Notes on Multiple Higher Category Theory
Camell Kachour

TL;DR
This paper explores the application of the Stretchings method, originating from Globular Geometry, to develop algebraic models of weakened strict multiple higher categories and groupoids, advancing the understanding of weak higher categorical structures.
Contribution
It adapts the Stretchings method to weakened strict multiple $ abla$-categories and constructs algebraic models of weak multiple $ abla$-groupoids, extending higher category theory techniques.
Findings
Development of algebraic models for weak multiple $ abla$-categories
Extension of Stretchings method to multiple $ abla$-categories
Framework for modeling weak multiple $ abla$-groupoids
Abstract
These notes follows the articles \cite{kamel, Cam, cam-cubique} which show how powerful can be the method of \textit{Stretchings} initiated with the \textit{Globular Geometry} by Jacques Penon in \cite{penon} , to weakened \textit{strict higher structures}. Here we adapt this method to weakened strict multiple -categories, strict multiple -categories, and in particular we obtain algebraic models of weak multiple -groupoids.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology
Notes on Multiple Higher Category Theory
Camell Kachour
Abstract
These notes follows the articles [4, 5, 8] which show how powerful can be the method of Stretchings initiated with the Globular Geometry by Jacques Penon in [10] , to weakened strict higher structures. Here we adapt this method to weakened strict multiple -categories, strict multiple -categories, and in particular we obtain algebraic models of weak multiple -groupoids.
**Keywords. weak multiple -categories, weak multiple -groupoids, computer sciences.
Mathematics Subject Classification (2010). 18B40,18C15, 18C20, 18G55, 20L99, 55U35, 55P15. **
Contents
Introduction
Strict multiple categories had been introduced by Charles Ehresman in [1] in order to produce higher generalization of categories. Surprisingly the multiple geometry used in it to produce a theory of higher category has not been studied as much as it deserve. In these notes we hope filling this gap where in particular we use the technology of Stretchings to produce algebraic models of multiple higher category theory. More specifically we shall introduce :
- •
Algebraic models of weak multiple -categories in 3
- •
Algebraic models of weak multiple -categories in 5
In particular we propose algebraic models of weak multiple -groupoids which are our models of weak multiple -categories.
1 Multiple Sets
1.1 Multiple Sets
Fix an integer , each finite sequence such that is called an -color, and all -colors form a set denoted . Also for a fix -color , denotes its underlying set of -colors . If , the set of -colors of is well understood, it has -colors : , i.e each is a subsequence of such that . The set of -colors of has a particular importance : for and we use the notation which means that the -color is i.e we delete the -color from . In fact if is an -color and if is an integer such that the -sequence is an -color, then we write this last -color by . Thus if is an -color then the notation ”minus” means the -color and the notation ”add” means that doesn’t belongs to and that must be seen as its corresponding -color, and possibly it can be reindexed if necessary.
An -colored set means a set colored by an -color like just above. An -multiple data means a countable set of -colored sets , and a multiple data means a countable set of -multiple datas if and a set of objects .
A multiple set is given by a multiple data such that for all , and all -colored set in it, this -colored set is equipped for all with sources and targets :
[TABLE]
such that for any -color with the following diagrams commute :
[TABLE]
We shall often use the short notation to denote a multiple set with sources and targets , and no confusion should appear with the langage of colors. If is another multiple set, a morphism of multiple sets :
[TABLE]
is given for all -color by a map of
[TABLE]
which is compatible with all sources and all targets :
[TABLE]
The category of multiple sets is denoted
1.2 Reflexive Multiple Sets
A reflexive multiple set is given by a multiple set such that for all -color () and all -color the -colored set is equipped with reflexions
[TABLE]
For consider an -color and such that , then we require the following commutative diagrams :
[TABLE]
If is another reflexive multiple set, then a morphism of reflexive multiple sets :
[TABLE]
is given by a morphism in which is compatible the reflexivity’s structures, that is for all -color and all -color , the following diagram commutes :
[TABLE]
The category of reflexive multiple sets is denoted
The first important monad of this article is the monad of reflexive multiple sets given by the forgetful functor
[TABLE]
and which is in fact monadic. The proof that is right adjoint and monadic come from the underlying projective sketches of and , and by the evident applications of the theorem of Foltz in [2] and Lair in [9].
2 Strict multiple -categories
A multiple -magma is given by a multiple set such that for all and all its underlying -colored sets are equipped with operations
[TABLE]
where are given by the following pullbacks
[TABLE]
and such that these operations follow the following positional axioms
- •
and
- •
and if
A multiple -magma shall be denoted with the shorter notation when no confusion appears. If is another multiple -magma, a morphism of multiple -magmas
[TABLE]
is given by a morphism such which respects the operations , that is for all -color and all -color we have . The category of multiple -magmas is denoted .
A reflexive multiple -magma is given by a reflexive multiple set and a multiple -magma such that
[TABLE]
Morphisms of reflexive multiple -magmas are those of which are also morphisms of . The category of reflexive multiple -magmas is denoted .
A strict multiple -categories is given by a reflexive multiple -magma such that operations are associative, are unital i.e if for all we have
[TABLE]
and follow the middle-four interchange axiom
[TABLE]
when compositions in both side are well defined.
Morphisms of strict multiple -categories are those of . The category of strict multiple -categories is denoted .
The second important monad of this article is the monad of strict multiple -categories given by the forgetful functor
[TABLE]
and which is in fact monadic. The proof that is right adjoint and monadic come from the underlying projective sketches of and which are not difficult to be described. A similar sketch is described in [8]. Then these results come from an application of the theorem of Foltz in [2] and Lair in [9].
3 Weak multiple -categories
3.1 Multiple categorical stretchings
A multiple categorical stretching is given by the following datas :
- •
A reflexive multiple -magma
- •
A strict multiple -category
- •
A morphism of reflexive multiple -magmas
{\definecolor{pgfstrokecolor}{rgb}{0,0,0}M}$${\definecolor{pgfstrokecolor}{rgb}{0,0,0}C}$$\scriptstyle{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pi}
- •
Operations :
[TABLE]
where and such that :
- –
and
- –
and
- –
A morphism of multiple categorical stretchings
[TABLE]
is given by the following commutative square in ,
[TABLE]
thus we also have the following square in for all -color
[TABLE]
and we require for all the following equality
[TABLE]
The category of multiple categorical stretchings is denoted .
Now consider the forgetful functor :
[TABLE]
given by :
[TABLE]
Proposition 1
The functor just above has a left adjoint which produces a monad on the category of multiple sets. □
The proof that is right adjoint comes from the underlying projective sketches of and which are not difficult to be described. A similar sketch is described in [8]. Then these results come from an application of the theorem of Foltz in [2].
Definition 1
A weak multiple -category is a -algebra □
4 Multiple -Sets
Consider a multiple set , and an -color and a -color . A -reversor on it is given by a map
[TABLE]
such that the following two diagrams commute :
[TABLE]
If for all , and all -color , and each they are such -reversor on then we say that it is a multiple -set. The family of maps is called a multiple -structure and in that case we shall say that is equipped with the multiple -structure . Seen as multiple -set we denote it by , or just for a shorter notation, where design its underlying multiple -structure.
If is another multiple -set, then a morphism of multiple -sets
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is given by a morphism of multiple sets such that for each , each -color and each we have the following commutative diagrams
[TABLE]
The category of multiple -sets is denoted -
Remark 1
The -structures that we used to define multiple -sets have globular and cubical analogues (see [5, 8]) that we called the minimal -structures. The multiple analogue of the globular maximal -structures as defined in [5] and of cubical maximal -structures as defined in [8] is as follow : for all , for each -color and for all -color , there exist maps
[TABLE]
[TABLE]
such that we have the following diagrams in which commute serially :
[TABLE]
[TABLE]
Also, as in [5, 8] respectively for the globular geometry or for the cubical geometry, it is possible to have a general notion of multiple -structure : this notion gives all possibilities of inverse structure between the minimal -structure and the maximal -structure. Thus we define it as follows : a multiple set is equipped with an -structure if for all , all -color , and for all -color , there exist an integer with , there exist a -color , and there exist maps
[TABLE]
[TABLE]
such that we have the following diagrams in which commute serially :
[TABLE]
[TABLE]
and that the minimal -structures are those with , and the maximal -structures are those with and where we have to consider all -colors . □
5 Weak multiple -categories
5.1 Strict multiple -categories
A multiple -magma is given by a multiple -magma equipped with an -structure in the sense of 4, and a morphism of -magma is a morphism of which is also a morphism of -. The category of multiple -magmas is denoted . A reflexive multiple -magma is given by a multiple -magma equipped with reflexivity and equipped with an -structure . A morphism of reflexive multiple -magmas is a morphism of which is also a morphism of -. The category of reflexive multiple -magmas is denoted .
A strict multiple -category is given by a strict multiple -category equipped with an -structure. As for the globular geometry or the cubical geometry (see [5, 8]) it is not difficult to show that such -structure is unique under this strictness. The category of strict multiple -categories is the full subcategory of spanned by the strict multiple -categories.
The third important monad of this article is the monad of strict multiple -categories given by the forgetful functor
[TABLE]
and which is in fact monadic. The proof that is right adjoint and monadic come from the underlying projective sketches of and which are not difficult to be described, and it is enough to use the result in 2 the sketch of reversors described in 4, and then these results come from an easy application of the theorem of Foltz in [2] and Lair in [9].
5.2 Multiple -categorical stretchings
A multiple -categorical stretching is given by the following datas :
- •
A reflexive multiple -magma
- •
A strict multiple –category
- •
A morphism of reflexive multiple -magma
{\definecolor{pgfstrokecolor}{rgb}{0,0,0}M}$${\definecolor{pgfstrokecolor}{rgb}{0,0,0}C}$$\scriptstyle{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pi}
- •
Operations :
[TABLE]
where such that (see 3.1) :
- –
and
- –
and
- –
A morphism of multiple -categorical stretchings
[TABLE]
is given by the following commutative square in ,
[TABLE]
thus we also have the following square in for all -color
[TABLE]
and we require for all the following equality
[TABLE]
The category of multiple -categorical stretchings is denoted .
Now consider the forgetful functor :
[TABLE]
given by :
[TABLE]
Proposition 2
The functor just above has a left adjoint which produces a monad on the category of multiple sets. □
The proof that is right adjoint comes from the underlying projective sketches of and which are not difficult to be described (see for example [8]). Then these results come from an application of the theorem of Foltz in [2].
Definition 2
A weak multiple -category is a -algebra. The category of weak multiple -categories is denoted . Also models of weak multiple -groupoids are given by the weak multiple -categories, and thus the category of weak multiple -groupoids is denoted □
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Charles Ehresmann, Problèmes universels relatifs aux catégories n 𝑛 n -aires , Comptes rendus de l’Académie des Sciences (1967), Volume 264, pages 273-276.
- 2[2] F. Foltz, Sur la catégorie des foncteurs dominés , Cahiers de Topologie et de Géométrie Différentielle Catégorique (1969), volume 11(2), pages 101–130).
- 3[3] Marco Grandis and Paré Robert, An introduction to multiple categories (On weak and lax multiple categories, I) , Cahiers de Topologie et de Géométrie Différentielle Catégorique, fascicule 2, volume LVII (2016).
- 4[4] Kamel Kachour, Définition algébrique des cellules non-strictes , Cahiers de Topologie et de Géométrie Différentielle Catégorique, volume 1 (2008), pages 1–68.
- 5[5] Camell Kachour, Algebraic definition of weak ( ∞ , n ) 𝑛 (\infty,n) -categories , Theory and Applications of Categories (2015), Volume 30, No. 22, pages 775-807
- 6[6] Camell Kachour, An algebraic approach to weak ω 𝜔 \omega -groupoids , Australian Category Seminar, 14 September 2011. http://web.science.mq.edu.au/groups/coact/seminar/cgi-bin/speaker-info.cgi?name=Camell+Kachour
- 7[7] Camell Kachour, ( ∞ , n ) 𝑛 (\infty,n) -ensembles cubiques , CLE Seminar, 23 novembre 2016. https://sites.google.com/site/logiquecategorique/Contenus/201611-kachour
- 8[8] Camell Kachour, Aspects of Cubical Higher Category Theory , Ar Xiv 1st February 2017. https://128.84.21.199/abs/1702.00336
