suggests that newly acquired declarative and non-declarative procedural memory traces are reprocessed offline within neural circuits involving the hippocampus and other learning-related cortical and subcortical brain structures during memory consolidation, and that this mnemonic process engages two complementary neurophysiological mechanisms: "synaptic" and "systems" consolidation processes [START_REF] Klinzing | Mechanisms of systems memory consolidation during sleep[END_REF][START_REF] Boutin | Transient synchronization of hippocampo-striato-thalamo-cortical networks during sleep spindle oscillations induces motor memory consolidation[END_REF][START_REF] Albouy | Both the hippocampus and striatum are involved in consolidation of motor sequence memory[END_REF][START_REF] Genzel | To Replay, Perchance to Consolidate[END_REF][START_REF] Pennartz | The ventral striatum in off-line processing: ensemble reactivation during sleep and modulation by hippocampal ripples[END_REF][START_REF] Sawangjit | The hippocampus is crucial for forming non-hippocampal long-term memory during sleep[END_REF][START_REF] Schapiro | The hippocampus is necessary for the consolidation of a task that does not require the hippocampus for initial learning[END_REF][START_REF] Fogel | How to become an expert: A new perspective on the role of sleep in the mastery of procedural skills[END_REF]. At the synaptic level, it has recently been demonstrated that, irrespective of the memory system, post-learning sleep is characterized by enhanced temporal synchrony between cortical SO and spindle activity, topographically restricted to the brain regions that were initially involved during learning [START_REF] Yordanova | Dynamic coupling between slow waves and sleep spindles during slow wave sleep in humans is modulated by functional presleep activation[END_REF][START_REF] Muehlroth | Precise Slow Oscillation-Spindle Coupling Promotes Memory Consolidation in Younger and Older Adults[END_REF][START_REF] Helfrich | Old Brains Come Uncoupled in Sleep: Slow Wave-Spindle Synchrony, Brain Atrophy, and Forgetting[END_REF]. More specifically, it has been proposed that the topographical cortical organization of spindle patterns, occurring independently at specific recording sites or nearly simultaneously across multiple cortical regions [START_REF] Andrillon | Sleep Spindles in Humans: Insights from Intracranial EEG and Unit Recordings[END_REF][START_REF] Nir | Regional Slow Waves and Spindles in Human Sleep[END_REF][START_REF] Contreras | Spindle oscillation in cats: The role of corticothalamic feedback in a thalamically generated rhythm[END_REF][START_REF] Contreras | Control of Spatiotemporal Coherence of a Thalamic Oscillation by Corticothalamic Feedback Author ( s ): Diego Contreras[END_REF], may relate respectively to the reprocessing of local memory units or to the synchronized concurrent reprocessing of interrelated memory units. Cortical SOs are assumed to underpin the temporally-coordinated reactivation of task-specific brain regions, and to orchestrate the synchronized local reprocessing of independent memory components, hence allowing the consolidation of combined memory representations by strengthening the associations between inter-connected brain areas and their related spindle activity (see 20,22, for reviews). This viewpoint is in accord with several sleep studies in the procedural domain, in which authors have reported a spatiotemporal distribution of SO and nested spindle activity over the cortex, as well as evidence of reactivation of the memory traces time-locked to spindles over localized cortical and subcortical brain regions recruited during the learning phase [START_REF] Boutin | Transient synchronization of hippocampo-striato-thalamo-cortical networks during sleep spindle oscillations induces motor memory consolidation[END_REF][START_REF] Fogel | Reactivation or transformation? Motor memory consolidation associated with cerebral activation timelocked to sleep spindles[END_REF][START_REF] Nishida | Daytime naps, motor memory consolidation and regionally specific sleep spindles[END_REF][START_REF] Huber | Local sleep and learning[END_REF][START_REF] Huber | Arm immobilization causes cortical plastic changes and locally decreases sleep slow wave activity[END_REF]. More research is needed in humans, however, to determine whether this topographical expression of sleep spindles reflects cortical and subcortical reactivation of newly acquired memory representations, and whether brain-wide patterns of concurrent spindles may constitute an informative metric for categorizing specific memory trace reactivations.

In addition to the repeated reactivation-induced strengthening and stabilization of the newly acquired memory trace at the local synaptic level, there is accumulating evidence supporting the existence of a gradual reorganization of temporary formed representations over distributed and long-lasting brain networks (e.g., [START_REF] King | Schema and Motor-Memory Consolidation[END_REF][START_REF] Hennies | Sleep Spindle Density Predicts the Effect of Prior Knowledge on Memory Consolidation[END_REF] ; (see also 9,16, for reviews). In accordance with this systems level concept, work from our laboratory looking at procedural memory consolidation has revealed that motor memory traces are not only strengthened during sleep, but undergo qualitative representational changes (e.g., plasticity within the striatum) time-locked to spindle activity as well [START_REF] Vahdat | Network-wide reorganization of procedural memory during NREM sleep revealed by fMRI[END_REF][START_REF] Fogel | Reactivation or transformation? Motor memory consolidation associated with cerebral activation timelocked to sleep spindles[END_REF]. This reorganization in brain activity further correlated with overnight performance improvements, thus suggesting that sleep-dependent strengthening and transformation of memory traces may be beneficial for enhanced integration and long-term retention of memory representations. By conducting EEG deep-brain source reconstruction and functional connectivity analyses (i.e., EEG coherence analyses) during sleep spindle epochs, Boutin et al. (2018) have also provided additional evidence that spindle oscillations may be involved in sleep-dependent motor memory consolidation by locally reactivating and functionally binding segregated, but inter-connected, brain regions such as the hippocampus, striatum, thalamus, and motor-related cortical areas. Together, these findings thus suggest that oscillatory synchrony in the spindle frequency band may underlie the reactivation, reorganization and consolidation of memory traces within broader subcorticalcortical networks during sleep (see Figure 1B for schematic illustration of the role of crossstructural functional connectivity in motor memory consolidation). Yet more research is needed in order to extend and generalize these findings to anatomically and functionally distinct memory systems, i.e., the declarative memory system. Here we define clusters of sleep spindles closely spaced in time as "trains", and operationalized spindle trains as groups of two or more consecutive and region-specific spindles interspaced by less than or equal to 6 seconds. Isolated spindles, in contrast, are those occurring more than 6 seconds apart from their region-specific neighbors in time. Recurring spindle "trains" cycle at an infraslow frequency scale of about 0.02Hz. In addition, a finely-tuned alternance between spindles and associated refractory periods within trains, oscillating at an intermediate mesoscale rhythm of about 0.2-0.3Hz, allows efficient and interference-free motor memory reprocessing and consolidation. (B) Oscillatory synchrony in the 11-16Hz spindle frequency band is thought to reflect cross-structural reactivation, reorganization and consolidation of motor memory traces within subcortical-cortical networks involving the hippocampus (yellow), striatum (putamen in orange and caudate nucleus in blue), thalamus (green) and motor-related cortical regions (red).

Cross-frequency interactions in the sleeping brain support memory consolidation

Accompanying the multi-scale oscillatory hierarchy of NREM sleep spindle activity described above, it has been proposed that cross-frequency coupling and hierarchical nesting of NREM sleep rhythms may be of particular importance for the consolidation of declarative memories (see 21,49, for reviews). Current trends postulate that SOs coordinate the bidirectional hippocampo-cortical information flow in a phase-dependent manner, with spindle oscillations nested in the depolarizing up-phase of the SOs, together with hippocampaldependent sharp wave-ripples (SWR; ~80-140 Hz) nested into spindle troughs [START_REF] Klinzing | Mechanisms of systems memory consolidation during sleep[END_REF][START_REF] Buzsáki | Hippocampal sharp wave-ripple: A cognitive biomarker for episodic memory and planning[END_REF][START_REF] Rasch | About sleep's role in memory[END_REF][START_REF] Staresina | Hierarchical nesting of slow oscillations, spindles and ripples in the human hippocampus during sleep[END_REF][START_REF] Latchoumane | Thalamic Spindles Promote Memory Formation during Sleep through Triple Phase-Locking of Cortical, Thalamic, and Hippocampal Rhythms[END_REF].

Initiated by SOs, synchronized spindle-ripple interactions are thought to enable hippocampal and associated coherent cross-structural reactivations and offline sequential replay of trainingrelated neuronal patterns in a time-compressed representation [START_REF] Buzsáki | Hippocampal sharp wave-ripple: A cognitive biomarker for episodic memory and planning[END_REF][START_REF] Genzel | To Replay, Perchance to Consolidate[END_REF][START_REF] Pennartz | The ventral striatum in off-line processing: ensemble reactivation during sleep and modulation by hippocampal ripples[END_REF][START_REF] Ramanathan | Sleep-Dependent Reactivation of Ensembles in Motor Cortex Promotes Skill Consolidation[END_REF][START_REF] Lansink | Hippocampus leads ventral striatum in replay of place-reward information[END_REF]. This triple phase-locking of cortical SO, thalamo-cortical spindle and hippocampal SWR is assumed to improve memory consolidation by facilitating the induction of persistent synaptic plastic changes [START_REF] Niethard | Cortical circuit activity underlying sleep slow oscillations and spindles[END_REF] and by mediating the long-term storage of hippocampal-dependent memory information into pre-existing networks [START_REF] Staresina | Hierarchical nesting of slow oscillations, spindles and ripples in the human hippocampus during sleep[END_REF]. The functional coupling of these three rhythms has been further shown to occur cross-regionally during NREM sleep (e.g., [START_REF] Staresina | Hierarchical nesting of slow oscillations, spindles and ripples in the human hippocampus during sleep[END_REF][START_REF] Latchoumane | Thalamic Spindles Promote Memory Formation during Sleep through Triple Phase-Locking of Cortical, Thalamic, and Hippocampal Rhythms[END_REF][START_REF] Xia | Parvalbumin-positive interneurons mediate neocortical-hippocampal interactions that are necessary for memory consolidation[END_REF]. Hence, crossfrequency coupling has been considered the neurophysiological mechanism underlying local synaptic potentiation and inter-regional communication, both required for memory consolidation (e.g., 9,10,16, for reviews) through global synchronization and communication within cortical [START_REF] Staresina | Hierarchical nesting of slow oscillations, spindles and ripples in the human hippocampus during sleep[END_REF][START_REF] Latchoumane | Thalamic Spindles Promote Memory Formation during Sleep through Triple Phase-Locking of Cortical, Thalamic, and Hippocampal Rhythms[END_REF][START_REF] Mölle | Hippocampus Whispering in Deep Sleep to Prefrontal Cortex-For Good Memories?[END_REF] and subcortical networks [START_REF] Pennartz | The ventral striatum in off-line processing: ensemble reactivation during sleep and modulation by hippocampal ripples[END_REF][START_REF] Lansink | Hippocampus leads ventral striatum in replay of place-reward information[END_REF]. Moreover, it has been suggested that the hippocampus is also involved in this coupling process by initiating and orchestrating cross-structural memory reactivation for the strengthening of declarative memory traces within subcortical-cortical networks during sleep [START_REF] Pennartz | The ventral striatum in off-line processing: ensemble reactivation during sleep and modulation by hippocampal ripples[END_REF][START_REF] Lansink | Hippocampus leads ventral striatum in replay of place-reward information[END_REF], while its contribution is suggested to diminish throughout the consolidation process [START_REF] Pinsard | Consolidation alters motor sequence-specific distributed representations[END_REF][START_REF] Hennies | Sleep Spindle Density Predicts the Effect of Prior Knowledge on Memory Consolidation[END_REF][START_REF] Norimoto | Hippocampal ripples down-regulate synapses[END_REF] (see also 82, for a review). This is consistent with previous procedural-memory studies, which have pointed out the role of the hippocampus in the temporary storage of newly acquired motor sequence knowledge and of its contribution to offline memory reactivation and reorganization within distributed networks during sleep, then followed by its progressive disengagement in motor sequence memory representation [START_REF] Boutin | Transient synchronization of hippocampo-striato-thalamo-cortical networks during sleep spindle oscillations induces motor memory consolidation[END_REF][START_REF] Vahdat | Network-wide reorganization of procedural memory during NREM sleep revealed by fMRI[END_REF][START_REF] Fogel | Reactivation or transformation? Motor memory consolidation associated with cerebral activation timelocked to sleep spindles[END_REF][START_REF] Pinsard | Consolidation alters motor sequence-specific distributed representations[END_REF][START_REF] Albouy | Interaction between hippocampal and striatal systems predicts subsequent consolidation of motor sequence memory[END_REF]. Hence, akin to theories proposed for declarative memories, we conjecture that the hippocampal to subcortico-cortical memory tradeoff associated with reactivations during spindle-ripples events may reflect the concomitant consolidation of motor memory representations and hippocampal trace clearance, in order to prevent synaptic saturation and the ability to learn new information during succeeding periods of wakefulness [START_REF] Balduzzi | What can neurons do for their brain? Communicate selectivity with bursts[END_REF][START_REF] Lubenov | Decoupling through Synchrony in Neuronal Circuits with Propagation Delays[END_REF][START_REF] Yoo | A deficit in the ability to form new human memories without sleep[END_REF]. Yet, direct experimental evidence supporting this mechanism in human is also lacking for both declarative and procedural memory systems.

Concluding remarks: A framework for sleep-dependent motor memory consolidation

Based upon substantial evidence accumulated from work investigating the role of NREM sleep oscillations in human motor memory consolidation, we propose here a framework that highlights the contribution of sleep spindle activity in this mnemonic process (see Figure 1). We first suggest that the clustering and temporally organized alternation between spindles and associated refractory periods during NREM stage 2 sleep are essential for efficient reactivation and interference-free reprocessing and consolidation of motor memories [START_REF] Boutin | Hierarchical organization of sleep spindles mediates motor memory consolidation[END_REF][START_REF] Boutin | Clustered and temporally organised occurrence of NREM-stage2 sleep spindles mediates motor memory consolidation[END_REF] (see [START_REF] Antony | Sleep Spindle Refractoriness Segregates Periods of Memory Reactivation[END_REF], for evidence in relation to declarative memories). More specifically, we conjecture that spindle-related memory reactivations follow two periodic rhythms that can be decomposed into a ~0.02Hz infraslow periodicity of spindle "trains" and a ~0.2-0.3Hz mesoscale rhythm for the iterative and region-specific occurrence of sleep spindles within trains. We further highlight the fact that such neurophysiological events show a topographical distribution over cortical and subcortical structures involved during the initial learning phase, and that their temporallycoordinated occurrence is assumed to reflect synchronized oscillatory activity (functional connectivity) across multiple learning-specific brain regions and associated integration of new information into pre-existing knowledge structures. At the systems level, it is then suggested that oscillatory synchrony in the 11-16Hz spindle frequency band underlies the cross-structural reactivation, reorganization and consolidation of memory traces within subcortical-cortical networks involving the hippocampus, striatum, thalamus and motor-related cortical regions for sequential motor memories [START_REF] Boutin | Transient synchronization of hippocampo-striato-thalamo-cortical networks during sleep spindle oscillations induces motor memory consolidation[END_REF][START_REF] Vahdat | Network-wide reorganization of procedural memory during NREM sleep revealed by fMRI[END_REF][START_REF] Fogel | Reactivation or transformation? Motor memory consolidation associated with cerebral activation timelocked to sleep spindles[END_REF], and potentially, the hippocampus and other cortical regions that were activated during the acquisition phase in the case of declarative memories.

Altogether, we thus argue that the integrity of the mechanisms supporting the synchronization and interaction of brain rhythms during sleep is fundamental to the proper consolidation and long-term storage of coherent memory representations. However, more research is needed to pinpoint the neurobiological mechanisms responsible for the endogenous neuromodulation of the infraslow and mesoscale rhythms in the human brain across the adult lifespan, and to determine from a developmental perspective whether these rhythms develop through childhood and adolescence along with learning/cognitive functions and brain maturation [START_REF] Fogel | The function of the sleep spindle: A physiological index of intelligence and a mechanism for sleep-dependent memory consolidation[END_REF][START_REF] Reynolds | Sleep spindles and cognitive performance across adolescence: A meta-analytic review[END_REF].

  demonstrated that the relationship between spindle trains and procedural memory consolidation is observed during NREM stage 2, but not stage 3, sleep. Hence, this suggests that spindles, which iteratively occur locally within clustered and temporally organized patterns during NREM stage 2 sleep, are most effective in reactivating and further consolidating specific motor memories. Interestingly, support for the latter pattern of results comes from Lecci and colleagues (2017) who have demonstrated that an infraslow rhythmicity of 0.02Hz in spindle activity (every ~50s), more pronounced during NREM stage 2 than stage 3 sleep, is related to offline declarative memory consolidation. Altogether, these findings thus suggest that such a NREM sleep infraslow periodicity in humans may constitute a critical autonomic regulatory mechanism for modulating arousability, and subdividing sleep into cycling periods of environmental alertness and offline memory consolidation. A perturbation of this infraslow periodic clustering of spindle activity during NREM sleep would therefore be expected to cause inappropriate arousals and memory impairments due to disturbed rhythmicity of spindle trains. Yet, the role of spindle clustering in sleep-dependent consolidation of both declarative and procedural memories awaits further experimental confirmation.At an intermediate mesoscale (~0.2-0.3 Hz) between the infraslow periodicities of spindle trains and spindle oscillations themselves (~11-16 Hz), there is also converging evidence that spindles within trains undergo periods of refractoriness of about 3-6 seconds[START_REF] Antony | Sleep Spindle Refractoriness Segregates Periods of Memory Reactivation[END_REF].

Figure 1 .

 1 Figure 1. Sleep spindle framework for motor memory consolidation. (A) Illustration of the clustering and hierarchical rhythmicity of spindle activity during NREM stage 2 sleep.Here we define clusters of sleep spindles closely spaced in time as "trains", and operationalized spindle trains as groups of two or more consecutive and region-specific spindles interspaced by less than or equal to 6 seconds. Isolated spindles, in contrast, are those occurring more than 6 seconds apart from their region-specific neighbors in time. Recurring spindle "trains" cycle at an infraslow frequency scale of about 0.02Hz. In addition, a finely-tuned alternance between spindles and associated refractory periods within trains, oscillating at an intermediate mesoscale rhythm of about 0.2-0.3Hz, allows efficient and interference-free motor memory reprocessing and consolidation. (B) Oscillatory synchrony in the 11-16Hz spindle frequency band is thought to reflect cross-structural reactivation, reorganization and consolidation of motor memory traces within subcortical-cortical networks involving the hippocampus (yellow), striatum (putamen in orange and caudate nucleus in blue), thalamus (green) and motor-related cortical regions (red).