Notes on Multiple Higher Category Theory

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 $\infty$-categories, strict multiple $(\infty,m)$-categories, and in particular we obtain algebraic models of weak multiple $\infty$-groupoids.

Equations

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$$\text{definecolor}pgfstrokecolorrgb0,0,0C_{\underline{i}-i_j}$$

$$\text{definecolor}pgfstrokecolorrgb0,0,0(C,s,t;1)$$

$$\text{definecolor}pgfstrokecolorrgb0,0,0C_{\underline{i}}$$

$$\text{definecolor}pgfstrokecolorrgb0,0,0{\mathbb{M}}_r\mathbb{S}\text{ets}$$

$$\text{definecolor}pgfstrokecolorrgb0,0,0M_{\underline{i}}\underset{M_{\underline{i}-i_j}}{\times }{M}_{\underline{i}}$$

$$\text{definecolor}pgfstrokecolorrgb0,0,0C_{\underline{i}}\underset{C_{\underline{i}-i_j}}{\times }{C}_{\underline{i}}$$

$$\text{definecolor}pgfstrokecolorrgb0,0,0(M,s,t;\circ )$$

$$1_{\underline{i}+i_k}^{\underline{i}}(a{\circ }_{i_j}^{\underline{i}}b)=1_{\underline{i}+i_k}^{\underline{i}}(a){\circ }_{i_j}^{\underline{i}+i_k}{1}_{\underline{i}+i_k}^{\underline{i}}(b)$$

$$a{\circ }_{\underline{i}-i_j}^{\underline{i}}{1}_{\underline{i}}^{\underline{i}-i_j}(t_{\underline{i}-i_j}^{\underline{i}}(a))=1_{\underline{i}}^{\underline{i}-i_j}(s_{\underline{i}-i_j}^{\underline{i}}(a)){\circ }_{\underline{i}-i_j}^{\underline{i}}=a$$

$$(a{\circ }_{\underline{i}-i_j}^{\underline{i}}b){\circ }_{\underline{i}-i_k}^{\underline{i}}(c{\circ }_{\underline{i}-i_j}^{\underline{i}}d)=(a{\circ }_{\underline{i}-i_k}^{\underline{i}}c){\circ }_{\underline{i}-i_j}^{\underline{i}}(b{\circ }_{\underline{i}-i_k}^{\underline{i}}d)$$

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$$\text{definecolor}pgfstrokecolorrgb0,0,0M$$

$$\text{definecolor}pgfstrokecolorrgb0,0,0M_{\underline{i}}$$

$$m_{\underline{i}+i_r}([\alpha ,\beta {]}_{\underline{i}+i_r}^{\underline{i}})=[m_{\underline{i}}(\alpha ),m_{\underline{i}}(\beta ){]}_{\underline{i}+i_r}^{\prime \underline{i}}$$

$$\text{definecolor}pgfstrokecolorrgb0,0,0\infty \text{-}\mathbb{ME}\text{tC}$$

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$$C_{\underline{i}}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$C_{\underline{i}}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\text{definecolor}pgfstrokecolorrgb0,0,0C_{\underline{i}}$$

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$$C_{\underline{i}}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

**Keywords. weak multiple $(\infty,n)$-categories, weak multiple $\infty$-groupoids, computer sciences.

Mathematics Subject Classification (2010). 18B40,18C15, 18C20, 18G55, 20L99, 55U35, 55P15. **

1. 1 Multiple Sets

2. 1.1 Multiple Sets

3. 1.2 Reflexive Multiple Sets

4. 2 Strict multiple $\infty$-categories

5. 3 Weak multiple $\infty$-categories

6. 3.1 Multiple categorical stretchings

7. 4 Multiple $(\infty,m)$-Sets

8. 5 Weak multiple $(\infty,m)$-categories

9. 5.1 Strict multiple $(\infty,m)$-categories

10. 5.2 Multiple $(\infty,m)$-categorical stretchings

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 $\infty$-categories in 3

• •

Algebraic models of weak multiple $(\infty,m)$-categories in 5

In particular we propose algebraic models of weak multiple $\infty$-groupoids which are our models of weak multiple $(\infty,0)$-categories.

1 Multiple Sets

1.1 Multiple Sets

Fix an integer $n\geq 1$, each finite sequence $\underline{i}=(i_{1},i_{2},...,i_{n})\in\mathbb{N}^{n}$ such that $1\leq i_{1}<i_{2}<...<i_{n}$ is called an $n$-color, and all $n$-colors form a set denoted $\mathfrak{I}_{n}$. Also for a fix $n$-color $\underline{i}\in\mathfrak{I}_{n}$, $S^{1}_{\underline{i}}=\{i_{1},...,i_{n}\}$ denotes its underlying set of $1$-colors $i_{j}$. If $1\leq k\leq n$, the set $S^{k}_{\underline{i}}$ of $k$-colors of $\underline{i}$ is well understood, it has $C^{n}_{k}=\frac{n!}{k!(n-k)!}$ $k$-colors : $S^{k}_{\underline{i}}=\{\underline{i}^{k}_{1},\underline{i}^{k}_{2},...,\underline{i}^{k}_{C^{n}_{k}}\}$, i.e each $\underline{i}^{k}_{j}$ is a subsequence $(i_{j_{1}},i_{j_{1}},...,i_{j_{k}})\in\mathbb{N}^{k}$ of $\underline{i}$ such that $1\leq i_{j_{1}}<i_{j_{2}}<...<i_{j_{k}}$. The set $S^{n-1}_{\underline{i}}$ of $(n-1)$-colors of $\underline{i}$ has a particular importance : for $S^{n-1}_{\underline{i}}=\{\underline{i}^{n-1}_{1},\underline{i}^{n-1}_{2},...,\underline{i}^{n-1}_{n}\}$ and $1\leq k\leq n$ we use the notation $\underline{i}^{n-1}_{k}=\underline{i}-i_{k}$ which means that the $(n-1)$-color $\underline{i}^{n-1}_{k}$ is $(i_{1},i_{2},...,\hat{i_{k}},...,i_{n})\in\mathbb{N}^{n-1}$ i.e we delete the $1$-color $i_{k}$ from $\underline{i}=(i_{1},i_{2},...,i_{n})\in\mathbb{N}^{n}$. In fact if $\underline{i}=(i_{1},i_{2},...,i_{n})\in\mathbb{N}^{n}$ is an $n$-color and if $l$ is an integer such that the $(n+1)$-sequence $(i_{1},i_{2},...,l,...,i_{n})$ is an $(n+1)$-color, then we write this last $(n+1)$-color by $\underline{i}+l$. Thus if $\underline{i}=(i_{1},i_{2},...,i_{n})\in\mathbb{N}^{n}$ is an $n$-color then the notation ”minus” $\underline{i}-i_{k}$ means the $(n-1)$-color $(i_{1},i_{2},...,\hat{i_{k}},...,i_{n})\in\mathbb{N}^{n-1}$ and the notation ”add” $\underline{i}+l$ means that $l$ doesn’t belongs to $S^{1}_{\underline{i}}$ and that $\underline{i}+l$ must be seen as its corresponding $(n+1)$-color, and possibly it can be reindexed if necessary.

An $n$-colored set means a set $C_{\underline{i}}$ colored by an $n$-color $\underline{i}$ like just above. An $n$-multiple data $C_{n}$ means a countable set $C_{n}$ of $n$-colored sets $C_{n}=(C_{\underline{i}})_{\underline{i}\in\mathfrak{I}_{n}}$, and a multiple data $C$ means a countable set $C=(C_{n})_{n\in\mathbb{N}}$ of $n$-multiple datas $C_{n}$ if $n\geq 1$ and a set of objects $C_{0}$.

A multiple set is given by a multiple data $C=(C_{n})_{n\in\mathbb{N}}$ such that for all $n\geq 1$, and all $n$-colored set $C_{\underline{i}}$ in it, this $n$-colored set is equipped for all $i_{j}\in S^{1}_{\underline{i}}$ with sources and targets :

[TABLE]

such that for any $n$-color $\underline{i}$ with $n\geq 2$ the following diagrams commute :

[TABLE]

We shall often use the short notation $(C,s,t)$ to denote a multiple set $C$ with sources $s$ and targets $t$, and no confusion should appear with the langage of colors. If $(C^{\prime},s^{\prime},t^{\prime})$ is another multiple set, a morphism of multiple sets :

[TABLE]

is given for all $n$-color $\underline{i}$ by a map $f_{\underline{i}}$ of $\mathbb{S}\text{ets}$

[TABLE]

which is compatible with all sources and all targets :

[TABLE]

The category of multiple sets is denoted $\mathbb{M}\mathbb{S}\text{ets}$

1.2 Reflexive Multiple Sets

A reflexive multiple set $(C,s,t,1)$ is given by a multiple set $(C,s,t)$ such that for all $n$-color $\underline{i}=(i_{1},...,i_{n})\in\mathfrak{I}_{n}$ ($n\geq 1$) and all $1$-color $i_{j}\in S^{1}_{\underline{i}}$ the $n$-colored set $C_{\underline{i}}$ is equipped with reflexions $1^{\underline{i}-i_{j}}_{\underline{i}}$

[TABLE]

For $n\geq 2$ consider an $n$-color $\underline{i}=(i_{1},...,i_{n})$ and $j,k\in\llbracket 1,...,n\rrbracket$ such that $j\neq k$, then we require the following commutative diagrams :

[TABLE]

If $(C^{\prime},s^{\prime},t^{\prime},1^{\prime})$ is another reflexive multiple set, then a morphism of reflexive multiple sets :

[TABLE]

is given by a morphism $f$ in $\mathbb{M}\mathbb{S}\text{ets}$ which is compatible the reflexivity’s structures, that is for all $n$-color $\underline{i}$ and all $1$-color $i_{j}\in S^{1}_{\underline{i}}$, the following diagram commutes :

[TABLE]

The category of reflexive multiple sets is denoted $\mathbb{M}_{r}\mathbb{S}\text{ets}$

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 $U$ is right adjoint and monadic come from the underlying projective sketches of $\mathbb{M}_{r}\mathbb{S}\text{ets}$ and $\mathbb{M}\mathbb{S}\text{ets}$, and by the evident applications of the theorem of Foltz in [2] and Lair in [9].

2 Strict multiple $\infty$-categories

A multiple $\infty$-magma $(M,s,t,(\circ^{\underline{i}}_{\underline{i}-i_{j}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n})}$ is given by a multiple set $(M,s,t)$ such that for all $n\geq 1$ and all $\underline{i}\in\mathfrak{I}_{n}$ its underlying $n$-colored sets $M_{\underline{i}}$ are equipped with operations

[TABLE]

where $M_{\underline{i}}\underset{M_{\underline{i}-i_{j}}}{\times}M_{\underline{i}}$ are given by the following pullbacks

[TABLE]

and such that these operations $\circ^{\underline{i}}_{i_{j}}$ follow the following positional axioms

• •

$s^{\underline{i}}_{\underline{i}-i_{j}}(a\circ^{\underline{i}}_{i_{j}}b)=s^{\underline{i}}_{\underline{i}-i_{j}}(b)$ and $t^{\underline{i}}_{\underline{i}-i_{j}}(a\circ^{\underline{i}}_{i_{j}}b)=t^{\underline{i}}_{\underline{i}-i_{j}}(a)$

• •

$s^{\underline{i}}_{\underline{i}-i_{k}}(a\circ^{\underline{i}}_{i_{j}}b)=s^{\underline{i}}_{\underline{i}-i_{k}}(a)\circ^{\underline{i}}_{i_{j}}s^{\underline{i}}_{\underline{i}-i_{k}}(b)$ and $t^{\underline{i}}_{\underline{i}-i_{k}}(a\circ^{\underline{i}}_{i_{j}}b)=t^{\underline{i}}_{\underline{i}-i_{k}}(a)\circ^{\underline{i}}_{i_{j}}t^{\underline{i}}_{\underline{i}-i_{k}}(b)$ if $j\neq k$

A multiple $\infty$-magma shall be denoted with the shorter notation $(M,s,t;\circ)$ when no confusion appears. If $(M^{\prime},s^{\prime},t^{\prime};\circ)$ is another multiple $\infty$-magma, a morphism of multiple $\infty$-magmas

[TABLE]

is given by a morphism $f\in\mathbb{M}\mathbb{S}\text{ets}$ such which respects the operations $\circ$, that is for all $n$-color $\underline{i}$ and all $1$-color $i_{j}\in S^{1}_{\underline{i}}$ we have $f_{\underline{i}}(a\circ^{\underline{i}}_{i_{j}}b)=f_{\underline{i}}(a)\circ^{\underline{i}}_{i_{j}}f_{\underline{i}}(b)$. The category of multiple $\infty$-magmas is denoted $\infty\text{-}\mathbb{M}\mathbb{M}\text{ag}$.

A reflexive multiple $\infty$-magma $(M,s,t,1,(\circ^{\underline{i}}_{\underline{i}-i_{j}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n})}$ is given by a reflexive multiple set $(M,s,t,1)$ and a multiple $\infty$-magma $(M,s,t,(\circ^{\underline{i}}_{\underline{i}-i_{j}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n})}$ such that

[TABLE]

Morphisms of reflexive multiple $\infty$-magmas are those of $\infty\text{-}\mathbb{M}\mathbb{M}\text{ag}$ which are also morphisms of $\mathbb{M}_{r}\mathbb{S}\text{ets}$. The category of reflexive multiple $\infty$-magmas is denoted $\infty\text{-}\mathbb{M}_{r}\mathbb{M}\text{ag}$.

A strict multiple $\infty$-categories is given by a reflexive multiple $\infty$-magma $(C,s,t;(\circ^{\underline{i}}_{\underline{i}-i_{j}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n})}$ such that operations $\circ^{\underline{i}}_{\underline{i}-i_{j}}$ are associative, are unital i.e if for all $a\in C_{\underline{i}}$ we have

[TABLE]

and follow the middle-four interchange axiom

[TABLE]

when compositions in both side are well defined.

Morphisms of strict multiple $\infty$-categories are those of $\infty\text{-}\mathbb{M}_{r}\mathbb{M}\text{ag}$. The category of strict multiple $\infty$-categories is denoted $\infty\text{-}\mathbb{M}\mathbb{C}\text{at}$.

The second important monad of this article is the monad $\mathbb{S}=(S,\lambda,\nu)$ of strict multiple $\infty$-categories given by the forgetful functor

[TABLE]

and which is in fact monadic. The proof that $U$ is right adjoint and monadic come from the underlying projective sketches of $\infty\text{-}\mathbb{M}\mathbb{C}\text{at}$ and $\mathbb{M}\mathbb{S}\text{ets}$ 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 $\infty$-categories

3.1 Multiple categorical stretchings

A multiple categorical stretching $\mathbb{E}=(M,C,(\pi_{\underline{i}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n}},([-,-]^{\underline{i}}_{\underline{i}+i_{r}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n}})$ is given by the following datas :

• •

A reflexive multiple $\infty$-magma $(M,s,t,1,(\circ^{\underline{i}}_{\underline{i}-i_{j}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n})}$

• •

A strict multiple $\infty$-category $(C,s,t,1,(\circ^{\underline{i}}_{\underline{i}-i_{j}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n})}$

• •

A morphism of reflexive multiple $\infty$-magmas

{\definecolor{pgfstrokecolor}{rgb}{0,0,0}M}$${\definecolor{pgfstrokecolor}{rgb}{0,0,0}C}$$\scriptstyle{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pi}

• •

Operations $[-,-]^{\underline{i}}_{\underline{i}+i_{r}}$ :

[TABLE]

where $\underline{M}_{\underline{i}}=\{(\alpha,\beta)\in M_{\underline{i}}\times M_{\underline{i}}:\pi_{\underline{i}}(\alpha)=\pi_{\underline{i}}(\beta)\}$ and such that :

• –

$s^{\underline{i}+i_{r}}_{i_{s}}[\alpha;\beta]^{\underline{i}}_{\underline{i}+i_{r}}=[s^{\underline{i}}_{i_{s}}(\alpha),s^{\underline{i}}_{i_{s}}(\beta)]^{\underline{i}-i_{s}}_{\underline{i}-i_{s}+i_{r}}$ and $t^{\underline{i}+i_{r}}_{i_{s}}[\alpha;\beta]^{\underline{i}}_{\underline{i}+i_{r}}=[t^{\underline{i}}_{i_{s}}(\alpha),t^{\underline{i}}_{i_{s}}(\beta)]^{\underline{i}-i_{s}}_{\underline{i}-i_{s}+i_{r}}$

• –

$s^{\underline{i}+i_{j}}_{i_{j}}([\alpha;\beta]^{\underline{i}}_{\underline{i}+i_{j}})=\alpha$ and $t^{\underline{i}+i_{j}}_{i_{j}}([\alpha;\beta]^{\underline{i}}_{\underline{i}+i_{j}})=\beta$

• –

$\pi_{\underline{i}+i_{r}}([\alpha;\beta]^{\underline{i}}_{\underline{i}+i_{r}})=1^{\underline{i}}_{\underline{i}+i_{r}}(\pi_{\underline{i}}(\alpha))=1^{\underline{i}}_{\underline{i}+i_{r}}(\pi_{\underline{i}}(\beta))$

A morphism of multiple categorical stretchings

[TABLE]

is given by the following commutative square in $\infty\text{-}\mathbb{M}_{\text{r}}\mathbb{M}\text{ag}$,

[TABLE]

thus we also have the following square in $\mathbb{S}\text{ets}$ for all $n$-color $\underline{i}\in\mathfrak{I}_{n}$ $(n\geq 1)$

[TABLE]

and we require for all $(\alpha,\beta)\in\underline{M}_{\underline{i}}$ the following equality

[TABLE]

The category of multiple categorical stretchings is denoted $\infty\text{-}\mathbb{M}\mathbb{E}\text{tC}$.

Now consider the forgetful functor :

[TABLE]

given by :

[TABLE]

Proposition 1

The functor $U$ just above has a left adjoint which produces a monad $\mathbb{M}=(M,\eta,\nu)$ on the category of multiple sets. □

The proof that $U$ is right adjoint comes from the underlying projective sketches of $\infty\text{-}\mathbb{M}\mathbb{E}\text{tC}$ and $\mathbb{M}\mathbb{S}\text{ets}$ 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 $\infty$-category is a $\mathbb{M}$-algebra □

4 Multiple $(\infty,m)$-Sets

Consider a multiple set $(C,s,t)$, and an $n$-color $\underline{i}$ and a $1$-color $i_{j}\in S^{1}_{\underline{i}}$. A $(\underline{i},i_{j})$-reversor on it is given by a map

[TABLE]

such that the following two diagrams commute :

[TABLE]

If for all $k>m$, and all $k$-color $\underline{i}$, and each $i_{j}\in S^{1}_{\underline{i}}$ they are such $(\underline{i},i_{j})$-reversor $j^{\underline{i}}_{i_{j}}$ on $(C,s,t)$ then we say that it is a multiple $(\infty,m)$-set. The family of maps $(j^{\underline{i}}_{i_{j}})_{k>m,\underline{i}\in\mathfrak{I}_{k},i_{j}\in S^{1}_{\underline{i}}}$ is called a multiple $(\infty,m)$-structure and in that case we shall say that $(C,s,t)$ is equipped with the multiple $(\infty,m)$-structure $j=(j^{\underline{i}}_{i_{j}})_{k>m,\underline{i}\in\mathfrak{I}_{k},i_{j}\in S^{1}_{\underline{i}}}$. Seen as multiple $(\infty,m)$-set we denote it by $(C,s,t;(j^{\underline{i}}_{i_{j}})_{k>m,\underline{i}\in\mathfrak{I}_{k},i_{j}\in S^{1}_{\underline{i}}})$, or just $(C,s,t;j)$ for a shorter notation, where $j$ design its underlying multiple $(\infty,m)$-structure.

If $(C^{\prime},s^{\prime},t^{\prime};(j^{\prime\underline{i}}_{i_{j}})_{k>m,\underline{i}\in\mathfrak{I}_{k},i_{j}\in S^{1}_{\underline{i}}})$ is another multiple $(\infty,m)$-set, then a morphism of multiple $(\infty,m)$-sets

{\definecolor{pgfstrokecolor}{rgb}{0,0,0}(C,s,t;(j^{\underline{i}}_{i_{j}})_{k>m,\underline{i}\in\mathfrak{I}_{k},i_{j}\in S^{1}_{\underline{i}}})}$${\definecolor{pgfstrokecolor}{rgb}{0,0,0}(C^{\prime},s^{\prime},t^{\prime};(j^{\prime\underline{i}}_{i_{j}})_{k>m,\underline{i}\in\mathfrak{I}_{k},i_{j}\in S^{1}_{\underline{i}}})}$$\scriptstyle{\definecolor{pgfstrokecolor}{rgb}{0,0,0}f}

is given by a morphism of multiple sets such that for each $k>m$, each $k$-color $\underline{i}$ and each $i_{j}\in S^{1}_{\underline{i}}$ we have the following commutative diagrams

[TABLE]

The category of multiple $(\infty,m)$-sets is denoted $(\infty,m)$-$\mathbb{M}\mathbb{S}\text{ets}$

Remark 1

The $(\infty,m)$-structures that we used to define multiple $(\infty,m)$-sets have globular and cubical analogues (see [5, 8]) that we called the minimal $(\infty,m)$-structures. The multiple analogue of the globular maximal $(\infty,m)$-structures as defined in [5] and of cubical maximal $(\infty,m)$-structures as defined in [8] is as follow : for all $k>m$, for each $k$-color $\underline{i}$ and for all $(k-m-1)$-color $\{i_{j_{1}},...,i_{j_{k-m-1}}\}\in S^{k-m-1}_{\underline{i}}$, there exist maps

[TABLE]

[TABLE]

such that we have the following diagrams in $\mathbb{S}\text{ets}$ 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 $(\infty,m)$-structure : this notion gives all possibilities of inverse structure between the minimal $(\infty,m)$-structure and the maximal $(\infty,m)$-structure. Thus we define it as follows : a multiple set $(C,s,t)$ is equipped with an $(\infty,m)$-structure if for all $k>m$, all $k$-color $\underline{i}$, and for all $1$-color $i_{j_{1}}\in S^{1}_{\underline{i}}$, there exist an integer $q$ with $1\leq q\leq k-m-1$, there exist a $q$-color $\{i_{j_{1}},...,i_{j_{q}}\}\in S^{q}_{\underline{i}}$, and there exist maps

[TABLE]

[TABLE]

such that we have the following diagrams in $\mathbb{S}\text{ets}$ which commute serially :

[TABLE]

[TABLE]

and that the minimal $(\infty,m)$-structures are those with $q=1$, and the maximal $(\infty,m)$-structures are those with $q=k-m-1$ and where we have to consider all $q$-colors $\{i_{j_{1}},...,i_{j_{q}}\}\in S^{q}_{\underline{i}}$. □

5 Weak multiple $(\infty,m)$-categories

5.1 Strict multiple $(\infty,m)$-categories

A multiple $(\infty,m)$-magma $(M,s,t;j)$ is given by a multiple $\infty$-magma $(M,s,t)$ equipped with an $(\infty,m)$-structure $j$ in the sense of 4, and a morphism of $(\infty,m)$-magma is a morphism of $\infty\text{-}\mathbb{M}\mathbb{M}\text{ag}$ which is also a morphism of $(\infty,m)$-$\mathbb{M}\mathbb{S}\text{ets}$. The category of multiple $(\infty,m)$-magmas is denoted $(\infty,m)\text{-}\mathbb{M}\mathbb{M}\text{ag}$. A reflexive multiple $(\infty,m)$-magma $(M,s,t;1;j)$ is given by a multiple $\infty$-magma $(M,s,t)$ equipped with reflexivity $1$ and equipped with an $(\infty,m)$-structure $j$. A morphism of reflexive multiple $(\infty,m)$-magmas is a morphism of $\infty\text{-}\mathbb{M}_{r}\mathbb{M}\text{ag}$ which is also a morphism of $(\infty,m)$-$\mathbb{M}\mathbb{S}\text{ets}$. The category of reflexive multiple $(\infty,m)$-magmas is denoted $(\infty,m)\text{-}\mathbb{M}_{r}\mathbb{M}\text{ag}$.

A strict multiple $(\infty,m)$-category is given by a strict multiple $\infty$-category $(C,s,t,(\circ^{\underline{i}}_{\underline{i}-i_{j}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n})}$ equipped with an $(\infty,m)$-structure. As for the globular geometry or the cubical geometry (see [5, 8]) it is not difficult to show that such $(\infty,m)$-structure is unique under this strictness. The category $(\infty,m)\text{-}\mathbb{M}\mathbb{C}\text{at}$ of strict multiple $(\infty,m)$-categories is the full subcategory of $(\infty,m)\text{-}\mathbb{M}_{r}\mathbb{M}\text{ag}$ spanned by the strict multiple $(\infty,m)$-categories.

The third important monad of this article is the monad $\mathbb{S}^{m}=(S^{m},\lambda^{m},\nu^{m})$ of strict multiple $(\infty,m)$-categories given by the forgetful functor

[TABLE]

and which is in fact monadic. The proof that $U$ is right adjoint and monadic come from the underlying projective sketches of $(\infty,m)\text{-}\mathbb{M}\mathbb{C}\text{at}$ and $\mathbb{M}\mathbb{S}\text{ets}$ 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 $(\infty,m)$-categorical stretchings

A multiple $(\infty,m)$-categorical stretching $(M,C,(\pi_{\underline{i}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n}},([-,-]^{\underline{i}}_{\underline{i}+i_{r}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n}})$ is given by the following datas :

• •

A reflexive multiple $(\infty,m)$-magma $(M,s,t;1;j;(\circ^{\underline{i}}_{\underline{i}-i_{j}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n})}$

• •

A strict multiple $(\infty,m)$–category $(C,s,t;1;j;(\circ^{\underline{i}}_{\underline{i}-i_{j}})_{n\geq 1,\underline{i}\in\mathfrak{I}_{n})}$

• •

A morphism of reflexive multiple $(\infty,m)$-magma

{\definecolor{pgfstrokecolor}{rgb}{0,0,0}M}$${\definecolor{pgfstrokecolor}{rgb}{0,0,0}C}$$\scriptstyle{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pi}

• •

Operations $[-,-]^{\underline{i}}_{\underline{i}+i_{r}}$ :

[TABLE]

where $\underline{M}_{\underline{i}}=\{(\alpha,\beta)\in M_{\underline{i}}\times M_{\underline{i}}:\pi_{\underline{i}}(\alpha)=\pi_{\underline{i}}(\beta)\}$ such that (see 3.1) :

• –

$s^{\underline{i}+i_{r}}_{i_{s}}[\alpha;\beta]^{\underline{i}}_{\underline{i}+i_{r}}=[s^{\underline{i}}_{i_{s}}(\alpha),s^{\underline{i}}_{i_{s}}(\beta)]^{\underline{i}-i_{s}}_{\underline{i}-i_{s}+i_{r}}$ and $t^{\underline{i}+i_{r}}_{i_{s}}[\alpha;\beta]^{\underline{i}}_{\underline{i}+i_{r}}=[t^{\underline{i}}_{i_{s}}(\alpha),t^{\underline{i}}_{i_{s}}(\beta)]^{\underline{i}-i_{s}}_{\underline{i}-i_{s}+i_{r}}$

• –

$s^{\underline{i}+i_{j}}_{i_{j}}([\alpha;\beta]^{\underline{i}}_{\underline{i}+i_{j}})=\alpha$ and $t^{\underline{i}+i_{j}}_{i_{j}}([\alpha;\beta]^{\underline{i}}_{\underline{i}+i_{j}})=\beta$

• –

$\pi_{\underline{i}+i_{r}}([\alpha;\beta]^{\underline{i}}_{\underline{i}+i_{r}})=1^{\underline{i}}_{\underline{i}+i_{r}}(\pi_{\underline{i}}(\alpha))=1^{\underline{i}}_{\underline{i}+i_{r}}(\pi_{\underline{i}}(\beta))$

A morphism of multiple $(\infty,m)$-categorical stretchings

[TABLE]

is given by the following commutative square in $(\infty,m)\text{-}\mathbb{M}_{r}\mathbb{M}\text{ag}$,

[TABLE]

thus we also have the following square in $\mathbb{S}\text{ets}$ for all $n$-color $\underline{i}\in\mathfrak{I}_{n}$ $(n\geq 1)$

[TABLE]

and we require for all $(\alpha,\beta)\in\underline{M}_{\underline{i}}$ the following equality

[TABLE]

The category of multiple $(\infty,m)$-categorical stretchings is denoted $(\infty,m)\text{-}\mathbb{M}\mathbb{E}\text{tC}$.

Now consider the forgetful functor :

[TABLE]

given by :

[TABLE]

Proposition 2

The functor $U^{m}$ just above has a left adjoint which produces a monad $\mathbb{M}^{m}=(M^{m},\eta^{m},\nu^{m})$ on the category of multiple sets. □

The proof that $U$ is right adjoint comes from the underlying projective sketches of $(\infty,m)\text{-}\mathbb{M}\mathbb{E}\text{tC}$ and $\mathbb{M}\mathbb{S}\text{ets}$ 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 $(\infty,m)$-category is a $\mathbb{M}^{m}$-algebra. The category of weak multiple $(\infty,m)$-categories is denoted $\mathbb{M}^{m}\text{-}\mathbb{A}\text{lg}$. Also models of weak multiple $\infty$-groupoids are given by the weak multiple $(\infty,0)$-categories, and thus the category of weak multiple $\infty$-groupoids is denoted $\mathbb{M}^{0}\text{-}\mathbb{A}\text{lg}$ □