aa r X i v : . [ m a t h . K T ] M a y Aspects of Cubical Higher Category Theory
Camell KachourMay 29, 2017
Abstract
In this article we show how to build aspects of articles [24, 25, 34] but with the cubical geometry.Thus we define a monad on the category CS ets of cubical sets which algebras are models of cubical weak ∞ -categories. Also for each n ∈ N we define a monad on CS ets which algebras are models of cubical weak ( ∞ , n ) -categories. And finally we define a monad on the category CS ets which algebras are models ofcubical weak ∞ -functors, and a monad on the category CS ets which algebras are models of cubical weaknatural ∞ -transformations. Keywords. cubical weak ( ∞ , n ) -categories, cubical weak ∞ -groupoids, computersciences. Mathematics Subject Classification (2010).
Contents ∞ -categories 6 ∞ -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 ∞ -categories 10 ∞ -categories . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Computations for low dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 ( ∞ , m ) -sets 225 Cubical weak ( ∞ , m ) -categories, m ∈ N ( ∞ , m ) -categories, m ∈ N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 The category of cubical weak ( ∞ , m ) -categories, m ∈ N . . . . . . . . . . . . . . . . . . . . . . . 25 In this "v2 arxived version" we have changed only p = 0 by p = m + 1 just before the section 5.2. CS ets means the cartesian product CS ets × CS ets, and CS ets n means the n -fold cartesian product of CS ets with itself. The category of cubical weak ∞ -functors 26 (0 , ∞ ) -magmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2 The category of cubical (0 , ∞ ) -categorical stretchings . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∞ -natural transformations 29 ∞ -natural transformations . . . . . . . . . . . . . . . . . . . . . . . 297.2 The category of cubical (1 , ∞ ) -magmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.3 The category of cubical (1 , ∞ ) -categorical stretchings . . . . . . . . . . . . . . . . . . . . . . . . . 33 Introduction
In this article we explain how to build algebraic models of • cubical weak ∞ -categories (see 3) • cubical weak ( ∞ , m ) -categories (see 5.2) • cubical weak ∞ -functors (see 6) • cubical weak natural ∞ -transformations (see 7)In particular cubical weak ( ∞ , -categories known as cubical weak ∞ -groupoids are very important for usbecause other models of cubical weak ∞ -groupoids exist but are defined in an non-algebraic way [6, 7, 9, 10, 13,17, 37], but more by considering kind of cubical Kan complexes.As a matter of fact a very important feature of cubical higher category theory is their flexible possibilityto have models of higher structures build by mimic simplicial methods for presheaves on the classical category ∆ , to presheaves on the reflexive cubical category C r of cubical sets with connections (see 1.2), and in an otherhand to have also models of higher structures build by mimic algebraic methods of the globular setting (see[24, 25, 34]).For this last point it is important to notice that cubical strict ∞ -categories (see 2) are very close in natureto their globular analogue : first datas of it are given by countable family of sets ( C n ) n ∈ N , equipped both withkind of sources and targets, and partial operations, and two kinds of reflexions on each set C n , subject to axioms.See [5, 12, 38].Cubical sets have richer structure than globular sets, analogue to simplicial sets, and this richness allows totranslate many definitions of simplicial higher category to cubical higher category (see [1, 17, 36]). But as weshall see, cubical higher category theory has the algebraic flexibility of globular higher category theory, which isa feature we have difficulty to see for simplicial higher category theory. This important aspect of cubical highercategory theory push to see them as a bridge between simplicial higher category theory and globular highercategory theory.We believe that our models of cubical weak ∞ -groupoids should opens new perspective to the Grothendieckconjecture on homotopy types of spaces, which is stated in the globular setting, and is as follow : Conjecture (Grothendieck)
The category of (some) models of globular weak ∞ -groupoids is equipped witha Quillen model structure which is Quillen equivalent to the category of spaces equipped with its usual Quillenmodel structure, i.e those which weak equivalences are given by the homotopy groups, and which fibrations areSerre fibrations. Finally it is important to notice that cubical strict higher structures have already applications and impactsin homology [2, 8, 9, 10] and in algebraic topology [11, 21, 36]. The use of connections with simplicial methodcan be found in [1, 17, 36].This article is devoted to several work which main steps are as follow :2
We define our own terminology in order to be as close as possible to the notation of the globular environ-ment, and in it we define the monad of cubical strict ∞ -categories on the category of cubical sets • We define the category of cubical categorical stretchings which is the cubical anologue of the category ofglobular categorical stretchings of [34]. The key ingredient is a cubical analogue of the globular contractionsbuild in [34]. Then we give a monad on the category of cubical sets which algebras are our models ofcubical weak ∞ -categories. This monad is the cubical analogue of the monad in [34], which algebras arethe globular weak ∞ -categories of Penon. • We define cubical ( ∞ , n ) -sets, which main tools are the cubical reversors . These are the cubical analogueof the globular ( ∞ , n ) -sets which has been defined in [25]. More precisely they are build by using cubicalanalogue of minimal ( ∞ , n ) -structures in the sense of [25]. • We define the category of cubical reflexive ∞ -magmas and then the category of cubical ( ∞ , n ) -categoricalstretchings, which is the cubical analogue of the globular ( ∞ , n ) -categorical stretchings which has beendefined in [25]. This allow us to build a monad on the category of cubical sets which algebras are ourmodels of cubical weak ( ∞ , n ) -categories. This monad is the cubical analogue of the monad on globularsets build in [25] and which algebras are globular models of weak ( ∞ , n ) -categories. • In the sections 6 and 7 we extend globular weak ∞ -functors and globular weak natural ∞ -transformationsto the cubical setting. In particular we shall see that the monad of cubical weak ∞ -functors act on thecategory CS ets × CS ets and the monad of cubical weak natural ∞ -transformations act on the category CS ets × CS ets × CS ets × CS ets. In these last sections some interesting internal -cubes appears in ∞ - CC ATwhich actually are all easy cubical strict -categories. Acknowledgement.
I thank mathematicians of the IRIF, and the good ambience provided in the labduring my postdoctoral position in this team, especially I want to mention Paul-André Méllies, Mai Gehrke,Thomas Ehrhard, Pierre-Louis Curien, Yves Guiraud, Jordi Lopez-Abad, Maxime Lucas, Albert Burroni andFrançois Métayer. I dedicate this work to Ronnie Brown. See also [1, 22] for more references on cubical sets.
Consider the small category C with integers n ∈ N as objects. Generators for C are, for all n ∈ N given by sources n s nn − ,j / / n − for each j ∈ { , .., n } and targets n t nn − ,j / / n − for each j ∈ { , .., n } suchthat for ≤ i < j ≤ n we have the following cubical relations(i) s n − n − ,i ◦ s nn − ,j = s n − n − ,j − ◦ s nn − ,i , (ii) s n − n − ,i ◦ t nn − ,j = t n − n − ,j − ◦ s nn − ,i , (iii) t n − n − ,i ◦ s nn − ,j = s n − n − ,j − ◦ t nn − ,i , (iv) t n − n − ,i ◦ t nn − ,j = t n − n − ,j − ◦ t nn − ,i These generators plus these relations give the small category C called the cubical category that we mayrepresent schematically with the low dimensional diagram : Cubical sets in our terminology are the precubical sets of Richard Steiner Also called ( ∞ , n ) -graphs in [25] Three months have been financially supported (December 2016 until February 2017) under an European Research CouncilProject called
Duall : . · · C C C C C s , t , s , t , s , t , s , t , s , t , s , t , s , t , s , t , s , t , s t and this category C gives also the sketch E S of cubical sets used especially in 2.2, 3 and 5.2 to produce themonads S = ( S, λ, µ ) , W = ( W, η, ν ) and W m = ( W m , η m , ν m ) on CS ets, which algebras are respectively cubicalstrict ∞ -categories, cubical weak ∞ -categories and cubical weak ( ∞ , m ) -categories. Definition 1
The category of cubical sets CS ets is the category of presheaves [ C ; S ets ] . The terminal cubicalset is denoted . ✷ Occasionally a cubical set shall be denoted with the notation C = ( C n , s nn − ,j , t nn − ,j ) ≤ j ≤ n, n ∈ N in case we want to point out its underlying structures. Reflexivity for cubical sets are of two sorts : one is "classical" in the sense that they are very similar to their globu-lar analogue; thus we shall use the notation (1 nn +1 ,j ) n ∈ N ,j ∈{ ,..,n } to denote these maps C ( n ) nn +1 ,j / / C ( n + 1) which formally behave like globular reflexivity ([25]); 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 nota-tion. However we do prefer to use instead the notation C ( n ) n,γn +1 ,j / / C ( n + 1) ( γ ∈ { + , −} ) in order to pointout the reflexive nature of connections.Consider the cubical category C . For all n ∈ N we add in it generators n − n − n,j / / n for each j ∈ { , .., n } subject to the relations :(i) nn +1 ,i ◦ n − n,j = 1 nn +1 ,j +1 ◦ n − n,i if ≤ i ≤ j ≤ n ;(ii) s nn − ,i ◦ n − n,j = 1 n − n − ,j − ◦ s n − n − ,i if ≤ i < j ≤ n ;(iii) s nn − ,i ◦ n − n,j = 1 n − n − ,j ◦ s n − n − ,i − if ≤ j < i ≤ n ;(iv) s nn − ,i ◦ n − n,j = id ( n − if i = j .(i) nn +1 ,i ◦ n − n,j = 1 nn +1 ,j +1 ◦ n − n,i if ≤ i ≤ j ≤ n ;(ii) t nn − ,i ◦ n − n,j = n − n − ,j − ◦ t n − n − ,i if ≤ i < j ≤ n ;(iii) t nn − ,i ◦ n − n,j = 1 n − n − ,j ◦ t n − n − ,i − if ≤ j < i ≤ n ;(iv) t nn − ,i ◦ n − n,j = id ( n − if i = j .These generators and relations give the small category C sr called the semireflexive cubical category where aquick look at its underlying semireflexive structure is given by the following diagram :4 C C C C · · · , , , , , , , , , Definition 2
The category of semireflexive cubical sets C sr S ets is the category of presheaves [ C sr ; S ets ] . Theterminal semireflexive cubical set is denoted sr ✷ Consider the semireflexive cubical category C sr . For all integers n ≥ we add in it generators n − n − ,γn,j / / n for each j ∈ { , .., n − } subject to the relations :(i) for ≤ j < i ≤ n , n,γn +1 ,i ◦ n − ,γn,j = 1 n,γn +1 ,j +1 ◦ n − ,γn,i ;(ii) for ≤ i ≤ n − , n,γn +1 ,i ◦ n − ,γn,i = 1 n,γn +1 ,i +1 ◦ n − ,γn,i ;(iii) for ≤ i, j ≤ n , ( n,γn +1 ,i ◦ n − n,j = 1 nn +1 ,j +1 ◦ n − ,γn,i if ≤ i < j ≤ n = 1 nn +1 ,j ◦ n − ,γn,i − if ≤ j < i ≤ n ;(iv) for ≤ j ≤ n , n,γn +1 ,j ◦ n − n,j = 1 nn +1 ,j ◦ n − n,j ;(v) for ≤ i, j ≤ n , ( s nn − ,i ◦ n − ,γn,j = 1 n − ,γn − ,j − ◦ s n − n − ,i if ≤ i < j ≤ n −
1= 1 n − ,γn − ,j ◦ s n − n − ,i − if ≤ j + 1 < i ≤ n ;and ( t nn − ,i ◦ n − ,γn,j = 1 n − ,γn − ,j − ◦ t n − n − ,i if ≤ i < j ≤ n −
1= 1 n − ,γn − ,j ◦ t n − n − ,i − if ≤ j + 1 < i ≤ n ;(vi) for ≤ j ≤ n − , s nn − ,j ◦ n − , − n,j = s nn − ,j +1 ◦ n − , − n,j = 1 n − and t nn − ,j ◦ n − , + n,j = t nn − ,j +1 ◦ n − , + n,j =1 n − ;(vii) for ≤ j ≤ n − , s nn − ,j ◦ n − , + n,j = s nn − ,j +1 ◦ n − , + n,j = 1 n − n − ,j ◦ s n − n − ,j ;(viii) for ≤ j ≤ n − , t nn − ,j ◦ n − , − n,j = t nn − ,j +1 ◦ n − , − n,j = 1 n − n − ,j ◦ t n − n − ,j .These generators and relations give the small category C r called the reflexive cubical category and in it,connections have the following shape : C C C C C · · · , − , , +2 , , − , , +3 , , − , , +3 , , − , , +4 , , − , , +4 , , − , , +4 , , − , , +5 , , − , , +5 , , − , , +5 , , − , , +5 , Definition 3
The category of reflexive cubical sets C r S ets is the category of presheaves [ C r ; S ets ] . The terminalreflexive cubical set is denoted r ✷ The category of strict cubical ∞ -categories Cubical strict ∞ -categories have been studied in [5, 12, 38].In [5] the authors proved that the category of cubical strict ∞ -categories with cubical strict ∞ -functorsas morphisms is equivalent to the category of globular strict ∞ -categories with globular strict ∞ -functors asmorphisms. Consider a cubical reflexive set ( C, (1 nn +1 ,j ) n ∈ N ,j ∈ J ,n +1 K , (1 n,γn +1 ,j ) n ≥ ,j ∈ J ,n K ) equipped with partial operations ( ◦ nj ) n ≥ ,j ∈ J ,n K where if a, b ∈ C ( n ) then a ◦ nj b is defined for j ∈ { , ..., n } if s nj ( b ) = t nj ( a ) . We also require these operations to follow the following axioms of positions :(i) For ≤ j ≤ n we have : s nn − ,j ( a ◦ nj b ) = s nn − ,j ( a ) and t nn − ,j ( a ◦ nj b ) = t nn − ,j ( a ) ,(ii) s nn − ,i ( a ◦ nj b ) = (cid:26) s nn − ,i ( a ) ◦ n − j − s nn − ,i ( b ) if ≤ i < j ≤ ns nn − ,i ( a ) ◦ n − j s nn − ,i ( b ) if ≤ j < i ≤ n (iii) t nn − ,i ( a ◦ nj b ) = (cid:26) t nn − ,i ( a ) ◦ n − j − t nn − ,i ( b ) if ≤ i < j ≤ nt nn − ,i ( a ) ◦ n − j t nn − ,i ( b ) if ≤ 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 themonad on CS ets 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 otherdiagrams and cones. • For ≤ i < j ≤ n we consider the following two cones : M n × M n − ,j M n M n M n M n − π n ,j π n ,j s nn − ,j t nn − ,j M n − × M n − ,j − M n − M n − M n − M n − π n − ,j − π n − ,j − s n − n − ,j − t n − n − ,j − and the following commutative diagram (definition of s nn − ,i × j,j − s nn − ,i ) M n × M n − ,j M n M n M n − × M n − ,j − M n − M n − M n M n − M n − π n ,j π n ,j s nn − ,i × j,j − s nn − ,i s nn − ,i π n − ,j − π n − ,j − s n − n − ,j − s nn − ,i t n − n − ,j − which gives the following commutative diagram M n × M n − ,j M n M n − × M n − ,j − M n − M n M n − ⋆ nj s nn − ,i × j,j − s nn − ,i ⋆ n − j − s nn − ,i • For ≤ j < i ≤ n we consider the following two cones :6 n × M n − ,j M n M n M n M n − π n ,j π n ,j s nn − ,j t nn − ,j M n − × M n − ,j M n − M n − M n − M n − π n − ,j π n − ,j s n − n − ,j t n − n − ,j and the following commutative diagram (definition of s nn − ,i × j,j s nn − ,i ) M n × M n − ,j M n M n M n − × M n − ,j M n − M n − M n M n − M n − π n ,j π n ,j s nn − ,i × j,j s nn − ,i s nn − ,i π n − ,j π n − ,j s n − n − ,j s nn − ,i t n − n − ,j The previous datas gives the following commutative diagram of axioms M n × M n − M n M n − × M n − M n − M n M n − ⋆ nj s nn − ,i × j,j s nn − ,i ⋆ n − j s nn − ,i and for ≤ j ≤ n we have the following commutative diagram of axioms M n × M n − M n M n M n M n − ⋆ nj π s nn − ,j s nn − ,j which actually complete the description of E M Definition 4
Cubical reflexive ∞ -magmas are cubical reflexive set equipped with partial operations like justabove which follow axioms of positions. A morphism between two cubical reflexive ∞ -magmas is a morphismof their underlying cubical reflexive sets. The category of cubical reflexive ∞ -magmas is noted ∞ - CM ag r Remark 1
Cubical ∞ -magmas are poorer structure : they are cubical sets equipped with partial operationslike above with these axioms of positions. A morphism between two cubical ∞ -magmas is a morphism of theirunderlying cubical sets.The category of cubical ∞ -magmas is noted ∞ - CM ag ✷ Strict cubical ∞ -categories are cubical reflexive ∞ -magmas such that partials operations are associative andalso we require the following axioms : 7i) The interchange laws : ( a ◦ ni b ) ◦ nj ( c ◦ ni d ) = ( a ◦ nj c ) ◦ ni ( b ◦ nj d ) whenever both sides are defined(ii) nn +1 ,i ( a ◦ nj b ) = 1 nn +1 ,i ( a ) ◦ n +1 j +1 nn +1 ,i ( b ) if ≤ i ≤ j ≤ n nn +1 ,i ( a ◦ nj b ) = 1 nn +1 ,i ( a ) ◦ n +1 j nn +1 ,i ( b ) if ≤ j < i ≤ n + 1 (iii) n,γn +1 ,i ( a ◦ nj b ) = 1 n,γn +1 ,i ( a ) ◦ n +1 j +1 n,γn +1 ,i ( b ) if ≤ i < j ≤ n n,γn +1 ,i ( a ◦ nj b ) = 1 n,γn +1 ,i ( a ) ◦ n +1 j n,γn +1 ,i ( b ) if ≤ j < i ≤ n (iv) First transport laws : for ≤ j ≤ n n, + n +1 ,j ( a ◦ nj b ) = (cid:20) n, + n +1 ,j ( a ) 1 nn +1 ,j ( a )1 nn +1 ,j +1 ( a ) 1 n, + n +1 ,j ( b ) (cid:21) (v) Second transport laws : for ≤ j ≤ n n, − n +1 ,j ( a ◦ nj b ) = (cid:20) n, − n +1 ,j ( a ) 1 nn +1 ,j +1 ( b )1 nn +1 ,j ( b ) 1 n, − n +1 ,j ( b ) (cid:21) (vi) for ≤ j ≤ n , n, + n +1 ,i ( x ) ◦ n +1 i n, − n +1 ,i ( x ) = 1 nn +1 ,i +1 ( x ) and n, + n +1 ,i ( x ) ◦ n +1 i +1 n, − n +1 ,i ( x ) = 1 nn +1 ,i ( x ) The category ∞ - CC AT of strict cubical ∞ -categories is the full subcategory of ∞ - CM ag r spanned by strictcubical ∞ -categories. A morphism in ∞ - CC AT is called a strict cubical ∞ -functor . We study it more specificallyin 6 with the perspective to weakened it and to obtain cubical model of weak ∞ -functors. ∞ -categories In this section we describe cubical strict ∞ -categories as algebras for a monad on CS ets. We hope it to be aspecific ingredient to compare globular strict ∞ -categories with cubical strict ∞ -categoriesConsider the forgetful functor : ∞ - CC AT U / / CS ets which associate to any strict cubical ∞ -category its underlying cubical set and which associate to any strict cubical ∞ -functor its underlying morphismof cubical sets. Proposition 1
The functor U is right adjoint ✷ Its left adjoint is denoted F Proof
The proof is very similar to those in [34] : Actually it is not difficult to see that the category ∞ - CC ATand the category CS ets are both projectively sketchable. Let us denote by E C the sketch of ∞ - CC AT and E S the sketch of CS ets. Main parts of E C are described just below and we see that E C contains E S , and that thisinclusion induces a forgetful functor ∞ - CC AT U / / CS ets which has a left adjunction thanks to thesheafification theorem of Foltz [20]. Now we have the commutative diagram M od ( E C ) / / iso (cid:15) (cid:15) M od ( E S ) iso (cid:15) (cid:15) ∞ - CC AT U / / CS etswhich 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 sketchbehind the interchange laws, which shall complete main parts of E C : • In the first generation of E C we start with three cones :8 n × Z n − ,i Z n Z n Z n Z n − ρ n ,i ρ n ,i s nn − ,i t nn − ,i Z n × Z n − ,j Z n Z n Z n Z n − ρ n ,j ρ n ,j s nn − ,j t nn − ,j E ijn Z n Z n Z n Z n Z n − Z n − Z n − π n π n π n π n t nn − ,i s nn − ,i s nn − ,j t nn − ,j t nn − ,i s nn − ,i • Then we consider the following commutative diagrams : E ijn Z n × Z n − ,i Z n Z n Z n Z nπ n p n π n ρ n ,i ρ n ,i s nn − ,i t nn − ,i E ijn Z n × Z n − ,i Z n Z n Z n Z nπ n p n π n ρ n ,i ρ n ,i s nn − ,i t nn − ,i E ijn Z n × Z n − ,j Z n Z n Z n Z nπ n p n π n ρ n ,j ρ n ,j s nn − ,j t nn − ,j E ijn Z n × Z n − ,j Z n Z n Z n Z nπ n p n π n ρ n ,j ρ n ,j s nn − ,j t nn − ,j We consider then (still in the first generation) the following two commutative diagrams : E ijn Z n × Z n − ,i Z n Z n × Z n − ,i Z n Z n × Z n − ,j Z n Z n Z n Z n − p n c n p n ⋆ nn − ,i ⋆ nn − ,i ρ n ,j ρ n ,j s nn − ,j t nn − ,j E ijn Z n × Z n − ,j Z n Z n × Z n − ,j Z n Z n × Z n − ,i Z n Z n Z n Z n − p n c n p n ⋆ nn − ,j ⋆ nn − ,j ρ n ,i ρ n ,i s nn − ,i t nn − ,i • Finally we consider the following commutative diagram of interchange laws E ijn Z n × Z n − Z n Z n × Z n − Z n Z nc n c n ⋆ nn − ,j ⋆ nn − ,i (cid:4) The monad of strict cubical ∞ -categories on cubical sets is denoted S = ( S, λ, µ ) . Here λ is the unit map of S : CS ets λ / / S and µ is the multiplication of S : S µ / / S ∞ -categories We defined the category ∞ - CM ag r of cubical reflexive ∞ -magmas in 2. With objects of this category pluscubical strict ∞ -categories, we are going to define the category ∞ - CE tC of cubical categorical stretchings. Thiscategory is the key to weakened cubical strict ∞ -categories as it was done in [34] for the globular setting. Ourcubical weak ∞ -categories are algebraic in the sense that they are algebras (5) for a monad on CS ets which isbuild by using the category of cubical categorical stretchings. Our way to build the category ∞ - CM ag r allowto weakened the whole structure of cubical strict ∞ -categories. As we shall see, the central notion of cubicalcontractions (see below) are more subtle than globular contractions of [34] : in particular they must be thoughtwith an inductive definition on the dimension n of the n -cells ( n ∈ N ).The category ∞ - CE tC of cubical categorical stretchings has as objects quintuples E = ( M, C, π, ([ − ; − ] nn +1 ,j ) n ∈ N ; j ∈{ ,...,n +1 } , ([ − ; − ] n,γn +1 ,j ) n ≥ j ∈{ ,...,n } ; γ ∈{− , + } ) where M is a cubical ∞ -magma, C is a cubical strict ∞ -category, π is a morphism in ∞ - CM ag r M π / / C ([ − ; − ] nn +1 ,j ) n ∈ N ; j ∈{ ,...,n +1 } , ([ − ; − ] n,γn +1 ,j ) n ≥ j ∈{ ,...,n } ; γ ∈{− , + } are extra structures called the cubical brack-eting structures , and which are the cubical analogue of the key structure of the Penon approach to weakeningthe axioms of strict ∞ -categories; it is for us the key structures which are going to weakening the axioms ofcubical strict ∞ -categories. Let be more precise about it :For n ≥ and for all integer k ≥ , consider the following subsets of M n × M n • M n = { ( α, β ) ∈ M n × M n : π n ( α ) = π n ( β ) }• M sn,j = { ( α, β ) ∈ M n × M n : s nn − ,j ( α ) = s nn − ,j ( β ) and π n ( α ) = π n ( β ) }• M tn,j = { ( α, β ) ∈ M n × M n : t nn − ,j ( α ) = t nn − ,j ( β ) and π n ( α ) = π n ( β ) } and also we consider M = { ( α, β ) ∈ M × M : α = β } Thus these extra structures are given by maps ([ − ; − ] nn +1 ,j : M n / / M n +1 ) n ∈ N ; j ∈{ ,...,n +1 } such that • If ≤ i < j ≤ n + 1 , then s n +1 n,i ([ α, β ] nn +1 ,j ) = [ s nn − ,i ( α ) , s nn − ,i ( β )] n − n,j − , and t n +1 n,i ([ α, β ] nn +1 ,j ) = [ t nn − ,i ( α ) , t nn − ,i ( β )] n − n,j − • If ≤ j < i ≤ n + 1 then s n +1 n,i ([ α, β ] nn +1 ,j ) = [ s nn − ,i − ( α ) , s nn − ,i − ( β )] n − n,j , and t n +1 n,i ([ α, β ] nn +1 ,j ) = [ t nn − ,i − ( α ) , t nn − ,i − ( β )] n − n,j • If i = j then s n +1 n,i ([ α, β ] nn +1 ,j ) = α and t n +1 n,i ([ α, β ] nn +1 ,j ) = β • π n +1 ([ α, β ] nn +1 ,j ) = 1 nn +1 ,j ( π n ( α )) = 1 nn +1 ,j ( π n ( β )) , • ∀ α ∈ M n , [ α, α ] nn +1 ,j = 1 nn +1 ,j ( α ) .and also are given by maps ([ − ; − ] n, − n +1 ,j : M sn,j / / M n +1 ) n ≥ j ∈{ ,...,n } and ([ − ; − ] n, + n +1 ,j : M + n,j / / M n +1 ) n ≥ j ∈{ ,...,n } such that • for ≤ 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 nn − ,j ( α ); s nn − ,j ( β )] n − 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 nn − ,j ( α ); t nn − ,j ( β )] n − n,j • for ≤ i, j ≤ n + 1 – s n +1 n,i ([ α ; β ] n,γn +1 ,j ) = ( [ s nn − ,i ( α ); s nn − ,i ( β )] n − ,γn,j − if ≤ i < j ≤ n [ s nn − ,i − ( α ); s nn − ,i − ( β )] n − ,γn,j if ≤ j + 1 < i ≤ n + 1 – t n +1 n,i ([ α ; β ] n,γn +1 ,j ) = ( [ t nn − ,i ( α ); t nn − ,i ( β )] n − ,γn,j − if ≤ i < j ≤ n [ t nn − ,i − ( α ); t nn − ,i − ( β )] n − ,γn,j if ≤ j + 1 < i ≤ n + 1 • π n +1 ([ α ; β ] n,γn +1 ,j ) = 1 n,γn +1 ,j ( π n ( α )) = 1 n,γn +1 ,j ( π n ( β )) • ∀ α ∈ M n , [ α, α ] n,γn +1 ,j = 1 n,γn +1 ,j ( α ) . 11 morphism of cubical categorical stretchings E ( m,c ) / / E ′ is given by the following commutative square in ∞ - CM ag r , M π (cid:15) (cid:15) m / / M ′ π ′ (cid:15) (cid:15) C c / / C ′ such that for all n ∈ N , and for all ( α, β ) ∈ f M n , m n +1 ([ α, β ] nn +1 ,j ) = [ m n ( α ) , m n ( β )] nn +1 ,j ( j ∈ { , ..., n + 1 } ) and m n +1 ([ α, β ] n,γn +1 ,j ) = [ m n ( α ) , m n ( β )] n,γn +1 ,j ( j ∈ { , ..., n } , γ ∈ { , ..., n } ) The category of cubical categorical stretchings is denoted ∞ - CE tCNow consider the forgetful functor: ∞ - CE tC U / / CS ets given by : ( M, C, π, ([ − ; − ] nn +1 ,j ) n ∈ N ; j ∈{ ,...,n } , ([ − ; − ] n,γn +1 ,j ) n ∈ N ; j ∈{ ,...,n } ; γ ∈{− , + } ) ✤ / / M ,
Proposition 2
The functor U just above has a left adjoint which produces a monad W = ( W, η, ν ) on thecategory of cubical sets. ✷ Proof
The proof is very similar to those in [24, 34] : Actually it is not difficult to see that the category ∞ - CE tCand the category CS ets are both projectively sketchable. The sketch of cubical sets is denoted by E S (see 1.1) andthe sketch of the cubical categorical stretchings is denoted by E E . Main parts of this sketch is described just below,and we see that E E contains E S , and is such that it induces a forgetful functor ∞ - CE tC U / / CS etswhich has a left adjunction thanks to the sheafification theorem of Foltz [20]. Now we have the commutativediagram M od ( E E ) / / iso (cid:15) (cid:15) M od ( E S ) iso (cid:15) (cid:15) ∞ - CE tC U / / CS etswhich shows that U is right adjoint.Actually in 2 we described the sketch E M of cubical ∞ -magmas, which where used to describe in 1 mainpart of the sketch E C of cubical strict ∞ -categories. Thus we have already some part of the sketch E E that wecomplete by sketching operations [ − ; − ] nn +1 ,j and [ − ; − ] n,γn +1 ,j plus their axioms. With previous descriptions ofsketches, and the one below, we shall see that we obtain the following inclusions of sketches : E S E M E C E E Description of E E • In the first generation we start with the following four cones : M n M n M n Z nπ n π n π n π n M sn,j M n M n Z nπ n,s π n,s s nn − ,j s nn − ,j tn,j M n M n Z nπ n,t π n,t t nn − ,j t nn − ,j M n × M n M n M np n p n • We consider the following commutative diagrams : M n M n × M n M n M nπ n π n j n p n p n M sn,j M n × M n M n M nπ n,s π n,s j n,s p n p n M tn,j M n × M n M n M nπ n,t π n,t j n,t p n p n We have the following commutative diagrams which define the ” s × s ” : • If ≤ i < j ≤ n + 1 – for sources M n M n M n − M n − M n M n − Z n − π n π n s nn − ,i × π s nn − ,i s nn − ,i π n − π n − π n − s nn − ,i π n − – for targets M n M n M n − M n − M n M n − Z n − π n π n t nn − ,i × π t nn − ,i t nn − ,i π n − π n − π n − t nn − ,i π n − If ≤ j < i ≤ n + 1 for sources M n M n M n − M n − M n M n − Z n − π n π n s nn − ,i − × π s nn − ,i − s nn − ,i − π n − π n − π n − s nn − ,i − π n − – for targets M n M n M n − M n − M n M n − Z n − π n π n t nn − ,i − × π t nn − ,i − t nn − ,i − π n − π n − π n − t nn − ,i − π n − In the following diagrams we have ρ ∈ { s, t }• If ≤ i < j ≤ n – for sources M ρn,j M n M ρn − ,j − M n − M n M n − M n − π n,ρ π n,ρ s nn − ,i × ρ s nn − ,i s nn − ,i π n − ,ρ π n − ,ρ s n − n − ,j − s nn − ,i s n − n − ,j − – for targets M ρn,j M n M ρn − ,j − M n − M n M n − M n − π n,ρ π n,ρ t nn − ,i × ρ t nn − ,i t nn − ,i π n − ,ρ π n − ,ρ t n − n − ,j − t nn − ,i t n − n − ,j − If ≤ j < i ≤ n + 1 – for sources 14 ρn,j M n M ρn − ,j M n − M n M n − M n − π n,γ π n,ρ s nn − ,i − × ρ s nn − ,i − s nn − ,i − π n − ,ρ π n − ,ρ s n − n − ,j s nn − ,i − s n − n − ,j – for targets M ρn,j M n M ρn − ,j M n − M n M n − M n − π n,ρ π n,ρ t nn − ,i − × ρ t nn − ,i − t nn − ,i − π n − ,ρ π n − ,ρ t n − n − ,j t nn − ,i − t n − n − ,j Second generation • We consider the two cones : M − n,j M n M sn,j M n × M nπ n, − π n, − j n j n,s M + n,j M n M tn,j M n × M nπ n, +1 π n, +0 j n j n,t • We denote k − = j n,s ◦ π n, − = j n ◦ π n, − et k + = j n,t ◦ π n, +0 = j n ◦ π n, +1 , and in this case we obtainthe following commutative diagrams : M − n,j M n × M n M n M nq n, − q n, − k − p n p n M + n,j M n × M n M n M nq n, +0 q n, +1 k + p n p n And we obtain the following commutative diagrams which give the definition of the s × s for sets M − n,j • If ≤ i < j ≤ n – for sources 15 − n,j M n M − n − ,j M n − M sn,j M sn − ,j M n − × M n − π n, − π n, − s nn − ,i × − s nn − ,i s nn − ,i × π s nn − ,i π n − , − π n − , − j n − s nn − ,i × s s nn − ,i j n − ,s – for targets M − n,j M n M − n − ,j M n − M sn,j M sn − ,j M n − × M n − π n, − π n, − t nn − ,i × − t nn − ,i t nn − ,i × π t nn − ,i π n − , − π n − , − j n − t nn − ,i × s t nn − ,i j n − ,s • If ≤ j < i ≤ n + 1 – for sources M − n,j M n M − n − ,j M n − M sn,j M sn − ,j M n − × M n − π n, − π n, − s nn − ,i − × − s nn − ,i − s nn − ,i − × π s nn − ,i − π n − , − π n − , − j n − s nn − ,i × s s nn − ,i j n − ,s – for targets M − n,j M n M − n − ,j M n − M sn,j M sn − ,j M n − × M n − π n, − π n, − t nn − ,i − × − t nn − ,i − t nn − ,i − × π t nn − ,i − π n − , − π n − , − j n − t nn − ,i × s t nn − ,i j n − ,s And we obtain the following commutative diagrams which give the definition of the s × s for sets M + n,j • If ≤ i < j ≤ n – for sources 16 + n,j M n M + n − ,j M n − M tn,j M tn − ,j M n − × M n − π n, +1 π n, +0 s nn − ,i × + s nn − ,i s nn − ,i × π s nn − ,i π n − , +0 π n − , +1 j n − s nn − ,i × t s nn − ,i j n − ,t – for targets M + n,j M n M + n − ,j M n − M tn,j M tn − ,j M n − × M n − π n, +1 π n, +0 t nn − ,i × + t nn − ,i t nn − ,i × π t nn − ,i π n − , +0 π n − , +1 j n − t nn − ,i × t t nn − ,i j n − ,t • If ≤ j < i ≤ n + 1 – for sources M + n,j M n M + n − ,j M n − M tn,j M tn − ,j M n − × M n − π n, +1 π n, +0 s nn − ,i − × + s nn − ,i − s nn − ,i − × π s nn − ,i − π n − , +0 π n − , +1 j n − s nn − ,i − × t s nn − ,i − j n − ,t – for targets M + n,j M n M + n − ,j M n − M tn,j M tn − ,j M n − × M n − π n, +1 π n, +0 t nn − ,i − × + t nn − ,i − t nn − ,i − × π t nn − ,i − π n − , +0 π n − , +1 j n − t nn − ,i − × t t nn − ,i − j n − ,t We consider the following commutative diagrams : • If ≤ i < j ≤ n + 1 n M n +1 M n − M ns nn − ,i × π s nn − ,i [ − ; − ] nn +1 ,j s nn − ,i [ − ; − ] n − n,j M n M n +1 M n − M nt nn − ,i × π t nn − ,i [ − ; − ] nn +1 ,j t nn − ,i [ − ; − ] n − n,j M γn,j M n +1 M n − M ns nn − ,i × γ s nn − ,i [ − ; − ] n,γn +1 ,j s nn − ,i [ − ; − ] n − ,γn,j M γn,j M n +1 M n − M nt nn − ,i × γ t nn − ,i [ − ; − ] n,γn +1 ,j t nn − ,i [ − ; − ] n − ,γn,j • If ≤ j < i ≤ n + 1 M n M n +1 M n − M ns nn − ,i − × π s nn − ,i − [ − ; − ] nn +1 ,j s nn − ,i [ − ; − ] n − n,j M n M n +1 M n − M nt nn − ,i − × π t nn − ,i − [ − ; − ] nn +1 ,j t nn − ,i [ − ; − ] n − n,j M γn,j M n +1 M γn − ,j M ns nn − ,i − × γ s nn − ,i − [ − ; − ] n,γn +1 ,j s nn − ,i [ − ; − ] n − ,γn,j M n M n +1 M γn − ,j M nt nn − ,i − × γ t nn − ,i − [ − ; − ] n,γn +1 ,j t nn − ,i [ − ; − ] n − ,γn,j • If i = j – for the operations [ − ; − ] nn +1 ,j M n M n +1 M nπ n [ − ; − ] nn +1 ,j s n +1 n,j M n M n +1 M nπ n [ − ; − ] nn +1 ,j t n +1 n,j – for the operations [ − ; − ] n,γn +1 ,j M γn,j M n +1 M nq n,γ [ − ; − ] n,γn +1 ,j s n +1 n,j M γn,j M n +1 M nq n,γ [ − ; − ] n,γn +1 ,j t n +1 n,j – Other possible diagram for this definition : M n M n +1 M n M nπ n π n [ − ; − ] nn +1 ,j s n +1 n,j t n +1 n,j γn,j M n +1 M n M nq n,γ q n,γ [ − ; − ] n,γn +1 ,j s n +1 n,j t n +1 n,j Commutative diagrams for axioms : • For operations [ − ; − ] nn +1 ,j M n M n +1 M n Z n Z n +1 π n [ − ; − ] nn +1 ,j π n +1 π n nn +1 M n M n +1 M n Z n Z n +1 π n [ − ; − ] nn +1 ,j π n +1 π n nn +1 • For operations [ − ; − ] n,γn +1 ,j M γn,j M n +1 M n Z n Z n +1 q n,γ [ − ; − ] n,γn +1 ,j π n +1 π n n,γn +1 M γn,j M n +1 M n Z n Z n +1 q n,γ [ − ; − ] n,γn +1 ,j π n +1 π n n,γn +1 Comeback to the first generationWe build the diagonal with the following commutative diagrams : M n M n M n M n Z nid δ n id π n π n π n π n n M sn,j M n M n M n − id δ ns id π n,s π n,s s nn − ,j s nn − ,j M n M tn,j M n M n M n − id δ nt id π n,t π n,t t nn − ,j t nn − ,j Comeback to the second generationThe previous diagrams generate the following commutative diagrams : M n M − n,j M n M sn,j M n × M nδ ns δ n − δ n π n, − π n, − j n j n,s M n M + n,j M n M tn,j M n × M nδ nt δ n + δ n π n, +1 π n, +0 j n j n,t Then we obtain the following commutative diagrams of first generation for axioms of reflexivity of the operations [ − ; − ] nn +1 ,j : M n M n +1 M n [ − ; − ] nn +1 ,j δ n nn +1 And the following commutative diagrams of second generation for axioms of reflexivity of the operations [ − ; − ] n,γn +1 ,j : M γn,j M n +1 M n [ − ; − ] n,γn +1 ,j δ nγ n,γn +1 (cid:4) Definition 5
Cubical weak ∞ -categories are algebras for the monad W above. ✷ Let us show with a simple example how cubical weak ∞ -categories provide a richer weakened structure thanthe one of globular weak ∞ -categories : for simplicity we show it inside an object E of ∞ - CE tC : E = ( M, C, π, ([ − ; − ] nn +1 ,j ) n ∈ N ; j ∈{ ,...,n +1 } , ([ − ; − ] n,γn +1 ,j ) n ∈ N ; j ∈{ ,...,n } ; γ ∈{− , + } ) Consider the following string in M (1) a b c d f g h and take the -cells x = ( h ◦ g ) ◦ f and y = h ◦ ( g ◦ f ) . Because these cells belong to M − , ∩ M +1 , we get thefollowing -cells 20 da d [ x, y ] , a ( h ◦ g ) ◦ f d h ◦ ( g ◦ f ) a ad d [ x, y ] , h ◦ g ) ◦ f a h ◦ ( g ◦ f )1 d a dd d [ x, y ] , − , h ◦ ( g ◦ f ) ( h ◦ g ) ◦ f d d a dd d [ x, y ] , +2 , a a h ◦ ( g ◦ f )( h ◦ g ) ◦ f Remark 2
We could have defined cubical categorical stretchings slightly differently than those just above bymeans of using just the operations ([ − ; − ] nn +1 ,j ) n ∈ N ; j ∈{ ,...,n } to weakened the structure of cubical strict ∞ -categories. Denote by ∞ - CE tC ′ the category of these slightly impoverished structures. We also have a forgetfulfunctor ∞ - CE tC ′ CS ets U ′ which is right adjoint and which produce an other monad W ′ = ( W ′ , η ′ , ν ′ ) which algebras could be also considered as enough good models of cubical weak ∞ -categories. Also we have anevident forgetful functor ∞ - CE tC ∞ - CE tC ′ U which is right adjoint and which produce a functor W - A lg W - A lg ′ U which shows that the models that we have chosen for our article are also modelsfor these impoverished structures. And our choice to add operations ([ − ; − ] n,γn +1 ,j ) n ∈ N ; j ∈{ ,...,n } ; γ ∈{− , + } to getour models of cubical weak ∞ -categories is similar to the one who choses cubical strict ∞ -categories withconnections instead of considering it without connections. We believe that our choice gives not only morerefined models than those of the category W - A lg ′ but also is in fact really necessary for a good approach ofcubical weak ∞ -categories, where formalism of connections are implicit and used in our weakened structures. ✷ Remark 3
In [5] the authors have proved that the category of cubical strict ∞ -categories is equivalent to thecategory of globular strict ∞ -categories. We suspect that such phenomena is still right in the world of weakmodels. Let us be more precise about what we are saying : denote by P the Penon’s monad on the categoryof globular sets (see [34]) which algebras are particularly nice models of globular weak ∞ -categories (see forexample [4, 14]). It is suspected (see [39]) that its category of algebras P - A lg can be equipped with a canonicalQuillen model structure similar to the one build in [33] for strict globular ∞ -categories, and we also suspectthat W - A lg can be equipped with such canonical Quillen model structure. Thus a weak version of the article [5]should be that the category P - A lg is Quillen equivalent to W - A lg when these categories are equipped with theircanonical Quillen model structure. ∞ -categories Consider a cubical weak ∞ -category W ( C ) C v . In this monadic presentation, W ( C ) has to be thoughtas the free cubical weak ∞ -category representing the underlying syntax with which all algebras with underlyingcubical set C are interpretations of it via their morphisms structural. For example here v is the morphismstructural which plays the role of interpreting in C the "syntax" W ( C ) , and thus put on C a structure of W -algebra. We shall distinguished well notations of operations inside W ( C ) and inside C in order to separatethe syntactic part from the model part of our algebras. For example the operations of compositions shall bedenoted ◦ nj in the models, whereas we shall use the notation ⋆ nj instead when we work in the free models. Thereflexions are denoted ι nn +1 ,j in the models and nn +1 ,j in the free models. The connections are denoted ι n,αn +1 ,j in the models and n,αn +1 ,j in the free models. Definitions of operations for models use those for free models andthe interpretative nature of W ( C ) C v emerges then with the axiomatic of algebras for monads : forexample we consider first the following definition of operations on C :(i) If a , b ∈ C ( n ) are such that s nj ( b ) = t nj ( a ) for j ∈ { , ..., n } then we put a ◦ nj b = v n ( η ( a ) ⋆ nj η ( b )) a ∈ C ( n ) is an n -cell then we put ι nn +1 ,j ( a ) = v n +1 (1 nn +1 ,j ( η ( a ))) , n ∈ N , j ∈ J , n + 1 K (iii) If a ∈ C ( n ) is an n -cell then we put ι n,γn +1 ,j ( a ) = v n +1 (1 n,γn +1 ,j ( η ( a ))) , n ≥ , j ∈ J , n K , γ ∈ {− , + } Thus v puts on C a cubical ∞ -magma structure and its interpretative nature is primarily expressed by thefact that it is a morphism of cubical ∞ -magmas between the free cubical ∞ -magma W ( C ) and this cubical ∞ -magma on C . It is the axioms of algebras which show us such important fact : actually we need to show that v ( a ⋆ nj b ) = v ( a ) ◦ nj v ( b ) , v (1 nn +1 ,j ( a )) = ι nn +1 ,j ( v ( a )) , v (1 n,γn +1 ,j ( a )) = ι n,γn +1 ,j ( v ( a )) . Let us show the first equality : v ( a ) ◦ nj v ( b ) = v ( η ( v ( a )) ⋆ nj η ( v ( b )))= v ( W ( v ) η W ( C ) ( a ) ⋆ nj W ( v ) η W ( C ) ( b ))= v ( W ( v )( η W ( C ) ( a ) ⋆ nj η W ( C ) ( b )))= v ( ν ( C )( η W ( C ) ( a ) ⋆ nj η W ( C ) ( b )))= v ( ν ( C )( η W ( C ) ( a )) ⋆ nj ν ( C )( η W ( C ) ( b )))= v ( a ⋆ nj b ) Other equalities are shown similarly. In [34] J.Penon called magmatic such properties of algebras. In particularthese shall be useful for concrete computations in any W -algebras. Definition 6
Consider a reflexive cubical set C ∈ C r S ets. It has dimension p ∈ N for reflexions if all its q -cells x ∈ C ( q ) for which q > p are of the form x = 1 q − q,j ( y ) and if there is at least one p -cell which is not of this form.It has dimension p ∈ N for connections if all its q -cells x ∈ C ( q ) for which q > p are of the form x = 1 q − ,γq,j ( y ) and if there is at least one p -cell which is not of this form. It has dimension p ∈ N , if it has dimension p ∈ N forreflexions and connections. ✷ Definition 7
Consider a W -algebra ( C, v ) . It has dimension p ∈ N for reflexion if its underlying reflexive setproduced by its underlying ∞ -magma structure (see 3.2) has dimension p ∈ N for reflexion. It has dimension p ∈ N for connection if its underlying reflexive set produced by its underlying ∞ -magma structure has dimension p ∈ N for connections. It has dimension p ∈ N , if it has dimension p ∈ N for reflexions and connections. ✷ Definition 8
A cubical bicategory is a -dimensional W -algebra. ✷ ( ∞ , m ) -sets Globular ( ∞ , m ) -sets have been defined in [25, 26] and represent main parts of the underlying sketchs forglobular models of ( ∞ , m ) -categories : it algebraically formalize the idea of inverses, inverses of inverses, etc.that is, thanks to these structures, ideas of inverses are encoded with operations and thus give very elegant andtractable algebraic models of ( ∞ , m ) -categories (see [25]). For example we obtain models of ( ∞ , m ) -categorieswhich are projectively sketchable, and this elegant categorical property is not clear for such models when it isbuild with simplicial methods. Also thanks to a result in [4] globular models that we obtain in [25] are algebrasfor the Batanin’s operad (see [3, 25]) and these models of globular weak ∞ -groupoids are probably very closeto those proposed by Grothendieck (see [3, 32]).Here we define cubical version of the formalism developed in [25]. This formalism of this cubical world isvery similar to its globular world analogue, however it is important to notice that other sketches are possible(see 5).Consider a cubical set C = ( C n , s nn − ,j , t nn − ,j ) ≤ j ≤ n . If k ≥ and ≤ j ≤ k , then a ( k, j ) -reversor on it isgiven by a map C k j kj / / C k such that the following two diagrams commute : Also called ( ∞ , m ) -graphs or ( ∞ , m ) -globular sets. k j kj / / s kk − ,j " " ❊❊❊❊❊❊❊❊ C kt kk − ,j | | ②②②②②②②② C k − C k j kj / / t kk − ,j " " ❊❊❊❊❊❊❊❊ C ms kk − ,j | | ②②②②②②②② C k − If for each k > m and for each ≤ j ≤ k , there are such ( k, j ) -reversor j kj on C , then we say that C is acubical ( ∞ , m ) -set. The family of maps ( j kj ) k>m, ≤ j ≤ k is called an ( ∞ , m ) -structure and in that case we shallsay that C is equipped with the ( ∞ , m ) -structure ( j kj ) k>m, ≤ j ≤ k . Seen as cubical ( ∞ , m ) -set we denote it by C = (( C n , s nn − ,j , t nn − ,j ) ≤ j ≤ n , ( j kj ) k>m, ≤ j ≤ k ) . If C ′ = (( C ′ n , s ′ nn − ,j , t ′ nn − ,j ) ≤ j ≤ n , ( j ′ kj ) k>m, ≤ j ≤ k ) is another ( ∞ , m ) -set, then a morphism of ( ∞ , m ) -sets C f / / C ′ is given by a morphism of cubical sets such that for each k > m and for each ≤ j ≤ k we have the followingcommutative diagrams C kf k (cid:15) (cid:15) j kj / / C kf k (cid:15) (cid:15) C ′ k j ′ kj / / C ′ k The category of cubical ( ∞ , m ) -sets is denoted ( ∞ , m ) - CS ets Remark 4
This structural approach of inverses is much more powerful that the simplicial methods becausewith it we are able to build any kind of reversible higher structure . For example in our framework it is a simpleexercise to build some exotic one which could be difficult to be build with simplicial method. For example
Remark 5
The ( ∞ , m ) -structures that we used to define cubical ( ∞ , m ) -sets have globular analogues (see [25])that we called the minimal ( ∞ , m ) -structures . The cubical analogue of the globular maximal ( ∞ , m ) -structures as defined in [25] is as follow : for all k > m and for each i m +1 , i m +2 , ..., i k such that ≤ i k ≤ k, ≤ i k − ≤ k − , ..., ≤ i m +2 ≤ m + 2 , ≤ i m +1 ≤ m + 1 , we have two diagrams in S ets each commuting serially : C k j kik / / s kk − ,ik (cid:15) (cid:15) t kk − ,ik (cid:15) (cid:15) C ks kk − ,ik (cid:15) (cid:15) t kk − ,ik (cid:15) (cid:15) C k − j k − ik − / / s k − k − ,ik − (cid:15) (cid:15) t k − k − ,ik − (cid:15) (cid:15) C k − s k − k − ,ik − (cid:15) (cid:15) t k − k − ,ik − (cid:15) (cid:15) C k − j k − ik − / / (cid:15) (cid:15) (cid:15) (cid:15) C k − (cid:15) (cid:15) (cid:15) (cid:15) C m +2 j m +2 im +2 / / s m +2 m +1 ,im +2 (cid:15) (cid:15) t m +2 m +1 ,im +2 (cid:15) (cid:15) C m +2 s m +2 m +1 ,im +2 (cid:15) (cid:15) t m +2 m +1 ,im +2 (cid:15) (cid:15) C m +1 j m +1 im +1 / / t m +1 m,im +1 $ $ ■■■■■■■■ C m +1 s m +1 m,im +1 z z ✉✉✉✉✉✉✉✉ C m C k j kik / / s kk − ,ik (cid:15) (cid:15) t kk − ,ik (cid:15) (cid:15) C ks kk − ,ik (cid:15) (cid:15) t kk − ,ik (cid:15) (cid:15) C k − j k − ik − / / s k − k − ,ik − (cid:15) (cid:15) t k − k − ,ik − (cid:15) (cid:15) C k − s k − k − ,ik − (cid:15) (cid:15) t k − k − ,ik − (cid:15) (cid:15) C k − j k − ik − / / (cid:15) (cid:15) (cid:15) (cid:15) C k − (cid:15) (cid:15) (cid:15) (cid:15) C m +2 j m +2 im +2 / / s m +2 m +1 ,im +2 (cid:15) (cid:15) t m +2 m +1 ,im +2 (cid:15) (cid:15) C m +2 s m +2 m +1 ,im +2 (cid:15) (cid:15) t m +2 m +1 ,im +2 (cid:15) (cid:15) C m +1 j m +1 im +1 / / s m +1 m,im +1 $ $ ■■■■■■■■ C m +1 t m +1 m,im +1 z z ✉✉✉✉✉✉✉✉ C m Such datas ( j ki k ) k>m, ≤ i k ≤ k is called a cubical maximal ( ∞ , n ) -structure .23ore generally, an ( ∞ , m ) -structure is given by the following datas : For each k > m , there exist p : m + 1 ≤ p < k and diagrams : C k j kik / / s kk − ,ik (cid:15) (cid:15) t kk − ,ik (cid:15) (cid:15) C ks kk − ,ik (cid:15) (cid:15) t kk − ,ik (cid:15) (cid:15) C k − j k − ik − / / s k − k − ,ik − (cid:15) (cid:15) t k − k − ,ik − (cid:15) (cid:15) C k − s k − k − ,ik − (cid:15) (cid:15) t k − k − ,ik − (cid:15) (cid:15) C k − j k − ik − / / (cid:15) (cid:15) (cid:15) (cid:15) C k − (cid:15) (cid:15) (cid:15) (cid:15) C p +2 j p +2 ip +2 / / s p +2 p +1 ,ip +2 (cid:15) (cid:15) t p +2 p +1 ,ip +2 (cid:15) (cid:15) C p +2 s p +2 p +1 ,ip +2 (cid:15) (cid:15) t p +2 p +1 ,ip +2 (cid:15) (cid:15) C p +1 j p +1 ip +1 / / t p +1 p,ip +1 ●●●●●●●● C p +1 s p +1 p,ip +1 { { ✇✇✇✇✇✇✇✇ C p C k j kik / / s kk − ,ik (cid:15) (cid:15) t kk − ,ik (cid:15) (cid:15) C ks kk − ,ik (cid:15) (cid:15) t kk − ,ik (cid:15) (cid:15) C k − j k − ik − / / s k − k − ,ik − (cid:15) (cid:15) t k − k − ,ik − (cid:15) (cid:15) C k − s k − k − ,ik − (cid:15) (cid:15) t k − k − ,ik − (cid:15) (cid:15) C k − j k − ik − / / (cid:15) (cid:15) (cid:15) (cid:15) C k − (cid:15) (cid:15) (cid:15) (cid:15) C p +2 j p +2 ip +2 / / s p +2 p +1 ,ip +2 (cid:15) (cid:15) t p +2 p +1 ,ip +2 (cid:15) (cid:15) C p +2 s p +2 p +1 ,ip +2 (cid:15) (cid:15) t p +2 p +1 ,ip +2 (cid:15) (cid:15) C p +1 j p +1 ip +1 / / s p +1 p,ip +1 ●●●●●●●● C p +1 t p +1 p,ip +1 { { ✇✇✇✇✇✇✇✇ C p with ≤ i j ≤ j for all j ∈ { p + 1 , ..., k } . These datas (cid:0) ( j li l ) k>m,m +1 ≤ p
A cubical strict ( ∞ , m ) -category C as above has a unique underlying ( ∞ , m ) -set. ✷ emark 6 Cubical strict ( ∞ , m ) -categories are richer that globular strict ( ∞ , m ) -categories. For example in[30] the author has shown that they are others and equivalent ways (see also [27]) to define inverses for strict ( ∞ , m ) -categories, and it seems then that we can imagine other ( ∞ , m ) -structures which could lead to otherapproach of algebraic models of cubical weak ( ∞ , m ) -categories as defined below 5.2. ✷ A cubical strict ∞ -functor preserve ( k, j ) -reversors. Thus morphisms between cubical strict ( ∞ , m ) -categoriesare just cubical strict ∞ -functors. The category of cubical strict ( ∞ , m ) -category is denoted ( ∞ , m ) - CC at. Asin 2.2 it is not difficult to show the following proposition Proposition 4
The evident forgetful functor ( ∞ , m ) - CC at U / / CS etsis right adjoint and monadic. ✷ The monad of cubical strict ( ∞ , m ) -category on cubical sets is denoted S m = ( S m , λ m , µ m ) . Here λ m is theunit map of S m : CS ets λ m / / S m and µ m is the multiplication of S m : ( S m ) µ / / S m ( ∞ , m ) -categories, m ∈ N A cubical reflexive ( ∞ , m ) -magma is an object of ∞ - CM ag r such that its underlying cubical set is equipped withan ( ∞ , m ) -structure. Morphisms between cubical reflexive ( ∞ , m ) -magmas are those of ∞ - CM ag r which arealso morphisms of ( ∞ , m ) - CS ets , i.e they preserve the underlying ( ∞ , m ) -structures. The category of cubicalreflexive ( ∞ , m ) -magmas is denoted ( ∞ , m ) - CM ag r .The category ( ∞ , m ) - CE tC of cubical ( ∞ , m ) -categorical stretchings has as objects quintuples E = ( M, C, π, ([ − ; − ] nn +1 ,j ) n ∈ N ; j ∈{ ,...,n } , ([ − ; − ] n,γn +1 ,j ) n ∈ N ; j ∈{ ,...,n } ; γ ∈{− , + } ) where M is a cubical reflexive ( ∞ , m ) -magma, C is a cubical strict ( ∞ , m ) -category, π is a morphism in ( ∞ , m ) - CM ag r M π / / C and ([ − ; − ] nn +1 ,j ) n ∈ N ; j ∈{ ,...,n } , ([ − ; − ] n,γn +1 ,j ) n ∈ N ; j ∈{ ,...,n } ; γ ∈{− , + } are the cubical bracketing structures whichhave already been defined in 3. A morphism of cubical ( ∞ , m ) -categorical stretchings E ( m,c ) / / E ′ is given by the following commutative square in ( ∞ , m ) - CM ag r , M π (cid:15) (cid:15) m / / M ′ π ′ (cid:15) (cid:15) C c / / C ′ such that for all n ∈ N , and for all ( α, β ) ∈ f M n , m n +1 ([ α, β ] nn +1 ,j ) = [ m n ( α ) , m n ( β )] nn +1 ,j ( j ∈ { , ..., n + 1 } ) and m n +1 ([ α, β ] n,γn +1 ,j ) = [ m n ( α ) , m n ( β )] n,γn +1 ,j ( j ∈ { , ..., n } , γ ∈ { , ..., n } ) The category of cubical ( ∞ , m ) -categorical stretchings is denoted ( ∞ , m ) - CE tC.Now consider the forgetful functor: ( ∞ , m ) - CE tC U / / CS etsgiven by ( M, C, π, ([ − ; − ] nn +1 ,j ) n ∈ N ; j ∈{ ,...,n } , ([ − ; − ] n,γn +1 ,j ) n ∈ N ; j ∈{ ,...,n } ; γ ∈{− , + } ) ✤ / / M This functor has a left adjoint which produces a monad W m = ( W m , η m , ν m ) on the category of cubical sets.25 efinition 9 Cubical weak ( ∞ , m ) -categories are algebras for the monad W m above. ✷ Thus the category of our models of cubical weak ( ∞ , m ) -categories is denoted W m - A lg.If it is evident to see that we have "an embedded" of ( ∞ , m ) - CC at in ∞ - CC at, and this also the case for theweak case, even it is more subtle : It comes from the forgetful functor ( ∞ , m ) - CE tC ∞ - CE tC U which forgets the underlying ( ∞ , m ) -structures. Also we have the following morphism of the category A dj ofadjunctions : ( ∞ , m ) - CE tC ∞ - CE tC CS ets CS ets V ⊣ U m ⊣ UidF F because U ◦ V = U m (see [24]), thus it produces a morphism V ∗ in the category M nd of monads ( CS ets , W ) ( CS ets , W m ) V ∗ and passing to algebras, gives the following functor A lg ( V ∗ ) which is the "embedded" we were looking for W m - A lg W - A lg A lg ( V ∗ ) Also each m > we have an evident forgetful functor ( ∞ , m − - CE tC ( ∞ , m ) - CE tC V m − and byusing the same technology as just above we obtain the following filtration in C AT W - A lg W - A lg W - A lg W m − - A lg W m - A lg · · · However it is important to notice that the filtered colimit of it in C AT doesn’t give W - A lg, even up toequivalence of categories. See [25] for a discussion of a similar phenomena for the globular approach. Definition 10
Cubical weak ( ∞ , -categories are our models of cubical weak ∞ -groupoids. In particular ourmodels are algebraic, in the sense that these cubical weak ∞ -groupoids are algebras for the monad W =( W , η , ν ) defined above on the category CS ets of cubical sets. ✷ Also the dimension of W m -algebras is defined as for W -algebras (see 3.3). We now propose a definition ofcubical bigroupoids. Definition 11
A cubical bigroupoid is a -dimensional W -algebra. ✷ ∞ -functors In [35] Jacques Penon had proposed algebraic models of globular weak ∞ -functors which were extended to allkind of globular weak higher transformations in [24]. The methods used in [24, 35] has consisted to use differentkind of stretchings used to weakened different kind of strict structure. For example in [35] he build a categoryof stretchings named in [24] the category of (0 , ∞ ) - categorical stretchings which were adapted to weakenedstrict ∞ -functors. And in [24] the author used the category of ( n, ∞ ) - categorical stretchings to weakened allkind of globular strict n -transformations for all n ≥ (strict natural ∞ -transformations correspond to n = 2 and strict ∞ -modifications correspond to n = 3 , etc.). As we are going to see, our models of cubical weak ∞ -functors are build with similar technology : we are going to define cubical functorial stretchings which contains (0 , ∞ ) -categorical stretchings must not be confused with ( ∞ , -categorical stretchings used in 5.2 to weakened cubical strict ( ∞ , -categories and which were used in [25] to weakened globular strict ( ∞ , -categories. of the structure behind cubical weak ∞ -functors. This structure produces a monad on thecategory CS ets × CS ets which algebras are our models of cubical weak ∞ -functors. In 7 we shall investigatesimilar constructions but for cubical weak natural ∞ -transformations.Cubical strict ∞ -functors have been defined in 2.1. A morphism between two cubical strict ∞ -functors C D F and C ′ D ′ F ′ is given by a commutative -cube in CC AT C C ′ D D ′ F c F ′ d The category of cubical strict ∞ -functors is denoted ∞ - CF unct (0 , ∞ ) -magmas A cubical (0 , ∞ ) -magma is given by a morphism M F M / / M of CS ets such that M and M areobjects of ∞ - CM ag r . Such object is denoted ( M , F M , M ) . A morphism between (0 , ∞ ) -magmas ( M , F M , M ) ( M ′ , F ′ M , M ′ ) m is given by two morphisms of ∞ - CM ag r : M M ′ m , M M ′ m , such that the followingdiagram commutes in CS ets M M M ′ M ′ m F M m F ′ M The category of cubical (0 , ∞ ) -magmas is denoted by (0 , ∞ ) - CM ag r (0 , ∞ ) -categorical stretchings A (0 , ∞ ) -stretching is given by a triple E = ( E , E , F M , F C ) such that E , E are cubical categorical stretchings given by E = ( M , C , π , ( [ − ; − ] nn +1 ,j ) n ∈ N ; j ∈{ ,...,n } , ( [ − ; − ] n,γn +1 ,j ) n ∈ N ; j ∈{ ,...,n } ; γ ∈{− , + } ) and E = ( M , C , π , ( [ − ; − ] nn +1 ,j ) n ∈ N ; j ∈{ ,...,n } , ( [ − ; − ] n,γn +1 ,j ) n ∈ N ; j ∈{ ,...,n } ; γ ∈{− , + } )( M , F M , M ) is an object of (0 , ∞ ) - CM ag r , C C F C is a strict cubical ∞ -functor, such that thefollowing square is commutative in CS ets : here we put these brackets, because these structural informations must be thought up to some weak equivalences, following theidea of models developed in any higher category theory M C C π F M π F C A morphism of (0 , ∞ ) -stretchings E = ( E , E , F M , F C ) E ′ = ( E ′ , E ′ , F ′ M , F ′ C ) is given by the following commutative diagram in CS ets : M ′ M ′ M M C ′ C ′ C C F ′ M π ′ π ′ m F M π m F ′ C c F C c π such that ( m , m ) is a morphism of (0 , ∞ ) - CM ag r , ( m , c ) and ( m , c ) are morphisms of ∞ - CE tC. Thecategory of (0 , ∞ ) -stretchings is denoted (0 , ∞ ) - CE tC.Now consider the forgetful functor: (0 , ∞ ) - CE tC U / / CS ets × CS etsgiven by E = ( E , E , F M , F C ) ✤ U / / ( M , M ) This functor has a left adjoint which produces a monad T = ( T , λ , µ ) on the category CS ets × CS ets. Definition 12
Cubical weak ∞ -functors are algebras for the monad T above. ✷ Thus a cubical weak ∞ -functor is given by a quadruple ( C , C , v , v ) such that if we note T ( C , C ) =( T ( C , C ) , T ( C , C ) then we get its underling morphisms of CS ets T ( C , C ) C v T ( C , C ) C v and these morphisms of CS ets put on ( C , C ) a structure of cubical weak ∞ -functor C C F defined by F = v ◦ F M ◦ λ ( C , C ) with 28 T ( C , C ) T ( C , C ) C C λ ( C ,C ) v F v v ◦ F M ◦ λ ∞ -natural transformations We finish this article on cubical higher category theory by building a monad on the category ( CS ets ) = CS ets × CS ets × CS ets × CS etswhich algebras are our models of cubical weak natural ∞ -transformations. In [24] we defined globular natural ∞ -transformations by using the structure given by an adapted category of stretchings, namely the category of(globular) (1 , ∞ ) -stretchings. Here we use similar technology by defining first the category of cubical (1 , ∞ ) - stretchings which contains the underlying structure needed to weakened cubical strict natural ∞ -transformations.In particular it leads to a monad on the category ( CS ets ) which algebras are our models of cubical weak natural ∞ -transformations. ∞ -natural transformations Cubical strict natural transformations were introduced in [23]. Here we give the evident strict and higher versionof it. A cubical strict ∞ -natural transformation is given by a -cube in ∞ - CC AT C , C , C , C , τ ⇓ H F GK which -cells corresponds to four cubical strict ∞ -categories C , , C , , C , , C , , which -cells correspondsto four cubical strict ∞ -functors F , G , H , K , and which the only -cell τ corresponds, for all -cells a in C , , toa -cell G ( F ( a )) K ( H ( a )) τ ( a ) such that for all -cells a b f of C , we have the followingcommutative diagram : G ( F ( a )) K ( H ( a )) G ( F ( b )) K ( H ( b )) G ( F ( f )) τ ( a ) K ( H ( f )) τ ( b ) A morphism between two cubical strict ∞ -natural transformations τ and τ ′ is given by a -cube in ∞ - CC AT29 ′ , C ′ , C , C , C ′ , C ′ , C , C , τ ′ ⇓ F ′ H ′ G ′ τ ⇓ c , FH c , K ′ c , K c , G such that c , F = F ′ c , , c , G = G ′ c , , c , H = H ′ c , and c , K = K ′ c , . The category of cubical strict ∞ -natural transformations is denoted (1 , ∞ ) - CT rans, and we obtain an internal -cube in C AT (1 , ∞ ) - CT rans ∞ - CF unct CC AT σ , σ , τ , τ , σ τ Proposition 5
The internal -cube of C AT just above can be structured in a strict cubical -category ✷ Proof
Consider the following object τ ∈ (1 , ∞ ) - CT rans : C , C , C , C , τ ⇓ H F GK such that σ , ( τ ) = F , σ , ( τ ) = H , τ , ( τ ) = K and τ , ( τ ) = G , and such that σ and τ are clearlydefined.Definition of the classical reflexivity : (1 , ∞ ) - CT rans ∞ - CF unct CC AT , , , ( F ) is given by C , C , C , C , , ( F ) ⇓ C , F C , F and is such that , ( F )( a ) = 1 ( F ( a )) for all -cells a ∈ C , (0) ,and also , ( F ) is given by 30 , C , C , C , , ( F ) ⇓ F C , F C , and is such that , ( F )( a ) = 1 ( F ( a )) for all -cells a ∈ C , (0) .Definition of the connections : (1 , ∞ ) - CT rans ∞ - CF unct , − , , +2 , , − , ( F ) is given by C , C , C , C , , − , ( F ) ⇓ F F C , C , and is such that , − , ( F )( a ) = 1 ( F ( a )) for all -cells a ∈ C , (0) ,and , +2 , ( F ) is given by C , C , C , C , , +2 , ( F ) ⇓ C , C , FF and is such that , +2 , ( F )( a ) = 1 ( F ( a )) for all -cells a ∈ C , (0) .The following shape of -cells C , C , C , C , C , C , τ ⇓ H F G ρ ⇓ H ′ K G ′ K ′ allows to define the composition ρ ◦ , τ , C , C , C , ρ ◦ , τ ⇓ H ′ ◦ H F G ′ ◦ GK by the formula : ( ρ ◦ , τ )( a ) = ρ ( H ( a )) ◦ G ′ ( τ ( a )) and the following shape of -cells C , C , C , C , C , C , τ ⇓ H F G ρ ⇓ F ′ G ′ K K ′ allows to define the composition ρ ◦ , τC , C , C , C , ρ ◦ , τ ⇓ H F ′ ◦ F GK ′ ◦ K by the formula : ( ρ ◦ , τ )( a ) = K ′ ( τ ( a )) ◦ ρ ( F ( a )) The proof that these datas put a structure of cubical strict -categories on the internal -cube of theproposition is left to the reader. (cid:4) (1 , ∞ ) -magmas A cubical (1 , ∞ ) -magma is an object with shape M , M , M , M , τ M ⇓ H M F M G M K M such that ( M , , F M , M , ) , ( M , , G M , M , ) , ( M , , H M , M , ) and ( M , , K M , M , ) are objects of (0 , ∞ ) - CM ag r ,and such that τ M is a map M , (0) M , (1) τ M which sends each -cells a of M , to an -cell τ M ( a ) ∈ M , (1) such that s ( τ M ( a )) = G M ( F M ( a )) and t ( τ M ( a )) = K M ( H M ( a )) . We want to avoid heavynotations and shall denote usually just by τ M such object of a category (1 , ∞ ) - CM ag r , where we have to thinkthis greek letter τ as the variable usually used for natural transformations and the subscript M in it just means"Magmatic". 32iven τ M and τ ′ M two objects of (1 , ∞ ) - CM ag r , a morphism between them is given by a commutative diagramin ∞ - CS ets M ′ , M ′ , M , M , M ′ , M ′ , M , M , τ ′ M ⇓ F ′ M H ′ M G ′ M τ M ⇓ m , F M H M m , K ′ M m , K M m , G M such that ( m , , m , ) , ( m , , m , ) , ( m , , m , ) , ( m , , m , ) are morphisms of (0 , ∞ ) - CM ag r . It is importantto note that commutativity of this diagram means also the equality m , ◦ τ M = τ ′ M ◦ m , .We obtain an internal -cube in C AT (1 , ∞ ) - CM ag r (0 , ∞ ) - CM ag r ∞ - CM ag r σ , σ , τ , τ , σ τ Proposition 6
The internal -cube of C AT just above can be structured in a cubical reflexive -magma ✷ Proof
The proof is easy and basic datas have been already defined in 5. (cid:4) (1 , ∞ ) -categorical stretchings A cubical (1 , ∞ ) -categorical stretching is given by a commutative diagram in ∞ - CS ets : M , M , M , M , C , C , C , C , K M π , π , τ M ⇒ H M F M π , G M K C τ C ⇒ H C F C G C π , such that ( π , , F M , F C , π , ) , ( π , , G M , G C , π , ) , ( π , , H M , H C , π , ) and ( π , , K M , K C , π , ) are objectsof (0 , ∞ ) - CE tC, and also τ M is an object of (1 , ∞ ) - CM ag r and τ C is an object of (1 , ∞ ) - CT rans. It is important tonote that commutativity of this diagram means also that the equality π , ◦ τ M = τ C ◦ π , holds. Such cubical (1 , ∞ ) -categorical stretching can be denoted ( τ M , τ C ) . Given an other cubical (1 , ∞ ) -categorical stretching ( τ ′ M , τ ′ C ) : 33 ′ , M ′ , M ′ , M ′ , C ′ , C ′ , C ′ , C ′ , K ′ M π ′ , π ′ , τ ′ M ⇒ H ′ M F ′ M π ′ , G ′ M K ′ C τ ′ C ⇒ H ′ C F ′ C G ′ C π ′ , a morphism ( τ M , τ C ) / / ( τ ′ M , τ ′ C ) of such cubical (1 , ∞ ) -categorical stretchings is given by • a morphism of (1 , ∞ ) - CM ag r underlied by ( m , , m , , m , , m , ) , a morphism of (1 , ∞ ) - CT rans underliedby ( c , , c , , c , , c , ) : M ′ , M ′ , M , M , M ′ , M ′ , M , M , τ ′ M ⇓ F ′ M H ′ M G ′ M τ M ⇓ m , F M H M m , K ′ M m , K M m , G M C ′ , C ′ , C , C , C ′ , C ′ , C , C , τ ′ C ⇓ F ′ C H ′ C G ′ C τ C ⇓ c , F C H C c , K ′ C c , K C c , G C • the following morphisms : (( m , , c , ) , ( m , , c , )) , (( m , , c , ) , ( m , , c , )) , (( m , , c , ) , ( m , , c , )) and (( m , , c , ) , ( m , , c , )) , of (0 , ∞ ) - CE tC M ′ , M ′ , M , M , C ′ , C ′ , C , C , F ′ M π ′ , π ′ , m , F M π , m , F ′ C c , F C c , π , M ′ , M ′ , M , M , C ′ , C ′ , C , C , H ′ M π ′ , π ′ , m , H M π , m , H ′ C c , H C c , π , ′ , M ′ , M , M , C ′ , C ′ , C , C , G ′ M π ′ , π ′ , m , G M π , m , G ′ C c , G C c , π , M ′ , M ′ , M , M , C ′ , C ′ , C , C , K ′ M π ′ , π ′ , m , K M π , m , K ′ C c , K C c , π , We denote (1 , ∞ ) - CE tC the category of cubical (1 , ∞ ) -categorical stretchings. Now we have a forgetfulfunctor : (1 , ∞ ) - CE tC U / / ( CS ets ) which sends the object ( τ M , τ C ) to the object ( M , , M , , M , , M , ) .This functor has a left adjoint which produces a monad T = ( T , λ , µ ) on the category ( CS ets ) . Definition 13
Cubical weak natural ∞ -transformations are algebras for the monad T above. ✷ Thus we obtain a -cube in the category A dj of pairs of adjunctions defined in [24] (1 , ∞ ) - CE tC (0 , ∞ ) - CE tC ∞ - CE tC ( CS ets ) ( CS ets ) CS ets σ , ⊣ U σ , τ , τ , ⊣ U σ τ ⊣ Uσ , F σ , τ , τ , F σ τ F which allow to obtain a -cocube in the category M nd of categories equipped with monads defined in [24] (( CS ets ) , T ) (( CS ets ) , T ) ( CS ets , W ) σ , σ , τ , τ , σ τ And finally it gives the following -cube in C AT T - A lg T - A lg W - A lg σ , σ , τ , τ , σ τ roposition 7 The internal -cube of C AT just above can be structured in a cubical weak -category ✷ Proof
Detail of the proof is quite long but is not difficult. For example basic datas of such structure aresimilar to those build in 5. (cid:4)
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