Camell Kachour

Introduction to Higher Cubical Operads

Keywords: cubical (∞, n)-categories, weak cubical ∞-groupoids, homotopy types. Mathematics Subject Classification (2010). 18B40, 18C15, 18C20, 18G55, 20L99, 55U35, 55P15

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Introduction to Higher Cubical Operads

Introduction

In this article, divided in two parts, we show how to build main aspects of the article [START_REF] Batanin | Monoidal globular categories as a natural environment for the theory of weak-n-categories[END_REF] but with the cubical geometry. This first part is devoted to build the contractible S-operad B 0 C equipped with a cubical C 0 -system, where S is the monad of free strict cubical ∞-categories on cubical sets. Actions of this monad are on cubical sets with no notions of reflexivities (the classical and the connections) in order to be sure that it is cartesian (see [START_REF] Kachour | The structure of generalised higher spans[END_REF]). In our approach, classical reflexivities plus connections, appear in the level of algebras. This operad is free on this C 0 -system (which itself is a specific cubical pointed S-collection). We exhibit some simple coherences cells of B 0 C and show how they provide more richness compare to its globular analogue (see [START_REF] Batanin | Monoidal globular categories as a natural environment for the theory of weak-n-categories[END_REF]). In the second part of this article (see [START_REF] Kachour | Introduction to Higher Cubical Operads. Second Part : The Functor of Fundamental Cubical Weak ∞-Groupoids for Spaces 2[END_REF]), we use it as a fundamental step to associate to any topological space X its fundamental cubical weak ∞-groupoids Π ∞ (X), and this endows a functor T op ∞-CGrp Π∞(-) which has a left adjoint functor CN ∞ . This pair of adjunction (CN ∞ , Π ∞ (-)) should put an equivalence between the homotopy category of homotopy types and the homotopy category of ∞-CGrp of cubical weak ∞-groupoids equipped with an adapted Quillen model structure. This was shown to be true but in the context of the Cisinski model structure on the category of cubical sets with connections (see [START_REF] Maltsiniotis | La catégorie cubique avec connections est une catégorie test stricte[END_REF]).

Acknowledgement. This work was doing thanks to the beautiful ambiance I got during my two months visiting in the IHES. I especially want to mention Maxim Kontsevich whose questions on my work were deep and push me to clarify a lot of aspects of my work. I want also to mention Pierre Cartier, who patiently, has taken his time to understand my work and who shared with me many of his very deep point of views of Higher Category Theory. During my stay, Vasily Pestun has shared with me some deep questions which relate mathematics and theoretical physics. I was especially impressed by the huge imagination that a physicist can have, starting with simple questions in mathematics, but which answers is highly non-trivial and should open new mathematics, that mathematicians cannot see when "jailed" in mathematics. I cannot also forget my friendly discussions with Thibault Damour ! I want also to point out how Laurent Lafforgue, Fanny Kassel, Emmanuel Ullmo and Elisabeth Jasserand, where all very kind with me. Many other mathematicians or physicists open my eyes on other point of views of science. I want to mention Thiago Araujo (Strings theory), Yang Liu (Non-commutative geometry), and Benjamin Ward (Operads and homology). Last but not least, this work wouldn't exist without my natural interactions with Ross Street and Mark Weber.

Cubical sets

See also [11] for more references on cubical sets.

The cubical category

Consider the small category C with integers n ∈ N as objects. Generators for C are, for all n ∈ N given by sources n s n n-1,j / / n -1 for each j ∈ {1, .., n} and targets n t n n-1,j / / n -1 for each j ∈ {1, .., n} such that for 1 ≤ i < j ≤ n we have the following cubical relations

(i) s n-1 n-2,i • s n n-1,j = s n-1 n-2,j-1 • s n n-1,i , (ii) s n-1 n-2,i • t n n-1,j = t n-1 n-2,j-1 • s n n-1,i , (iii) t n-1 n-2,i • s n n-1,j = s n-1 n-2,j-1 • t n n-1,i , (iv) t n-1 n-2,i • t n n-1,j = t n-1 n-2,j-1 • t n n-1,i
These generators plus these relations give the small category C called the cubical category that we may represent schematically with the low dimensional diagram : and this category C gives also the sketch E S of cubical sets used especially in 2.2, ?? and ?? to produce the monads S = (S, λ, µ), W = (W, η, ν) and W m = (W m , η m , ν m ) on CSets, which algebras are respectively cubical strict ∞-categories, cubical weak ∞-categories and cubical weak (∞, m)-categories.

• • • C 4 C 3 C 2 C 1 C 0 s 4 3,1 t 4 3,1 s 4 3
Definition 1 The category of cubical sets CSets is the category of presheaves [C; Sets]. The terminal cubical set is denoted 1.

2

Occasionally a cubical set shall be denoted with the notation C = (C n , s n n-1,j , t n n-1,j ) 1≤j≤n, n∈N in case we want to point out its underlying structures.

Reflexive cubical sets

Reflexivity for cubical sets are of two sorts : one is "classical" in the sense that they are very similar to their globular analogue; thus we shall use the notation (1 n n+1,j ) n∈N,j∈{1,..,n} to denote these maps C(n)

1 n n+1,j / / C(n + 1)
which formally behave like globular reflexivity ( [START_REF] Kachour | Algebraic definition of weak (∞, n)-categories[END_REF]); the others are called connections and are given by maps

C(n) Γ / / C(n + 1)
where the notation using the greek letter "Gamma" seems to be the usual notation.

However we do prefer to use instead the notation C(n)

1 n,γ n+1,j / / C(n + 1) (γ ∈ {+, -}) in order to point out the reflexive nature of connections.

Consider the cubical category C. For all n ∈ N we add in it generators n -1

1 n-1 n,j
/ / n for each j ∈ {1, .., n} subject to the relations :

(i) 1 n n+1,i • n-1 n,j = 1 n n+1,j+1 • 1 n-1 n,i if 1 ≤ i ≤ j ≤ n; (ii) s n n-1,i • n-1 n,j = 1 n-2 n-1,j-1 • s n-1 n-2,i if 1 ≤ i < j ≤ n; (iii) s n n-1,i • n-1 n,j = 1 n-2 n-1,j • s n-1 n-2,i-1 if 1 ≤ j < i ≤ n; (iv) s n n-1,i • n-1 n,j = id(n -1) if i = j. (i) 1 n n+1,i • n-1 n,j = 1 n n+1,j+1 • 1 n-1 n,i if 1 ≤ i ≤ j ≤ n; (ii) t n n-1,i • 1 n-1 n,j =1 n-2 n-1,j-1 • t n-1 n-2,i if 1 ≤ i < j ≤ n; (iii) t n n-1,i • 1 n-1 n,j = 1 n-2 n-1,j • t n-1 n-2,i-1 if 1 ≤ j < i ≤ n; (iv) t n n-1,i • 1 n-1 n,j = id(n -1) if i = j.
These generators and relations give the small category C sr called the semireflexive cubical category where a quick look at its underlying semireflexive structure is given by the following diagram : Consider the semireflexive cubical category C sr . For all integers n ≥ 1 we add in it generators n -1

C 0 C 1 C 2 C 3 C 4 • • • 1 0 1 1 1 2,1 1 1 2,2 1 2 3,1 1 2 3,2 1 2 3,3 1 3 4,1 1 3 4,2 1 3 4
1 n-1,γ n,j
/ / n for each j ∈ {1, .., n -1} subject to the relations :

(i) for 1 ≤ j < i ≤ n, 1 n,γ n+1,i • 1 n-1,γ n,j = 1 n,γ n+1,j+1 • 1 n-1,γ n,i ; (ii) for 1 ≤ i ≤ n -1, 1 n,γ n+1,i • 1 n-1,γ n,i = 1 n,γ n+1,i+1 • 1 n-1,γ n,i ; (iii) for 1 ≤ i, j ≤ n, 1 n,γ n+1,i • 1 n-1 n,j = 1 n n+1,j+1 • 1 n-1,γ n,i if 1 ≤ i < j ≤ n = 1 n n+1,j • 1 n-1,γ n,i-1 if 1 ≤ j < i ≤ n ; (iv) for 1 ≤ j ≤ n, 1 n,γ n+1,j • 1 n-1 n,j = 1 n n+1,j • 1 n-1 n,j ; (v) for 1 ≤ i, j ≤ n, s n n-1,i • 1 n-1,γ n,j = 1 n-2,γ n-1,j-1 • s n-1 n-2,i if 1 ≤ i < j ≤ n -1 = 1 n-2,γ n-1,j • s n-1 n-2,i-1 if 2 ≤ j + 1 < i ≤ n ; and t n n-1,i • 1 n-1,γ n,j = 1 n-2,γ n-1,j-1 • t n-1 n-2,i if 1 ≤ i < j ≤ n -1 = 1 n-2,γ n-1,j • t n-1 n-2,i-1 if 2 ≤ j + 1 < i ≤ n ; (vi) for 1 ≤ j ≤ n-1, s n n-1,j •1 n-1,- n,j = s n n-1,j+1 •1 n-1,- n,j = 1 n-1 and t n n-1,j •1 n-1,+ n,j = t n n-1,j+1 •1 n-1,+ n,j = 1 n-1 ; (vii) for 1 ≤ j ≤ n -1, s n n-1,j • 1 n-1,+ n,j = s n n-1,j+1 • 1 n-1,+ n,j = 1 n-2 n-1,j • s n-1 n-2,j ; (viii) for 1 ≤ j ≤ n -1, t n n-1,j • 1 n-1,- n,j = t n n-1,j+1 • 1 n-1,- n,j = 1 n-2 n-1,j • t n-1 n-2,j .
These generators and relations give the small category C r called the reflexive cubical category and in it, connections have the following shape : Cubical strict ∞-categories have been studied in [START_REF] Steiner | Thin fillers in the cubical nerves of omega-categories[END_REF][START_REF] Ali | Multiple Categories: The Equivalence of a Globular and a Cubical Approach[END_REF].

C 1 C 2 C 3 C 3 C 4 • • • 1 1,- 2,1 1 1,+ 2,1
In [START_REF] Ali | Multiple Categories: The Equivalence of a Globular and a Cubical Approach[END_REF] the authors proved that the category of cubical strict ∞-categories with cubical strict ∞-functors as morphisms is equivalent to the category of globular strict ∞-categories with globular strict ∞-functors as morphisms. Consider a cubical reflexive set

(C, (1 n n+1,j ) n∈N,j∈ 1,n+1 , (1 n,γ n+1,j ) n≥1,j∈ 1,n ) equipped with partial operations (• n j ) n≥1,j∈ 1,n where if a, b ∈ C(n) then a • n j b is defined for j ∈ {1, ..., n} if s n j (b) = t n j (a)
. We also require these operations to follow the following axioms of positions :

(i) For 1 ≤ j ≤ n we have :

s n n-1,j (a • n j b) = s n n-1,j (a) and t n n-1,j (a • n j b) = t n n-1,j (a), (ii) s n n-1,i (a • n j b) = s n n-1,i (a) • n-1 j-1 s n n-1,i (b) if 1 ≤ i < j ≤ n s n n-1,i (a) • n-1 j s n n-1,i (b) if 1 ≤ j < i ≤ n (iii) t n n-1,i (a • n j b) = t n n-1,i (a) • n-1 j-1 t n n-1,i (b) if 1 ≤ i < j ≤ n t n n-1,i (a) • n-1 j t n n-1,i (b) if 1 ≤ j < i ≤ n
The following sketch E M of axioms of positions as above shall be used in 2.2 to justify the existence of the monad on CSets of cubical strict ∞-categories. It is important to notice that the sketch just below has only one generation which means that diagrams and cones involved in it are not build with previous data of other diagrams and cones.

• For 1 ≤ i < j ≤ n we consider the following two cones :

M n × Mn-1,j M n M n M n M n-1 π n 0,j π n 1,j s n n-1,j t n n-1,j M n-1 × Mn-2,j-1 M n-1 M n-1 M n-1 M n-2 π n-1 0,j-1 π n-1 1,j-1 s n-1 n-2,j-1 t n-1 n-2,j-1
and the following commutative diagram (definition of

s n n-1,i × j,j-1 s n n-1,i ) M n × Mn-1,j M n M n M n-1 × Mn-2,j-1 M n-1 M n-1 M n M n-1 M n-2 π n 1,j π n 0,j s n n-1,i × j,j-1 s n n-1,i s n n-1,i π n-1 0,j-1 π n-1 1,j-1 s n-1 n-2,j-1 s n n-1,i t n-1 n-2,j-1
which gives the following commutative diagram

M n × Mn-1,j M n M n-1 × Mn-2,j-1 M n-1 M n M n-1 n j s n n-1,i × j,j-1 s n n-1,i n-1 j-1 s n n-1,i
• For 1 ≤ j < i ≤ n we consider the following two cones :

M n × Mn-1,j M n M n M n M n-1 π n 0,j π n 1,j s n n-1,j t n n-1,j M n-1 × Mn-2,j M n-1 M n-1 M n-1 M n-2 π n-1 0,j π n-1 1,j s n-1 n-2,j t n-1 n-2,j
and the following commutative diagram (definition of

s n n-1,i × j,j s n n-1,i ) M n × Mn-1,j M n M n M n-1 × Mn-2,j M n-1 M n-1 M n M n-1 M n-2 π n 1,j π n 0,j s n n-1,i × j,j s n n-1,i s n n-1,i π n-1 0,j π n-1 1,j s n-1 n-2,j s n n-1,i t n-1 n-2,j
The previous datas gives the following commutative diagram of axioms

M n × Mn-1 M n M n-1 × Mn-2 M n-1 M n M n-1 n j s n n-1,i × j,j s n n-1,i n-1 j s n n-1,i
and for 1 ≤ j ≤ n we have the following commutative diagram of axioms

M n × Mn-1 M n M n M n M n-1 n j π1 s n n-1,j s n n-1,j
which actually complete the description of E M Definition 4 Cubical reflexive ∞-magmas are cubical reflexive set equipped with partial operations like just above which follow axioms of positions. A morphism between two cubical reflexive ∞-magmas is a morphism of their underlying cubical reflexive sets. The category of cubical reflexive ∞-magmas is noted ∞-CMag r Remark 1 Cubical ∞-magmas are poorer structure : they are cubical sets equipped with partial operations like above with these axioms of positions. A morphism between two cubical ∞-magmas is a morphism of their underlying cubical sets.The category of cubical ∞-magmas is noted ∞-CMag 2

Definition

Strict cubical ∞-categories are cubical reflexive ∞-magmas such that partials operations are associative and also we require the following axioms :

(i) The interchange laws : (a

• n i b) • n j (c • n i d) = (a • n j c) • n i (b • n j d) whenever both sides are defined (ii) 1 n n+1,i (a • n j b) = 1 n n+1,i (a) • n+1 j+1 1 n n+1,i (b) if 1 ≤ i ≤ j ≤ n 1 n n+1,i (a • n j b) = 1 n n+1,i (a) • n+1 j 1 n n+1,i (b) if 1 ≤ j < i ≤ n + 1 (iii) 1 n,γ n+1,i (a • n j b) = 1 n,γ n+1,i (a) • n+1 j+1 1 n,γ n+1,i (b) if 1 ≤ i < j ≤ n 1 n,γ n+1,i (a • n j b) = 1 n,γ n+1,i (a) • n+1 j 1 n,γ n+1,i (b) if 1 ≤ j < i ≤ n (iv) First transport laws : for 1 ≤ j ≤ n 1 n,+ n+1,j (a • n j b) = 1 n,+ n+1,j (a) 1 n n+1,j (a) 1 n n+1,j+1 (a) 1 n,+ n+1,j (b) (v) Second transport laws : for 1 ≤ j ≤ n 1 n,- n+1,j (a • n j b) = 1 n,- n+1,j (a) 1 n n+1,j+1 (b) 1 n n+1,j (b) 1 n,- n+1,j (b) (vi) for 1 ≤ j ≤ n, 1 n,+ n+1,i (x) • n+1 i 1 n,- n+1,i (x) = 1 n n+1,i+1 (x) and 1 n,+ n+1,i (x) • n+1 i+1 1 n,- n+1,i (x) = 1 n n+1,i (x)
The category ∞-CCAT of strict cubical ∞-categories is the full subcategory of ∞-CMag r spanned by strict cubical ∞-categories. A morphism in ∞-CCAT is called a strict cubical ∞-functor. We study it more specifically in ?? with the perspective to weakened it and to obtain cubical model of weak ∞-functors.

The monad of cubical strict ∞-categories

In this section we describe cubical strict ∞-categories as algebras for a monad on CSets. We hope it to be a specific ingredient to compare globular strict ∞-categories with cubical strict ∞-categories Consider the forgetful functor : ∞-CCAT U / / CSets which associate to any strict cubical ∞category its underlying cubical set and which associate to any strict cubical ∞-functor its underlying morphism of cubical sets.

Proposition 1 The functor U is right adjoint 2 Its left adjoint is denoted F Proof We are going to use Sketch Theory as explain in [START_REF] Coppey | Leçons de théorie des esquisses[END_REF] : Actually it is not difficult to see that the category ∞-CCAT and the category CSets are both projectively sketchable. Let us denote by E C the sketch of ∞-CCAT and E S the sketch of CSets. Main parts of E C are described just below and we see that E C contains E S , and that this inclusion induces a forgetful functor ∞-CCAT U / / CSets which has a left adjunction thanks to the sheafification theorem of Foltz [START_REF] Foltz | Sur la catégorie des foncteurs dominés[END_REF]. Now we have the commutative diagram

M od(E C ) / / iso M od(E S ) iso ∞-CCAT U / / CSets which shows that U is right adjoint.
The description of E C started with the description of E M in 2. We carry on to it in describing the sketch behind the interchange laws, which shall complete main parts of E C :

• In the first generation of E C we start with three cones :

Z n × Zn-1,i Z n Z n Z n Z n-1 ρ n 0,i ρ n 1,i s n n-1,i t n n-1,i Z n × Zn-1,j Z n Z n Z n Z n-1 ρ n 0,j ρ n 1,j s n n-1,j t n n-1,j E ijn Z n Z n Z n Z n Z n-1 Z n-1 Z n-1 π n 00 π n 10 π n 01 π n 11 t n n-1,i s n n-1,i s n n-1,j t n n-1,j t n n-1,i s n n-1,i
• Then we consider the following commutative diagrams :

E ijn Z n × Zn-1,i Z n Z n Z n Z n π n 00 p n 1000 π n 10 ρ n 1,i ρ n 0,i s n n-1,i t n n-1,i E ijn Z n × Zn-1,i Z n Z n Z n Z n π n 01 p n 1101 π n 11 ρ n 1,i ρ n 0,i s n n-1,i t n n-1,i E ijn Z n × Zn-1,j Z n Z n Z n Z n π n 11 p n 1011 π n 10 ρ n 1,j ρ n 0,j s n n-1,j t n n-1,j E ijn Z n × Zn-1,j Z n Z n Z n Z n π n 01 p n 0001 π n 00 ρ n 1,j ρ n 0,j s n n-1,j t n n-1,j
• We consider then (still in the first generation) the following two commutative diagrams :

E ijn Z n × Zn-1,i Z n Z n × Zn-1,i Z n Z n × Zn-1,j Z n Z n Z n Z n-1 p n 1000 c n 1 p n 1101 n n-1,i n n-1,i ρ n 1,j ρ n 0,j s n n-1,j t n n-1,j E ijn Z n × Zn-1,j Z n Z n × Zn-1,j Z n Z n × Zn-1,i Z n Z n Z n Z n-1 p n 1011 c n 2 p n 0001 n n-1,j n n-1,j ρ n 1,i ρ n 0,i s n n-1,i t n n-1,i
• Finally we consider the following commutative diagram of interchange laws

E ijn Z n × Zn-1 Z n Z n × Zn-1 Z n Z n c n 1 c n 2 n n-1,j n n-1,i
The monad of strict cubical ∞-categories on cubical sets is denoted S = (S, λ, µ). Here λ is the unit map of S : 1 CSets λ / / S and µ is the multiplication of S : S 2 µ / / S 3 The cubical higher operad of cubical weak ∞-categories

The monoidal category of cubical pointed S-collections

In [START_REF] Kachour | The structure of generalised higher spans[END_REF] we will prove that the free cubical strict ∞-categories monad S = (S, λ, µ) on CSets built in 2.2 is cartesian. Thanks to this cartesianess we can build the monoidal category S-Coll p of pointed S-collections If S is a cartesian monad on a category G then S-collections are kind of S-graphs defined in [START_REF] Leinster | Higher Operads, Higher Categories[END_REF], where their domains of arities is an object S(1) such that 1 is a terminal object of the category G. The category of S-collections is denoted S-Coll. The category of pointed S-collections is also defined in [START_REF] Leinster | Higher Operads, Higher Categories[END_REF] and is denoted S-Coll p . In this section we accept the following result Conjecture The monad S = (S, λ, µ) (see 2.2) of strict cubical ∞-categories on cubical sets is cartesian Thus we can work with the locally finitely presentable category S-Coll p of pointed S-collections (n ∈ N). An object of S-Coll p is denoted (C, a, c; p), and described by a commutative diagram in CSets

1 λ } } p id S(1) C a o o c / / 1
The category S-Coll p is monoidal and described in [START_REF] Leinster | Higher Operads, Higher Categories[END_REF]. Monoids in it are S-operads.

Definition 5

The category of S-operads is given by the category of monoids of the monoidal category S-Coll p . We denote is by COper.

2

Sometimes we shall call it cubical operads in order to make clear the geometry involved for this kind of higher operads.

Cubical contractions

Consider a pointed S-collection (C, a, c; p), and for each n ≥ 1 and for all integer k ≥ 1, we define the following subsets of

C n × C n • C n = {(α, β) ∈ C n × C n : a n (α) = a n (β)} • C s n,j = {(α, β) ∈ C n × C n : s n n-1,j (α) = s n n-1,j (β) and a n (α) = a n (β)} • C t n,j = {(α, β) ∈ C n × C n : t n n-1,j (α) = t n n-1,j (β) and a n (α) = a n (β)}
and also we consider

C 0 = {(α, β) ∈ C 0 × C 0 : α = β}
Then (C, a, c; p) is equipped with a cubical contractibility structure if they are extra structures given by maps :

([-; -] n n+1,j : C n / / C n+1 ) n∈N;j∈{1,...,n+1} ([-; -] n,- n+1,j : C s n,j / / C n+1 ) n≥1;j∈{1,...,n} , ([-; -] n,+ n+1,j : C t n,j / / C n+1 ) n≥1;j∈{1,...,n} such that • If 1 ≤ i < j ≤ n + 1, then s n+1 n,i ([α, β] n n+1,j ) = [s n n-1,i (α), s n n-1,i (β)] n-1 n,j-1 , and t n+1 n,i ([α, β] n n+1,j ) = [t n n-1,i (α), t n n-1,i (β)] n-1 n,j-1 • If 1 ≤ j < i ≤ n + 1 then s n+1 n,i ([α, β] n n+1,j ) = [s n n-1,i-1 (α), s n n-1,i-1 (β)] n-1 n,j , and t n+1 n,i ([α, β] n n+1,j ) = [t n n-1,i-1 (α), t n n-1,i-1 (β)] n-1 n,j • If i = j then s n+1 n,i ([α, β] n n+1,j ) = α and t n+1 n,i ([α, β] n n+1,j ) = β • a n+1 ([α, β] n n+1,j ) = 1 n n+1,j (a n (α)) = 1 n n+1,j (a n (β)), • ∀α ∈ C n , [α, α] n n+1,j = 1 n n+1,j (α).
and such that

• for 1 ≤ j ≤ n we have :

-s n+1 n,j ([α; β] n,- n+1,j ) = α and s n+1 n,j+1 ([α; β] n,- n+1,j ) = β -t n+1 n,j ([α; β] n,+ n+1,j ) = α and t n+1 n,j+1 ([α; β] n,+ n+1,j ) = β -s n+1 n,j ([α; β] n,+ n+1,j ) = s n+1 n,j+1 ([α; β] n,+ n+1,j ) = [s n n-1,j (α); s n n-1,j (β)] n-1 n,j -t n+1 n,j ([α; β] n,- n+1,j ) = t n+1 n,j+1 ([α; β] n,- n+1,j ) = [t n n-1,j (α); t n n-1,j (β)] n-1 n,j • for 1 ≤ i, j ≤ n + 1 -s n+1 n,i ([α; β] n,γ n+1,j ) = [s n n-1,i (α); s n n-1,i (β)] n-1,γ n,j-1 if 1 ≤ i < j ≤ n [s n n-1,i-1 (α); s n n-1,i-1 (β)] n-1,γ n,j if 2 ≤ j + 1 < i ≤ n + 1 -t n+1 n,i ([α; β] n,γ n+1,j ) = [t n n-1,i (α); t n n-1,i (β)] n-1,γ n,j-1 if 1 ≤ i < j ≤ n [t n n-1,i-1 (α); t n n-1,i-1 (β)] n-1,γ n,j if 2 ≤ j + 1 < i ≤ n + 1 • a n+1 ([α; β] n,γ n+1,j ) = 1 n,γ n+1,j (a n (α)) = 1 n,γ n+1,j (a n (β)) • ∀α ∈ C n , [α, α] n,γ n+1,j = 1 n,γ n+1,j (α).
Such S-collection (C, a, c; p) is called contractible where its contractibilty structure is usually denoted by :

([-; -] n n+1,j ) n∈N;j∈{1,...,n+1} , ([-; -] n,γ n+1,j ) n≥1;j∈{1,...,n};γ∈{-,+}
A morphism of pointed contractible S-collections is given by a morphism of S-Coll p :

(C, a, c; p) (C , a , c ; p ) f which preserves their structures of contractibility, i.e it is given by a map :

C C f such that : f ([α; β] n n+1,j ) = [f (α); f (β)] n n+1,j and f ([α; β] n,γ n+1,j ) = [f (α); f (β)] n,γ n+1,j
The category of pointed contractible S-collections is denoted CS-Coll p .

Proposition 2 (G.M. Kelly) Let K be a locally finitely presentable category, and M nd f (K) the category of finitary monads on K and strict morphisms of monads. Then M nd f (K) is itself locally finitely presentable. If T and S are object of M nd f (K), then the coproduct T S is algebraic, which means that

K T × K K S is equal to K T S
and the diagonal of the pullback square Lets denote by B the monad on S-Coll p which algebras are cubical higher operads, and denote by T V the monad on S-Coll p which algebras are pointed contractible S-collections. We are in the situation of the above proposition, which shows that algebras of the sum B T V is the following pullback in CAT :

K T × K K S p1 / / p2 K S U K T V / / K is the forgetful functor K T S -→ K. Furthermore the projections K T × K K S -→ K T and K T × K K S -→ K S are monadic.
COper × S-Collp CS-Coll p CS-Coll p COper S-Coll p V U
This pullback is denoted CCOper for short. It is the category of contractible cubical higher operads. The left adjoint functor F of the monadic forgetful functor :

CCOper S-Coll p W
gives free contractible cubical higher operads. In particular it gives the free contractible cubical higher operad B 0 C on the specific pointed S-collection (C 0 , a 0 , c 0 ; p 0 ) that we shall describe in the next section. This operad B 0 C is the cubical analogue of the operad of Michael Batanin, the one which algebras are the globular weak ∞-categories. Similarly algebras for this cubical operad B 0 C are cubical weak ∞-categories.

The cubical operad of cubical weak ∞-categories

Cubical pasting diagrams or Cubical trees are cells of the free cubical strict ∞-category S(1) on the terminal object 1 of CSets. For example 1(n) n j 1(n) for j ∈ {1, ..., n} are basic examples of cubical trees, and they are such that

s n n-1,i (1(n) n j 1(n)) = 1(n -1) n-1 j-1 1(n -1) if 1 ≤ i < j ≤ n 1(n -1) n-1 j 1(n -1) if 1 ≤ j < i ≤ n
Now we are going to build a specific pointed S-collection (C 0 , a 0 , c 0 ; p 0 ) which is the underlying pointed S-collection of the contractible cubical higher operad B 0 C which algebras are weak cubical ∞-categories. This collection is build as follow :

• C 0 (1) contains a cell u 1 such that s1 0 (u 1 ) = t 1 0 (u 1 ) = u 0 , and for each integer n ≥ 2 we have an n-cell

u n ∈ C 0 (n) which is such that : ∀j ∈ {1, ..., n}, s n n-1,j (u n ) = t n n-1,j (u n ) = u n-1 .
Arities and coarities of such cells are easy : ∀n ∈ N,

a 0 n (u n ) = c 0 n (u n ) = 1(n) • C 0
n contains, for all n ≥ 1 and all j ∈ {1, ..., n} an n-cell µ n j which is such that :

-      s n n-1,j (µ n j ) = u n-1 , t n n-1,j (µ n j ) = u n-1 s n n-1,i (µ n j ) = t n n-1,i (µ n j ) = µ n-1 j-1 if 1 ≤ i < j ≤ n s n n-1,i (µ n j ) = t n n-1,i (µ n j ) = µ n-1 j if 1 ≤ j < i ≤ n -a n (µ n j ) = 1(n) n j 1(n) • The pointing C 0 S(1) 1
1 a 0 c 0 λ(1) id p 0 is given by p 0 n (1(n)) = u n . Definition 6
The free contractible cubical higher operad B 0 C on the pointed S-collection (C 0 , a 0 , c 0 ; p 0 ) described just above, is the operad for cubical weak ∞-categories. Its underlying monad is denoted

(B 0 C , η 0 , ν 0 ) 1 . The category of cubical weak ∞-categories is denoted B 0 C -Alg 2 Let us give simple examples of cells in B 0 C : consider the 1-cells x = γ(µ 1 0 ; µ 1 0 1 0 u 1 ) and y = γ(µ 1 0 ; u 1 1 0 µ 1 0 ). Because the couple (x, y) belongs to C - 1,0 ∩ C + 1,0 we get the following 2-cells in B 0 C : u 0 u 0 u 0 u 0 [x, y] 1 2,1 [u0,u0] 0 1 γ(µ 1 0 ;µ 1 0 1 0 u1) [u0,u0] 0 1 γ(µ 1 0 ;u1 1 0 µ 1 0 ) u 0 u 0 u 0 u 0 [x, y] 1 2,2 γ(µ 1 0 ;µ 1 0 1 0 u1) [u0,u0] 0 1 γ(µ 1 0 ;u1 1 0 µ 1 0 ) [u0,u0] 0 1 u 0 u 0 u 0 u 0 [x, y] 1,- 2,1 γ(µ 1 0 ;u1 1 0 µ 1 0 ) γ(µ 1 0 ;µ 1 0 1 0 u1) [u0,u0] 0 1 [u0,u0] 0 1 u 0 u 0 u 0 u 0 [x, y] 1,+ 2,1 [u0,u0] 0 1 [u0,u0] 0 1 γ(µ 1 0 ;u1 1 0 µ 1 0 ) γ(µ 1 0 ;µ 1 0 1 0 u1)
These coherences show that if G is an object of CSets, and if :

B 0 C (G) G v is a B 0 C -algebra, then for any string in G(1) : a b c d f g h
where :

X = (h • 1 0 g) • 1 0 f := v(γ(µ 1 0 ; µ 1 0 1 0 u 1 ); η 0 (h) 1 0 η 0 (g) 1 0 η 0 (f )) Y = h • 1 0 (g • 1 0 f ) := v(γ(µ 1 0 ; u 1 1 0 µ 1 0 ); η 0 (h) 1 0 η 0 (g) 1 0 η 0 (f )) the contractions of the operad B 0 C derive the following 2-cubes in G : a d a d [X, Y ] 1 2,1 1a (h• 1 0 g)• 1 0 f 1 d h• 1 0 (g• 1 0 f ) a a d d [X, Y ] 1 2,2 (h• 1 0 g)• 1 0 f 1a h• 1 0 (g• 1 0 f ) 1 d a d d d [X, Y ] 1,- 2,1 h• 1 0 (g• 1 0 f ) (h• 1 0 g)• 1 0 f 1 d 1 d a d d d [X, Y ] 1,+ 2,1 1a 1a h• 1 0 (g• 1 0 f ) (h• 1 0 g)• 1 0 f
These simple examples show us how cubical weak ∞-categories provide more richness of coherences than the globular weak ∞-categories. [START_REF] Brown | double modules?, double categories and groupoids, and a new homotopy double groupoid[END_REF] The Functor of Fundamental Cubical Weak ∞-Groupoids for Spaces 20 years ago Michael Batanin in [START_REF] Batanin | Monoidal globular categories as a natural environment for the theory of weak-n-categories[END_REF] had described the functor of fundamental globular weak ∞-groupoids for spaces in order to give a rigorous formulation of the Grothendieck conjecture on homotopy types [START_REF] Grothendieck | Pursuing Stacks (A la poursuite des Champs)[END_REF] : in particular he built a functor from the category Top of spaces to the category of globular weak ∞-groupoids. In order to do that he built an operadic approach of globular weak ∞-categories, that is his globular weak ∞-categories are algebra for a specific operad B 0 C . Two major steps for higher category theory were achieved in [START_REF] Batanin | Monoidal globular categories as a natural environment for the theory of weak-n-categories[END_REF] :

• he builts a higher dimensional approach of non-symmetric operads à la Peter May;

• his definition of weak ∞-categories is more general than simplicial models of (∞, 1)-categories. For example it is proved in [START_REF] Kachour | Algebraic definition of weak (∞, n)-categories[END_REF] that some algebraic models of (∞, 1)-categories are embedded in his weak ∞-categories.

In order to built the functor of fundamental globular weak ∞-groupoids for spaces he proved that the globular object D • in Top consisting of topological disks :

D 0 D 1 D 2 • • • D n-1 D n • • • s 1 0 t 1 0 s 2 1 t 2 1 s n n-1 t n n-1
is a B 0 C -coalgebra, which implication is the construction of the fundamental globular weak ∞-groupoid functor

Top ∞-Grp Π∞(-)
In [START_REF] Leinster | Higher Operads, Higher Categories[END_REF] Tom Leinster gave a simplification of the orginal definition of higher operads by Michael Batanin. However the very important examples of (co)endomorphism globular operads are built very naturally with globular monoidal categories, and this is not clear for us that the T-categorial framework of Leinster can capture such natural point of view of (co)endomorphism globular operads. It seems that in [START_REF] Leinster | Higher Operads, Higher Categories[END_REF], he succeeded to define such (co)endomorphism globular operads through T-categories, but only in the context of locally cartesian closed categories. For example if C is a category with pullbacks and if E is a global object in the monoidal globular category Span(C) consisting of globular higher spans in C, it is possible to define its globular endomorphism operad END(E) by using the theory of Batanin (see also [START_REF] Street | The role of Michael Batanin's monoidal globular categories, Higher Category Theory[END_REF]), but this is not clear for us how to get such operad END(E) with T-categories. Thus in order to write the first part of the article [START_REF] Kachour | Introduction to Higher Cubical Operads. First Part : The Cubical Operad of Cubical Weak ∞-Categories[END_REF] we used the Leinster approach to build the operad which algebras are cubical weak ∞-categories, but to define cubical higher operads of endomorphism we found that the cubical analogue of the globular monoidal categories was much more natural.

In this article, which is the second part of [START_REF] Kachour | Introduction to Higher Cubical Operads. First Part : The Cubical Operad of Cubical Weak ∞-Categories[END_REF], we use the cubical operad B 0 C of cubical weak ∞-categories (built in [START_REF] Kachour | Introduction to Higher Cubical Operads. First Part : The Cubical Operad of Cubical Weak ∞-Categories[END_REF]) as a fundamental step to associate to any topological space X its fundamental cubical weak ∞-groupoids Π ∞ (X), and this endows a functor Top ∞-CGrp

Π∞(-)
which has a left adjoint functor CN ∞ . This pair of adjunction (CN ∞ , Π ∞ (-)) should put an equivalence between the homotopy category of homotopy types and the homotopy category of ∞-CGrp of cubical weak ∞-groupoids with connections equipped with an adapted Quillen model structure. This was shown to be true but in the context of the Cisinski model structure on the category of cubical sets with connections (see [START_REF] Maltsiniotis | La catégorie cubique avec connections est une catégorie test stricte[END_REF]). It is also important to know that non-operadical approach have been considered in [START_REF] Brown | A new higher homotopy groupoid: the fundamental globular ω-groupoid of a filtered space[END_REF][START_REF] Grandis | Higher fundamental groupoids for spaces[END_REF] to define other higher groupoid constructions.

Important tools to build this functor Π ∞ (-) come from 2-category theory and especially thanks to the work of Mark Weber ( [START_REF] Weber | Yoneda structures from 2-toposes[END_REF][START_REF] Weber | Operads Within Monoidal Pseudo-algebras[END_REF]) and Ross Street ([44]) : pseudo-algebras for 2-monads and a generalization of the Span construction have been successfully considered for this interaction between elementary 2-topos and cubical geometry. An important feature of this article is also to show how the 2-categorical tools developed in [START_REF] Street | The petit topos of globular sets[END_REF][START_REF] Street | The role of Michael Batanin's monoidal globular categories, Higher Category Theory[END_REF][START_REF] Weber | Yoneda structures from 2-toposes[END_REF][START_REF] Weber | Operads Within Monoidal Pseudo-algebras[END_REF] can lead to generalization of the original theory of Michael Batanin's higher operads. It is Plan :

• In the first section we define cubical monoidal categories as pseudo S-algebras, where S is the 2-monad of free strict cubical monoidal categories on cubical categories.

• In the second section we state an important result of [START_REF] Kachour | The structure of generalised higher spans[END_REF] which shows that for general situations the Span-construction leads to pseudo algebraic structure. Then we give a nice combinatorial description of the cubical (co)spans coming from Marco Grandis ([31]). Then we define (co)endomorphisms operads by using the 2-categorical point of view of Ross Street and Mark Weber in [START_REF] Street | The petit topos of globular sets[END_REF][START_REF] Street | The role of Michael Batanin's monoidal globular categories, Higher Category Theory[END_REF][START_REF] Weber | Yoneda structures from 2-toposes[END_REF][START_REF] Weber | Operads Within Monoidal Pseudo-algebras[END_REF]. Our 2-categorical point of view of (co)endomorphisms operads can be adapted in the general context of pseudo algebras, and this is very important for a 2-categorical generalisation of the theory of Batanin.

• In the third section we proved that the cocubical object "box" (as defined in [START_REF] Brown | Non abelian Algebraic Topology[END_REF]) in Top :

I 0 I 1 I 2 I 3 I 4 • • • I n-1 I n • • • s 0 1 t 0 1 s 2 1,1 t 2 1,1 s 2 1,2 t 2 1,2 s 3 2,1 t 3 2,1 s 3 2,2 t 3 2,2
s n n-1,n-1 t n n-1,n-1 s n n-1,i t n n-1,i s n n-1,1 t n n-1,1
is a B 0 C -coalgebra, where B 0 C is the S-operad which algebras are cubical weak ∞-categories. Then we show how to "glue" the K i -functors of Quillen in order to obtain a functor :

Rings

∞-CGrp

K∞

• The fourth and last section is a short "manifesto" for the following slogan : "coalgebraic structures govern different higher category theory". In particular we explain the main steps to get the cubical weak ∞-category of cubical weak ∞-categories.

5 Cubical monoidal categories as Pseudo-algebras

The cubical category

Consider the small category C with integers n ∈ N as objects. Generators for C are, for all n ∈ N given by sources n

s n n-1,j
/ / n -1 for each j ∈ {1, .., n} and targets n

t n n-1,j
/ / n -1 for each j ∈ {1, .., n} such that for 1 ≤ i < j ≤ n we have the following cubical relations

(i) s n-1 n-2,i • s n n-1,j = s n-1 n-2,j-1 • s n n-1,i , (ii) s n-1 n-2,i • t n n-1,j = t n-1 n-2,j-1 • s n n-1,i , (iii) t n-1 n-2,i • s n n-1,j = s n-1 n-2,j-1 • t n n-1,i , (iv) t n-1 n-2,i • t n n-1,j = t n-1 n-2,j-1 • t n n-1,i
These generators plus these relations give the small category C called the cubical category that we may represent schematically with the low dimensional diagram :

• • • C 4 C 3 C 2 C 1 C 0 s 4 3,1 t 4 3,1 s 4 3,2 t 4 3,2 s 4 3,3 t 4 3,3 s 4 3,4 t 4 3,4 s 3 2,1 t 3 2,1 s 3 2,2 t 3 2,2 s 3 2,3 t 3 2,3 s 2 1,1 t 2 1,1 s 2 1,2 t 2 1,2 s 1 0 t 1 0
and this category C gives also the sketch E S of cubical sets used especially in [START_REF] Kachour | Aspects of Cubical Higher Category Theory[END_REF] to produce the monads S = (S, λ, µ), which algebras are cubical strict ∞-categories. 2

In particular it is shown in [START_REF] Kachour | Aspects of Cubical Higher Category Theory[END_REF] that the category ∞-CCat of strict cubical ∞-categories is sketchable by a projective sketch. Thus we put the following definition of cubical strict monoidal categories : Definition 9 Strict cubical monoidal categories are internal cubical strict ∞-categories in CAT. They form a strict 2-category CM s C where :

• 0-cells are internal cubical strict ∞-categories in CAT;

• 1-cells are internal cubical strict ∞-functors in CAT;

• 2-cells are internal globular 2 strict ∞-natural transformations in CAT.

2

In [START_REF] Kachour | Aspects of Cubical Higher Category Theory[END_REF] we denoted by (S, η, µ) the monad on CSets of cubical strict ∞-categories, and cubical n-trees are just n-cells of S(1). We shall prove in [START_REF] Kachour | The structure of generalised higher spans[END_REF] that this monad is cartesian, and we denote again by (S, η, µ) its corresponding 2-monad on the 2-category ∞-CCAT. Also the following 2-forgetful functor is 2-monadic : CM s C ∞-CCAT , because the forgetful functor ∞-CCat CSets is monadic and the 2-functor

CAT pull 2-CAT CAT(-)
, which takes a category X with pullbacks to the 2-category CAT(X) of internal categories preserves (finite) limits, thus preserves adjunctions and Eilenberg-Moore constructions. Thus we prefer to denote S-Alg s this 2-category CM s C of strict cubical monoidal categories. This 2-monad (S, η, µ) gives weaker notions of algebras, and we recall it for any 2-monad (S, η, µ) on a 2-category K (see [START_REF] Blackwell | Two-dimensional monad theory[END_REF][START_REF] Weber | Operads Within Monoidal Pseudo-algebras[END_REF]). In particular we shall need the notion of pseudo S-algebra in order to define cubical monoidal categories below.

Definition 10 Let (S, η, µ) be a 2-monad on a 2-category K. A pseudo-algebra structure (a, α 0 , α) on an object A ∈ K is given by a 1-cell S(A) A a and two invertible 2-cells in K :

S 2 (A) S(A) S(A) A S(a) µ(A) a a α =⇒ A S(A) A 1 A η A a α =⇒
such that the following equalities hold : 2 that is they are 2-globes between two cubical strict ∞-functors, whereas cubical strict ∞-natural transformations are 2-cubes with faces, four cubical strict ∞-functors. See [START_REF] Kachour | Aspects of Cubical Higher Category Theory[END_REF] The triple (A, α 0 , α) is called a pseudo S-algebra. If α 0 is an identity the pseudo algebra is said to be normal. If α 0 and α are identities then we recover the usual notion of S-algebra, and in that case we say that A is equipped with a strict S-algebra structure.

S 2 (A) S(A) S 3 (A) S 2 (A) A S 2 (A) S(A) µ(A) a µ(S(A)) S 2
Definition 11 Let (A, α 0 , α) and (A , α 0 , α ) two pseudo S-algebras. A strong S-morphism structure for a 1-cell

A A
f is given by an invertible 2-cell :

S(A) A S(A ) A S(f ) a f a f =⇒
, such that we have the following equalities :

S(A) A S 2 (A) S(A) A S 2 (A ) S(A ) a f µ(A) S(a) S 2 (f ) a S(f ) S(a ) a α ⇐= f⇑ S( f ) =⇒ = S(A) A S 2 (A) S(A ) A S 2 (A ) S(A ) a f µ(A) S 2 (f ) a µ(A ) S(a ) a α ⇐= f =⇒ = and

S(A)

A A

S(A )

A A a η(A) f f 1 A f =⇒ α 0 ⇑ = = A S(A) A A 1 A η(A) a f α 0 =⇒
Definition 12 Let f and f be strong S-morphisms :

(a, α 0 , α) (a , α 0 , α ) f f . A 2-cell f f
ψ is an algebra 2-cell if the following equality holds :

S(A) A S(A ) A S(f ) S(f ) a f a S(ψ) =⇒ f =⇒ = S(A) A S(A ) A S(f ) a f f a f =⇒ ψ =⇒
Let us denote by Ps-S-Alg the 2-category which objects are pseudo S-algebras, whose 1-cells are strong S-morphisms and whose 2-cells are algebra 2-cells. The full sub-2-category of Ps-S-Alg consisting of the normal pseudo-algebras is denoted Ps 0 -S-Alg, and the locally full sub-2-category of Ps-S-Alg consisting of the strict algebras and strict morphisms is denoted S-Alg s .

Remark 2 Ps 0 -S-Alg blabla bla : les P sn modeles les MG-cat, mais les Psn sont equivalentes au Ps. Donc on utilisera que les Ps pour modeler les MC-cat.

2 Now let us comeback to the 2-monad S = (S, λ, µ) on the 2-category of cubical categories ∞-CCAT as described above, which strict 2-algebras are strict cubical monoidal categories.

Definition 13

The 2-category of monoidal cubical categories consists of the 2-category Ps-S-Alg of pseudo S-algebras 2 Also by using the theorem 5.1 and the theorem 5.12 of [START_REF] Blackwell | Two-dimensional monad theory[END_REF] we get the following biadjunction, similar to the one described in [START_REF] Street | The role of Michael Batanin's monoidal globular categories, Higher Category Theory[END_REF] :

Corollary 1 The forgetful 2-functor U : Ps-S-Alg ∞-CCAT U F
such that :

• Ps-S-Alg is the 2-category of pseudo S-algebras;

• ∞-CCAT is the 2-category of cubical categories;

• F builds the free strict monoidal cubical categories functor.

exhibits a biadjunction which restricts to a 2-adjunction on the strict monoidal cubical categories. 6 Cubical Higher Spans and Cubical Higher Cospans • objects are pairs (c, x) where c ∈ C and x ∈ X(c).

• morphisms :

(c, x) (d, y) , are pairs (f, α) where d c f is in C and X(f )(x) y α is in X(d).
• compositions and identities come from C and the categories X(c).

Suppose now that T = (T, η, µ) is a cartesian monad on [C op , Sets], and let us denote again by T = (T, η, µ) its extension to a 2-monad on [C op , CAT]. In fact, for any category E with pullbacks it is proved in [START_REF] Kachour | The structure of generalised higher spans[END_REF] that :

Theorem 1 (Kachour,Weber) Span C (E) is a pseudo T-algebra 2
In fact we can dualize such construction and produce a similar result which says that Cospan C (E) is a pseudo T-algebra if E is a category with pushouts, and these produce the following diagram of functors :

CAT push CAT Ps-S-Alg ∞-CCAT CAT pull CAT j (-) op

Cospan(-) (-) op

Cospan(-)

i

Span(-)

k

Span(-)

This result has two essential virtues : first it convince the reader that actually the structure behind the spans and the cospans construction are really of pseudo-algebraic nature; secondly it shows, and this is we believe the main fact, that probably not only globular and cubical higher category theory need such structures, but other useful higher category theory could need it.

However because of the "cubical scopes" of this article, we are going to describe cubical spans and cubical cospans in a more combinatorial way because this concrete point of view has the advantage to see it unpacked, and thus gives an accurate idea of what these cubical spans and cubical cospans looks like. This combinatorial description has been described first by Marco Grandis in [START_REF] Grandis | Higher cospans and weak cubical categories (Cospans in algebraic topology, I)[END_REF], and it is instructive to compare it with the Batanin's combinatorial construction of globular spans and globular cospans [START_REF] Batanin | Monoidal globular categories as a natural environment for the theory of weak-n-categories[END_REF]. The only new tools here are the connections on cubical (co)spans which are accurately describe.

In order to formalize cubical higher spans and cubical higher cospans we will use the formal span category V or the formal cospan category Λ used by Marco Grandis (see [START_REF] Grandis | Higher cospans and weak cubical categories (Cospans in algebraic topology, I)[END_REF]). For simplicity we will explain only constructions for cubical higher spans, which use this small category V :

-1 1 0 f g because for cubical higher cospans, constructions are duals, and use the small category Λ :

0 -1 1 f g
Definition 14 Let C be a category. The category Span n (C) of cubical n-spans in C is the category of functors [V n ; C] and natural transformations between them.

2

The combinatoric description of the category V n shall be useful : each objects of V n are n-uplets (m 1 , ..., m n ) ∈ {0, -1, 1} n . Also the category V n underlies a n-cube structure, such that the object (0, ..., 0) represents the n-face, and the n-uplets (m 1 , ..., m n ) ∈ {0, -1, 1} n which countains exactly p integers m j which are equal to zero, represent p-faces. Consider (m 1 , ..., m n ) a (p + 1)-face and suppose m ji = 0 for 1 ≤ i ≤ p + 1. Thus we get two morphisms in V n :

(m 1 , ..., m ji , ..., m n ) (m 1 , ..., m ji-1 , mji , m ji+1 , ..., m n ) (m1,...,mj i -1,f,mj i +1,...,mn) (m1,...,mj i -1,g,mj i +1,...,mn )

such that (m 1 , ..., m ji-1 , f, m ji+1 , ..., m n ) switch the value m ji to the value mji = -1 and (m 1 , ..., m ji-1 , g, m ji+1 , ..., m n ) switch the value m ji to the value mji = 1.

Remark 3 Intuitively such map (m 1 , ..., m ji-1 , f, m ji+1 , ..., m n ) is a kind of s p+1 p,ji and the map (m 1 , ..., m ji-1 , g, m ji+1 , ..., m n ) is a kind of t p+1 p,ji .

2

In particular the following arrows in V n : (0, ..., 0) (0, ..., 0, -1, 0, ..., 0), (0,...,0,f,0,...,0) (0, ..., 0) (0, ..., 0, 1, 0, ..., 0) (0,...,0,g,0,...,0)

shall be important for an accurate description of the projective cone below, when we will describe the pseudo-algebraic structure produced by cubical higher spans in a category with pullbacks. Now we want to put a cubical category structure on cubical spans. For that we just recall the constructions of Marco Grandis (see [START_REF] Grandis | Higher cospans and weak cubical categories (Cospans in algebraic topology, I)[END_REF]).

• The formal source functor is given by 1 V

s

, where 1 = { } is the terminal category and s sends to -1. Similarly the formal target functor is given by 1 V t where t sends to 1.

These give the source functors

V n-1 V n s n n-1,i , given by s n n-1,i := V i-1 × s × V n-i for 1 ≤ i ≤ n,
and the target functors V n-1

V n t n n-1,i
, given by t n n-1,i := V i-1 × t × V n-i for 1 ≤ i ≤ n, and then we get the cubical category of spans in C :

• • • [V 4 ; C] [V 3 ; C] [V 2 ; C] [V ; C] C s 4 3,1 t 4
[V n ; C] [V n-1 ; C] s n n-1,i t n n-1,i
• The formal reflexivity functor is given by the unique functor V 1 ! , and this gives for

1 ≤ i ≤ n the reflexivity functors V n V n-1 1 n-1 n,i
, given by 1 n-1 n,i := V i-1 ×! × V n-i , and then we get a reflexivity structure on the cubical category of spans in C :

C [V 1 ; C] [V 2 ; C] [V 3 ; C] [V 4 ; C] • • • 1 0 1 1 1 2,1 1 1 2,2 1 2 3,1 1 2 3,2 1 2 3,3 1 3 4,1 1 3 4,2 1 3 4,3 1 3 4,4
where for each 1 ≤ i ≤ n, 1 n-1 n,i is a functor :

[V n-1 ; C] [V n ; C] 1 n-1 n,i
• Connections for cubical higher (co)spans are not defined in Grandis [START_REF] Grandis | Higher cospans and weak cubical categories (Cospans in algebraic topology, I)[END_REF], thus we need to formalize it properly. V 2 may be seen as the following cubical 2-span :

(0, 0) (-1, 0) (0, 1) (0, -1) (1, 0) (-1, 1) (-1, -1) (1, 1) (1, -1) and if A B f is an 1-cell, then the 2-cell : A B B B 1 1,- 2,1 (f ) f f 1 B 1 B
is represented by the following 2-span :

1 1,- 2,1 (f ) f 1 B f 1 B B A B B
and the 2-cell

A A A B 1 1,+ 2,1 (f ) 1 A 1 A f f
is represented by the following 2-span :

1 1,+ 2,1 (f ) 1 A f 1 A f A A B A
These show us how to formalise connections for cubical higher spans : the formal connection functors are thus given by : V 2 V,

1 - V 2 V 1 +
defined on objects 3 of V 2 by (0, 0), (-1, 0), (0, -1) 0, (0, 1), (1, 0), (-1, 1), (1, 1), (1, -1) 1, (-1, -1) -1,

1 - 1 - 1 -

and

(0, 0), (0, 1), (1, 0) 0, (-1, 0), (0, -1), (-1, 1), (-1, -1), (1, -1) -1, (1, 1) 1.

1 + 1 + 1 +
These give the connection functors :

V n+1 V n , V n+1 V n , 1 n,- n+1,i 1 n,+ n+1,i
given by 1 n,- n+1,i :=

V i-1 × 1 -× V n-i and 1 n,+ n+1,i := V i-1 × 1 + × V n-i
, and then we get the structure of connections on the cubical category of spans in C : 3 Of course, these definition on objects give the one on arrows of V 2 where for each 1 ≤ i ≤ n, 1 n,- n+1,i , 1 n,+ n+1,i are functors :

[V 1 ; C] [V 2 ; C] [V 3 ; C] [V 4 ; C] [V 5 ; C] • • • 1 1,- 2,1 1 1,+ 2,1 1 2,- 3,1 1 2,+ 3,1 1 2,- 3,2 1 2,+ 3,2 1 3,- 4,1 1 3,+ 4,1 1 3,- 4,2 1 3,+ 4,2 1
[V n ; C] [V n+1 ; C] 1 n,- n+1,i 1 n,+ n+1,i
Now suppose that C is a category equipped with pullbacks. This context allows to put a pseudo-algebra structure on cubical higher spans in C. In fact this cubical monoidal structure shall be given by these pullbacks. We will follow the definition of Grandis (see [START_REF] Grandis | Higher cospans and weak cubical categories (Cospans in algebraic topology, I)[END_REF]) with a small variation on projective sketch. Our goal, for each n ≥ 1 and each 1 ≤ i ≤ n, is to build functors :

[V n ; C] × [V n-1 ;C] [V n ; C] [V n ; C] ⊗ n i such that [V n ; C] × [V n-1 ;C] [V n ; C] comes from the pullback : [V n ; C] × [V n-1 ;C] [V n ; C] [V n ; C] [V n ; C] [V n-1 ; C] s n n-1,i t n n-1,i
• First we consider the category V 2 given by the following pushout :

{ } V V V 2 s t k + k -
Thus V 2 is given by the category :

a c -1 b 1
that we extend to the category V2 :

0 a c -1 b 1
Also the following subcategory W of V 2 shall be considered :

a c
b and the natural transformation ∆(0) :

W V2 , ∆ (0)
F τ where ∆(0) is the constant functor with value 0. This allow to see the category V2 as the category V 2 equipped with a cone over W , that is V2 is a projective sketch equipped with the cone ∆(0) F τ ; also we have the concatenation functor : V V2 k which sends 0 to 0, and -1 to -1, and finally 1 to 1, from the category V to the projective sketch V2 . Now for each n ≥ 1 and each 1 ≤ i ≤ n, consider the pushout diagram :

V n-1 V n V n V n i s n n-1,i t n n-1,i k + i k - i where k - i = V i-1 × k -× V n-i , k + i = V i-1 × k + × V n-i and V n i = V i-1 × V 2 × V n-i . The category V n
i may be thought as the gluing of itself along the functors s n n-1,i and t n n-1,i , and also the category

V n i := V i-1 × V2 × V n-i
may be thought as the category V n i equipped with a cone over its following subdiagram :

(0, ..., 0) (0, ..., 0) (0, ..., 0, 1, 0, ..., 0) ∼ (0, ..., 0, -1, 0, ..., 0) (0,...,0,f,0,...,0) (0,...,0,g,0,...,0)

in this subdiagram the symbol ∼ means the identification of (0, ..., 0, 1, 0, ..., 0) and (0, ..., 0, -1, 0, ..., 0) under the pushout, and the cone is formally described by the natural transformation :

V i-1 × W × V n-i V i-1 × V2 × V n-i V i-1 × ∆(0) × V n-i V i-1 × F × V n-i τ =V i-1 ×τ ×V n-i
• Now consider two cubical n-spans x and y such that s n n-1,i (x) = t n n-1,i (y) in category C equipped with pullbacks :

V n C. x y
We are in the following situation where we get the unique functor [x, y] i :

V n-1 V n V n V n i C s n n-1,i t n n-1,i k + i y k - i x [x,y]i thus we get the functor [x, y] i : V n i C [x,y] i
which is the extension of the functor [x, y] i on the category V n i , which sends the cone τ i = V i-1 × τ × V n-i to the following pullback in C :

•

x(0, ..., 0) y(0, ..., 0)

x(0, ..., 0, 1, 0, ..., 0) = y(0, ..., 0, -1, 0, ..., 0)

x(0,...,0,f,0,...,0) y(0,...,0,g,0,...,0)

• Thus we obtain the diagram :

V n V n i C, ki [x,y] i
where

k i = V i-1 × k × V n-i comes from the concatenation functor : V V2 , k
and we put : y

⊗ n i x = [x, y] i • k i .
As for globular higher spans, these tensor products on arrows comes from universality of these pullbacks. Thus for each n ≥ 1 and each 1 ≤ i ≤ n, we built functors :

[V n ; C] × [V n-1 ;C] [V n ; C] [V n ; C]
⊗ n i which put on Span(C) a pseudo-algebra structure.

Of course the description of the pseudo-algebra Cospan(C), where C is a category with pushouts, is obtained by dualizing these constructions.

B 0

C -algebras and B 0 C -coalgebras Definition 15 If C is a cubical monoidal category then a global object of C is given by a morphism :

1 C E in the category ∞-CCAT of cubical categories.
2

By the pseudo-universality of 1 S(1)

η(1)
we get the following morphism [E] of cubical monoidal categories :

S(1) C 1 [E] η (

S(C)

[E]

S(E) v

This morphism [E] is denoted End(E) for the case of the cubical monoidal category C = Span(C) where C is a category with pullbacks; thus a global object in it : 1 Span(C) E produces such extension

S(1) Span(C)

End(E)

, and furthermore this morphism End(E) contains all informations we need to define the S-operad of endomorphism END(E) associated to the global object E in Span(C) :

Definition 16 For all n ∈ N, n-cells of END(E) consist of elements of the set hom Span n (C) (End(E)(t), E(n)), for each cubical n-tree t ∈ S(1). These n-cells form the set END(E)(n), and the corresponding cubical set END(E) underlies an S-operad where the multiplication of it is defined as follow : if (x, y) is an n-cell of S(END(E)) ×

S(1)

END(E), and is such that4 µ(1)(S(a)(x)) = t and a(y) = t :

S(END(E)) ×

S(1)

END(E) S(END(E))

END(E)

S(1) 2 S (1) 1 S(1) END(E)
End(E)(t ) End(E)(t) E(n) v(x) y
where

S(Span(C)) Span(C)

v is the structural map of the pseudo S-algebra Span(C); the unity of it is given, for each n ∈ N, by the singleton

1 E(n) ∈ hom Span n (C) (E(n), E(n)).
The axiom of associativity of the multiplication of END(E) comes from the associativity of compositions of each categories Span n (C) (n ∈ N), and we have the similar result for the axiom of unities. , contains all informations we need to define the S-operad of coendomorphism COEND(E) associated to the global object E in Cospan(C) :

Definition 18 For all n ∈ N, n-cells of COEND(E) consist of elements of the set hom Cospan n (C) (E(n), Coend(E)(t)), for each cubical n-tree t ∈ S(1). These n-cells form the set COEND(E)(n), and the corresponding cubical set COEND(E) underlies an S-operad where the multiplication of it is defined as follow : if (x, y) is an n-cell of S(COEND(E)) ×

S(1)

COEND(E), and is such that µ(1)(S(a)(x)) = t and a(y) = t :

S(COEND(E)) × S(1)

COEND(E) S(COEND(E))

COEND(E)

S(1) 2 S (1) 1 S(1)

COEND(E)

Coend(E)(t ) Coend(E)(t) E(n) v(x) y
where

S(Cospan(C)) Cospan(C)

v is the structural map of the pseudo S-algebra Cospan(C); the unity of it is given, for each n ∈ N, by the singleton

1 E(n) ∈ hom Cospan n (C) (E(n), E(n)).
The axiom of associativity of the multiplication of COEND(E) comes from the associativity of compositions of each categories Cospan n (C) (n ∈ N), and we have the similar result for the axiom of unities.

2 Also we have the following easy result :

Corollary 3 A global object 1 Cospan(C) E
is the same thing as to give a cocubical object, still denoted by E, internal to the category C :

C op C E 2
Now we are ready to define B 0 C -coalgebras :

Definition 19 Consider a category C with pushouts, plus a cocubical object E in it :

C op C E . E is equipped with a B 0 C -coalgebra structure if there is a morphism of S-operads : B 0 C COEND(E) f . 2
Also it is easy to check that for each global object E of Span(C) where C has pullbacks, the construction of END(E) endows a functor, and also for each global object E of Cospan(C) where C has pushouts, the construction of COEND(E) is also functorial. Recall from 6.1 that we got the following diagram of functors :

CAT push CAT Ps-S-Alg ∞-CCAT CAT pull CAT j (-) op Cospan(-) (-) op Cospan(-) i Span(-) k Span(-)
and we have the following result :

Corollary 4 • If C D F
is a morphism of the category CAT pull , and if

1 Span(C) Span(D) E E Span(F )
is a morphism of the category (1 ↓ i • Span(-)), then it produces the morphism of S-operads :

END(E) END(E ) END(Span(F ))
Furthemore this construction is functorial and gives the functor

(1 ↓ i • Span(-)) S-Oper END(-) • If C D F
is a morphism of the category CAT push , and if

1 Cospan(C) Cospan(D) E E Cospan(F )
is a morphism of the category (1 ↓ i • Cospan(-)), then it produces the morphism of S-operads :

COEND(E) COEND(E )

COEND(Cospan(F ))

Furthemore this construction is functorial and gives the functor

(1 ↓ i • Cospan(-)) S-Oper COEND(-) 7
I 0 I 1 I 2 I 3 I 4 • • • I n-1 I n • • • s 0 1 t 0 1 s 2 1,1 t 2 1,1 s 2 1,2 t 2 1,2 s 3 2,1 t 3 2,1 s 3 2,2 t 3 2,2 s 3 2,3 t 3 2,3 s 4 3,4 t 4 3,4 s 4 3,3 t 4 3,3 s 4 3,2 t 4 3,2 s 4 3,1 t 4 3,1 s
n n-1,n-1 t n n-1,n-1 s n n-1,i t n n-1,i s n n-1,1 t n n-1,1
defined by : s n n-1,i (x 1 , ..., x i-1 , x i , ..., x n-1 ) = (x 1 , ..., x i-1 , 0, x i , ..., x n-1 )

t n n-1,i (x 1 , ..., x i-1 , x i , ..., x n-1 ) = (x 1 , ..., x i-1 , 1, x i , ..., x n-1
). This is a global object of the pseudo-algebra Cospan(Top). Following the notation in [START_REF] Brown | Non abelian Algebraic Topology[END_REF], this global object I • shall be called the box object. Thanks to the pseudo-universality of 1 S(

we get the following commutative (up to isomorphisms) diagram :

S(1)

Cospan(Top)

1

Coend(I • ) η(1) I •
and from the cubical monoidal functor Coend(I • ) we get the S-operad COEND(I • ) (6.2). The next section is devoted to prove that I • is a B 0 C -coalgebra, i.e that the S-operad COEND(I • ) is contractible and is equipped with a composition system in the sense of cubical higher operads [START_REF] Kachour | Introduction to Higher Cubical Operads. First Part : The Cubical Operad of Cubical Weak ∞-Categories[END_REF].

7.2 I • is a B 0 C -coalgebra 7.2.1 Composition systems on COEND(I • )
The cubical (n -1)-sphere S n-1 c is given by the sums :

S n-1 c := 1≤i≤n (I i-1 × {0} × I n-i I i-1 × {1} × I n-i )
and we have the inclusion : S n-1 c

I n For all 1 ≤ i ≤ n we are going to build by induction maps :

I n I n i I n-1 I n µ n i such that I n i I n-1
I n is the following pushout :

I n-1 I n I n I n i I n-1 I n s n n-1,i t n n-1,i
that is, we start with I 0 I 0 , id and we suppose that the maps

I n-1 I n-1 i I n-2 I n-1 µ n-1 i are already defined for 1 ≤ i ≤ n -1. We glue S n-1 c
with itself along the same face and we obtain the inclusion i :

I n-1 S n-1 c I n S n-1 c S n-1 c i I n-1 S n-1 c I n I n i I n-1 I n s n n-1,i s n n-1,i t n n-1,i t n n-1,i i
In order to build µ n i we are going to build first its interior

• µ n i : S n-1 c S n-1 c i I n-1 S n-1 c . • µ n i
It is defined by the following induction :

• If i = j then we put I i-1 × {0} × I n-i I i-1 × {0} × I n-i , id
where the identity map id sends the

(n -1)-faces I i-1 × {0} × I n-i of the first copy S n-1 c in S n-1 c i I n-1 S n-1 c to the (n -1)-faces I i-1 × {0} × I n-i
of the second copy S n-1 c , and we put :

I i-1 × {1} × I n-i I i-1 × {1} × I n-i , id
where the identity map id sends the (n -1)-faces

I i-1 × {1} × I n-i of the first copy S n-1 c in S n-1 c i I n-1 S n-1 c
to the (n -1)-faces I i-1 × {1} × I n-i of the second copy S n-1 c .

• If 1 ≤ j < i ≤ n then we put : I j-1 × {0} × I n-j I j-1 × {0} × I n-j i-1

I n-2 I j-1 × {0} × I n-j , µ n-1 i-1
and I j-1 × {1} × I n-j I j-1 × {1} × I n-j i-1

I n-2 I j-1 × {1} × I n-j , µ n-1 i-1
where the codomains are given by the following pushout :

I n-2 I n-1 I n I n-1 i-1 I n-2 I n-1 s n-1 n-2,i-1 t n-1 n-2,i-1 • If 1 ≤ i < j ≤ n then we put : I j-1 × {0} × I n-j I j-1 × {0} × I n-j j-1 I n-2 I j-1 × {0} × I n-j , µ n-1 j-1 and I j-1 × {1} × I n-j I j-1 × {1} × I n-j j-1 I n-2 I j-1 × {1} × I n-j , µ n-1 j-1
where the codomains are given by the following pushout :

I n-2 I n-1 I n I n-1 j-1 I n-2 I n-1 s n-1 n-2,j-1 t n-1 n-2,j-1
Thus we obtain the desired extension µ n i of • µ n i :

I n S n-1 c S n-1 c i I n-1 S n-1 c I n i I n-1 I n µ n i • µ n i i 7.2.2 Contractibility of COEND(I • )
Consider two maps in Top :

I n-1 X, f g
such that f and g are two (n -1)-cells of the operad

COEND(I •

). Thus X is described as an iterated pushouts of the topological n-cubes I n (n ∈ N) given by the global object I • in the pseudo-algebra Cospan(Top); and in particular X is contractible. We are going to build the contraction [f, g] n-1 n,j by induction. Thus we suppose that for all 1 ≤ j ≤ n -1 the maps

I n-1 X [f,g] n-2 n-1,j
exist, and we start our induction with an easy choice of extension [f, g] 0 1 , where f and g define here two points of X : I X.

[f,g] 0 1

The contraction [f, g] n-1 n,j is given by a continuous map I n X.

[f,g] n-1 n,j

In order to do that, for all 1 ≤ j ≤ n, we need first to define the map : S n-1 c X.

f,g n-1

n,j

This map f, g n-1 n,j has the following definition :

• for i = j we put :

I j-1 × {0} × I n-j X, f and I j-1 × {1} × I n-j X g • If 1 ≤ i < j ≤ n then I i-1 × {0} × I n-i X, [s n-1 n-2,i (f ),s n-1 n-2,i (g)] n-2 n-1,j-1 and I i-1 × {1} × I n-i X [t n-1 n-2,i (f ),t n-1 n-2,i (g)] n-2 n-1,j-1 • If 1 ≤ j < i ≤ n then I i-1 × {0} × I n-i X, [s n-1 n-2,i-1 (f ),s n-1 n-2,i-1 (g)] n-2 n-1,j and I i-1 × {1} × I n-i X [t n-1 n-2,i-1 (f ),t n-1 n-2,i-1 (g)] n-2 n-1,j
then we obtain the desired extension :

I n S n-1 c X [f,g] n-1 n,j f,g n-1 n,j
Now consider two (n -1)-cells of COEND(I • ) :

I n-1 X, f g such that for 1 ≤ j ≤ n -1 we have f • s n-1 n-2,j = g • s n-1 n-2,j : I n-2 I n-1 X, s n-1 n-2,j f g that is s n-1 n-2,j (f ) = s n-1
n-2,j (g). We are going to build the contraction [f, g] n-1,- n,j by induction. Thus we suppose that for all 1 ≤ j ≤ n -2 the maps

I n-1 X [f,g] n-2,- n-1,j
exist, and we start our induction with an easy choice of extension [f, g] 1,- 2,1 , where f and g define here two paths in X :

I 2 X [f,g] 1,- 2,1
The map [f, g] n-1,- n,j is given by a continuous map I n X.

[f,g] n-1,- n,j

In order to do that, for all 1 ≤ j ≤ n -1, we need first to define the map : S n-1 c X.

f,g n-1,- n,j

This map f, g n-1,- n,j has the following definition :

• if i = j we put :

I j-1 × {0} × I n-j X, f and I j × {0} × I n-j-1 X, g and I j-1 × {1} × I n-j X, [t n-1 n-2,j (f ),t n-1 n-2,j (g)] n-2,- n-1,j
and I j × {1} × I n-j-1 X.

[t n-1 n-2,j (f ),t n-1 n-2,j (g)] n-2,- n-1,j

• If 1 ≤ i, j ≤ n then we put :

-if 1 ≤ i < j ≤ n -1 then I i-1 × {0} × I n-i X, [s n-1 n-2,i (f ),s n-1 n-2,i (g)] n-2,- n-1,j-1
and

I i-1 × {1} × I n-i X. [t n-1 n-2,i (f ),t n-1 n-2,i (g)] n-2,- n-1,j-1 -if 2 ≤ j + 1 < i ≤ n then I i-1 × {0} × I n-i X, [s n-1 n-2,i-1 (f ),s n-1 n-2,i-1 (g)] n-2,- n-1,j and I i-1 × {1} × I n-i X [t n-1 n-2,i-1 (f ),t n-1 n-2,i-1 (g)] n-2,- n-1,j
then we obtain the desired extension :

I n S n-1 c X [f,g] n-1,- n,j f,g n-1,- n,j
Now consider two (n -1)-cells of COEND(I • ) :

I n-1 X, f g
such that X is contractible, and such that for 1 ≤ j ≤ n -1 we have

f • t n-1 n-2,j = g • t n-1 n-2,j : I n-2 I n-1 X, t n-1 n-2,j f g that is t n-1 n-2,j (f ) = t n-1 n-2,j (
g). We are going to build the contraction [f, g] n-1,+ n,j by induction. Thus we suppose that for all 1 ≤ j ≤ n -2 the maps

I n-1 X [f,g] n-2,+ n-1,j
exist, and we start our induction with an easy choice of extension [f, g] 1,+ 2,1 , where f and g define here two paths in X :

I 2 X. [f,g] 1,+ 2,1
The map [f, g] n-1,- n,j is given by a continuous map I n X.

[f,g] n-1,+ n,j

In order to do that, for all 1 ≤ j ≤ n -1, we need first to define the map : S n-1 c X.

f,g n-1,+ n,j

This map f, g n-1,+ n,j has the following definition :

• if i = j we put :

I j-1 × {1} × I n-j X, f and I j × {1} × I n-j-1 X, g and I j-1 × {0} × I n-j X, [s n-1 n-2,j (f ),s n-1 n-2,j (g)] n-2,- n-1,j and I j × {0} × I n-j-1 X, [s n-1 n-2,j (f ),s n-1 n-2,j (g)] n-2,- n-1,j
• If 1 ≤ i, j ≤ n then we put :

-if 1 ≤ i < j ≤ n -1 then I i-1 × {0} × I n-i X, [s n-1 n-2,i (f ),s n-1 n-2,i (g)] n-2,+ n-1,j-1 and I i-1 × {1} × I n-i X. [t n-1 n-2,i (f ),t n-1 n-2,i (g)] n-2,+ n-1,j-1 -if 2 ≤ j + 1 < i ≤ n then I i-1 × {0} × I n-i X, [s n-1 n-2,i-1 (f ),s n-1 n-2,i-1 (g)] n-2,+ n-1,j and I i-1 × {1} × I n-i X. [t n-1 n-2,i-1 (f ),t n-1 n-2,i-1 (g)] n-2,+ n-1,j
Then we obtain the desired extension :

I n S n-1 c X [f,g] n-1,+ n,j f,g n-1,+ n,j
and it is then straightforward to check the different axioms of contractions for such extensions

[f, g] n-1 n,j , [f, g] n-1,- n,j and [f, g] n-1,+ n,j
. With 7.2.1 and 7.2.2 we thus have proved the : Also we have the functor :

Theorem 2 I • is a B 0 C -coalgebra.
(1 ↓ i • Cospan(-)) S-Oper COEND(-)
which sends the following morphism Map(-, X) of (1 ↓ i • Cospan(-)) :

1

Cospan(Top) Cospan(SET op ) I • Map(I • ,X) op

Cospan(Map(-,X))

to the morphism of operads :

COEND(I • ) COEND(Map(I • , X) op ) END(Map(I • , X)) COEND(Cospan(Map(-,X)))
this shows that Map(I • , X) is an algebra for COEND(I • ). But we proved in 7.2 that I • is also a B 0 C -coalgebra, which means that we have a morphism of operads : is equipped with a structure of weak cubical ∞-category. This weak cubical ∞-category Π ∞ (X) is in fact a weak cubical ∞-groupoid (see [START_REF] Kachour | Aspects of Cubical Higher Category Theory[END_REF] for the definition of cubical weak ∞-groupoids), called the fundamental cubical weak ∞-groupoid of X. Also if X Y f is a continuous map between X and Y , then from our functorial constructions we get the following commutative diagram :

• • • Map(I n , X) Map(I n-1 , X) • • • Map(I 4 , X) Map(I 3 , X) Map(I 2 , X) Map(I, X) Map(I 0 , X) s n n-1,1 t n n-1,1 s n n-1,i t n n-1,i s n n-1,n t n n-1,n s 4 3,1 t 4
END(Map(I • , X)) B 0 C COEND(I • ) END(Map(I • , Y )) END(Map(I • ,f )) ! COEND(Cospan(Map(-,X)))•! COEND(Cospan(Map(-,Y )))•!
which exhibits the fundamental cubical weak ∞-groupoid functor :

Top ∞-CGrp Π∞(-)
which has a left adjoint functor CN ∞ . This pair of adjunction (CN ∞ , Π ∞ (-)) should put an equivalence between the homotopy category of homotopy types and the homotopy category of ∞-CGrp of cubical weak ∞-groupoids with connections equipped with an adapted Quillen model structure. This was shown to be true but in the context of the Cisinski model structure on the category of cubical sets with connections (see [START_REF] Maltsiniotis | La catégorie cubique avec connections est une catégorie test stricte[END_REF]).

Pursuing stacks : [START_REF] Grothendieck | Pursuing Stacks (A la poursuite des Champs)[END_REF] Comparaison globulaire-cubique : [27]

Application for higher K-theory

The functor Π ∞ (-) could be intuitively thought as the gluing of all the homotopy groups functor π i together, and because the π i are cohomologies, Π ∞ (-) could be thought as a higher dimensional cohomology, that is a functor between ∞-categories, or an ∞-functor which behave like cohomologies. It seems that such objects are of interest for the Stolz-Teichner program 5 [START_REF] Stolz | Supersymmetric field theories and generalized cohomology[END_REF] who try to investigate ideas from physic (TQFT=Topological Quantum Field Theory) through cohomologies, and also ETQFT (Extended TQFT) through higher dimensional cohomologies and vice-versa.

In this section we explain how to "glue" algebraic K i -functors (i ∈ N) of Quillen :

Rings Top

Ki into a single functor K ∞ , where here Rings is the category of rings with unit.

5 These ideas take their roots in the work of Graham Segal on Conformal Field Theory.

But first let us recall some basic facts which are defined more accurately in [START_REF] Rosenberg | Algebraic K-theory and Its Applications[END_REF] : the functors K i are defined by the compostion :

Rings

Top Grp

BGL(-) + ×K0(-) πi(-)
where for any rings R with unit :

• GL(R) = ∞ n=1 GL(n,R)
• BGL(R) is the classifying space of the group GL(R)

• the +-construction on BGL(R) is taken relative to the perfect subgroup E(R) (elementary matrices) of GL(R)

• K 0 (R) is given the discrete topology Thus we get the functor K ∞ which is given by the composition :

Rings Top ∞-CGrp BGL(-) + ×K0(-) Π∞(-)
8 Importance of Coalgebraic structures for Globular and Cubical Higher Category Theory

The Batanin's construction and the author's construction

In the article [START_REF] Batanin | Monoidal globular categories as a natural environment for the theory of weak-n-categories[END_REF], Michael Batanin has built the contractible operad B 0 C which algebras are globular weak ∞-categories. He also proved that the globular object D • in Top consisting of topological disks :

D 0 D 1 D 2 • • • D n-1 D n • • • s 1 0 t 1 0 s 2 1 t 2 1 s n n-1 t n n-1
is a B 0 C -coalgebra, which implication is the construction of the fundamental globular weak ∞-groupoid functor

Top

∞-Grp

Π∞(-)

In the other hand, in the article [START_REF] Kachour | Steps toward the weak higher category of weak higher categories in the globular setting[END_REF] the author built a coglobular object B

• C B 0 C B 1 C B 2 C • • • B n-1 C B n C • • • s 1 0 t 1 0 s 2 1 t 2 1 s n n-1 t n n-1
such that B 0 C is the contractible operad just above of Michael Batanin, B 1 C is the contractible operad which algebras are globular weak ∞-functors, B 2 C is the contractible operad which algebras are globular weak ∞-natural transformations, etc. Also we have the surprising fact : if B • C is a B 0 C -coalgebra then it implies that the globular weak ∞-category of globular weak ∞-categories exists. We didn't prove yet this fact6 , however this is an important improvement for globular higher category theory for two main reasons :

• in the beginning it was non-trivial to know why globular weak ∞-categories, globular weak ∞-functors, globular weak ∞-natural transformations, etc. organize in a globular weak ∞-category. Now we have replaced this very complex combinatorial question by a precise statement : the coendomorphism operad COEND(B • C ) should be contractible7 , like its topological little son8 COEND(D • ).

• it brings a spectacular analogy between topological spaces and globular higher categories, which was hope by Grothendieck and Thomason. Let us gives a first smell of such analogy :

-Consider the following 1-cell in the operad COEND(D • ) of topological disks : such that B 0 C is the S 0 -operad which algebras are cubical weak ∞-categories. Also the S 1 -operad B 1 C which algebras are cubical weak ∞-functors, the S 2 -operad B 2 C which algebras are cubical weak ∞-natural transformations, etc. where S 1 is the cartesian monad which algebras are cubical strict ∞-functors, S 2 is the cartesian monad which algebras are cubical strict ∞-natural transformations, etc. are not difficult to be built. For example the underlying combinatorics of the S 1 -collection of B 1 C comes easily from the monad of cubical weak ∞-functors as defined in [START_REF] Kachour | Aspects of Cubical Higher Category Theory[END_REF] and the underlying combinatorics of the S 2 -collection of B 2 C comes easily from the monad of cubical weak ∞-natural transformations as defined in [START_REF] Kachour | Aspects of Cubical Higher Category Theory[END_REF]. Also according to the cubical combinatorics it is straightforward to see that the cartesian monad S n of cubical strict n-transformations act on the category CSets 2 n , the cartesian product 2 n times in CAT of the category of cubical sets with itself. In order to build these contractible S n -operads B n C we have different technics to do it. We can use for example the formalism of the T-categorical stretchings as developed in [START_REF] Kachour | Aspects of Globular Higher Category Theory[END_REF], or we can use the theory of Garner [START_REF] Garner | Understanding the small object argument[END_REF] to build a fibrant replacement of the S n , or we can more classically just use the technology developed in [START_REF] Batanin | Monoidal globular categories as a natural environment for the theory of weak-n-categories[END_REF][START_REF] Kachour | Introduction to Higher Cubical Operads. First Part : The Cubical Operad of Cubical Weak ∞-Categories[END_REF]. This internal cocubical object B • of C is a global object of the pseudo-algebra Cospan(C), where here we deal with cubical higher cospans. Thanks to the functor defined in 6.2 .

D 1 D 1 D 0 D 1
B 0 C B 1 C B 2 C B 3 C B 4 C • • • B n-1 C B n C • • • s 0 1 t 0 1 s 2
This shows that Alg(B • C ) is an algebra for COEND(B • C ). Now suppose B • C is also a B 0 C -coalgebra. In fact we put the following conjecture :

Conjecture The operad of coendomorphism COEND(B • C ) is contractible. Contractibility here means the cubical one, as developed in [START_REF] Kachour | Introduction to Higher Cubical Operads. First Part : The Cubical Operad of Cubical Weak ∞-Categories[END_REF], where we consider contractions similar to their globular analogue, plus the "connections-contractions" which are for contractions what connections are for cubical ∞-categories.

If we accept this conjecture then it means that we have a morphism of operads : is equipped with a structure of weak cubical ∞-category. This is the cubical weak ∞-category of cubical weak ∞-categories. Like for globular higher category theory, we thus have an amazing analogy between topological spaces and cubical higher categories, up to these conjectures related to coalgebraicity. Thanks to it we can mimic the globular approach of weak Grothendieck ∞-topos as described in [START_REF] Kachour | Globular perspective for Grothendieck ∞-topos and Grothendieck (∞, n)-topos[END_REF] to have a real smell of what is a cubical weak Grothendieck ∞-topos.

Cet article montre a quel point le materiel 2-categorique developper dans [START_REF] Street | The petit topos of globular sets[END_REF][START_REF] Street | The role of Michael Batanin's monoidal globular categories, Higher Category Theory[END_REF][START_REF] Weber | Yoneda structures from 2-toposes[END_REF][START_REF] Weber | Operads Within Monoidal Pseudo-algebras[END_REF] permet de generaliser la theorie de Batanin. And we can imagine other geometry for higher groupoids associated to topological spaces

,3 1 3 4, 4 Definition 2

 342 The category of semireflexive cubical sets C sr Sets is the category of presheaves [C sr ; Sets]. The terminal semireflexive cubical set is denoted 1 sr 2

2

  But the following forgetful functors are monadic :COper S-Coll p U CS-Coll p S-Coll p V

Definition 7 Definition 8

 78 The category CSets of cubical sets is the category of presheaves [C; Sets]. The terminal cubical set is denoted 1. The 2-category ∞-CCAT of cubical categories is the 2-category of prestacks [C; CAT]. The terminal cubical category is also denoted 1.

2

  Also we shall denote by S-Algs ∞-CCAT V Fthe underlying strict 2-adjunction of this biadjunction.

6. 1 where

 1 The pseudo-algebraic structure of Span(C) Let us first recall the Span construction ([44, 48]) : for any small category C there is a 2-adjunction : CAT [C op , CAT] Span C (E)(c) = [(C/c) op , E] and the category EL(X) has the following definition :

  structural map of the pseudo S-algebra C. It is important to notice that the freeness of S(1) describes this extension [E] as the composition v • S(E) :

  x; y) is given by the composition y • v(x) in Span n (C) :

2 2 C 2

 22 Also we have the following easy result :Corollary 2 A global object 1 Span(C) E is the same thing as to give a cubical object, still denoted by E, internal to the category C : C C E Now we are ready to define B 0 C -algebras : Definition 17 Consider a category C with pullbacks, plus a cubical object E in it : C C E . E is equipped with a B 0 C -algebra structure if there is a morphism of S-operads : B 0 Operads of coendomorphisms and coalgebraic structures are defined similarly and dually, but because of their importance we prefer to give their precise dual definition : if C is a category with pushouts, thus C = Cospan(C) is a cubical monoidal category, and if : 1 Cospan(C) E is a global object in it, then the corresponding extension Coend(E) to S(1) : S(1) Cospan(C) Coend(E)

  x; y) is given by the composition y • v(x) in Cospan n (C) :

Higher Cospans in Top 7 . 1

 71 The global object box : I • Here I = [0, 1] is the usual interval of R. Consider the following internal cocubical object in Top :

2 7. 2 . 3

 223 The fundamental morphism of operadsLet us fix a topological space X ∈ Top. From it we get a functor Top SET opMap(-,X)in CAT push , thus from the functor : CAT push Ps-S-AlgCospan(-)we get the following morphism in Ps-S-Alg :

!

  which shows that we have a morphism of operads :B 0 C END(Map(I • , X))COEND(Cospan(Map(-,X)))•!that is the cubical set Map(I • , X) :

µ 1 0- 1 0 1 0

 111 and consider a topological space X. With these we get the following 1-cell of the fundamental weak ∞-groupoid Π ∞ (X) : Suppose that COEND(B • C ) is contractible. It is then possible to consider the following 1-cell in the operad COEND(B • C ) of operadical disks :and with this 1-cell of COEND(B • C ), we get the following 1-cell in the suspected globular weak ∞-category of globular weak ∞-categories : which is the composition of globular weak ∞-functors !8.2 Steps toward the cubical weak ∞-category of cubical weak ∞-categoriesConsider the following internal cocubical object in a subcategory C 9 of the category Mnd of monads, such that C has pushouts.

( 1 ↓ 1 Cospan

 11 i • Cospan(Alg(-) of (1 ↓ i • Cospan(-)) : the morphism of operads :COEND(B • C ) COEND(Alg(B • C ) op ) END(Alg(B • C ))COEND(Cospan(Alg(-)))

We use this short notation, but the reader has to have in mind that it means in particular that the underlying cubical set of the operad B 0 C is the value of this monad on the terminal object 1 of CSets

In this diagram S is seen as a monad on the category of cubical sets. See[START_REF] Kachour | Introduction to Higher Cubical Operads. First Part : The Cubical Operad of Cubical Weak ∞-Categories[END_REF] for the definition of S-operads.

Because of lack of financial support, because the author ignore how to fill applications, how to interact with persons in order to be enough persuasive, etc.

which imply that it is equipped with a composition system. See[START_REF] Kachour | Steps toward the weak higher category of weak higher categories in the globular setting[END_REF]

or little brother ...

Such subcategory exists according to a private communication with Ross Street and John Bourke who give me such accurate construction. We won't describe it here because of lack of time.

For each n ∈ N, Alg(B n C )(0) means the class of objects of the category Alg(B n C ).