Theory and Applications of Categories, Vol. 30, No. 22, 2015, pp. 775–807.

AN ALGEBRAIC DEFINITION OF (∞, N)-CATEGORIES

CAMELL KACHOUR

Abstract. In this paper we define a sequence of monads 𝕋^{(∞,n)}(n ∈ ℕ) on the category ∞-𝔾r of ∞-graphs. We conjecture that algebras for 𝕋^{(∞,0)}, which are defined in a purely algebraic setting, are models of ∞-groupoids. More generally, we conjecture that 𝕋^{(∞,n)}-algebras are models for (∞, n)-categories. We prove that our (∞, 0)-categories are bigroupoids when truncated at level 2.

Introduction

The notion of weak (∞, n)-category can be made precise in many ways depending on our approach to higher categories. Intuitively this is a weak ∞-category such that all its cells of dimension greater than n are equivalences.

Models of weak (∞, 1)-categories (case n = 1) are diverse: for example there are the quasicategories studied by Joyal and Tierney (see [24]), but also there are other models which have been studied like the Segal categories, the complete Segal spaces, the simplicial categories, the topological categories, the relative categories, and there are known to be equivalent (a survey of models of weak (∞, 1)-categories can be found in [11]).

For any n ∈ ℕ, models of weak (∞, n)-categories have been studied especially by Segal, Simpson (based on Segal’s idea; see [23, 30, 38]), Rezk (Θ_{n}-categories; see [36], but also see [35] for another approach), Bergner (see [10, 12]), and Barwick (who calls them n-fold complete Segal spaces; his approach is described in [5, 29]; see also [3]). It is known that some of these models are equivalent in an appropriate sense (see [35, 10, 12, 3]; see also the recent work of Dimitri Ara [2]).

However, these models of (∞, n)-categories are not of an algebraic nature. In this article we propose the first purely algebraic definition of weak (∞, n)-categories (or models of (∞, n)-categories to be more precise) in the globular setting, meaning that we describe these objects as algebras for some monad with good categorical properties. In particular these models are algebras for Batanin’s ω-operad. We conjecture that the models of the (∞, n)-categories that we propose here, are equivalent to other existing models in a precise sense explained below. Grothendieck also proposed an algebraic definition of ∞-groupoids in Pursuing Stacks, and Maltsiniotis (see [32]) showed that the latter could be extended to a definition of weak ∞-categories resembling to Batanin’s definition based on ω-operads. However, the algebraic nature of our definition of ∞-groupoid is stronger than in Grothendieck’s approach, because for him, ∞-groupoids are models for a limit theory, while our concept gives algebras over a monad.

Received by the editors 2014-03-14 and, in revised form, 2015-05-28.

Transmitted by Clemens Berger. Published on 2015-06-01.

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

Key words and phrases: (∞, n)-categories, weak ∞-groupoids, homotopy types.

\begin{matrix}c \\ ◯ \end{matrix} Camell Kachour, 2015. Permission to copy for private use granted.

Our main motivation for introducing an algebraic model of (∞, n)-categories came from our wish to build a machinery which would lead to a proof of the “Grothendieck conjecture on homotopy types” and, possibly, it generalisation. This conjecture of Grothendieck (see [22, 7]), claims that weak ∞-groupoids encode all homotopical information about their associated topological spaces. In his seminal article (see [7]), Michael Batanin gave an accurate formulation of this conjecture by building a fundamental weak ∞-groupoid functor between the category of topological spaces 𝕋op to the category of the weak ∞- groupoids in his sense. This conjecture is not solved yet, and a good direction to solve it should be to build first a Quillen model structure on the category of weak ω-groupoids in the sense of Michael Batanin, and then show that his fundamental weak ∞-groupoid functor is the right part of a Quillen equivalence. One obstacle for building such a model structure is that the category of Batanin ∞-groupoids is defined in a nonalgebraic way. An important property of the category of weak ∞-groupoids 𝔸lg(𝕋^{(∞,0)}) (see Section 3.9) that we propose here is to be locally presentable (see Section 3). Therefore, we hope that this will allow us in the future to use Smith’s theory on combinatorial model categories in our settings (see [9]).

More generally, we expect that it is be possible to build an adapted combinatorial model category structure for each category 𝔸lg(𝕋^{(∞,n)}) of these models of weak (∞, n)- categories (see Section 3.9) for arbitrary n ∈ ℕ, in order to be able to prove the existence of Quillen equivalences between our models of (∞, n)-categories and other models of (∞, n)- categories. This should be considered as a generalization of the Grothendieck conjecture for higher integers n > 0.

The aim of our present paper is to lay a categorical foundation for this multistage project. The model theoretical aspects of this project will be considered in future papers (but see Remark 3.7 about possible approaches).

Our algebraic description of weak (∞, n)-categories is an adaptation of the “philosophy” of categorical stretchings as developed by Jacques Penon in [34] to describe his weak ∞-categories (see also [16, 25]). Here we add the key concept of (∞, n)-graphs (see Section 1).

Weak ∞-categories in the sense of Penon can be seen as algebras for a specific ωoperad in the sense of Batanin, called by Batanin “the Penon operad” (see [8]). This result is deep and involves the complex machinery of higher computads. The goal of this article of Batanin was to prove that his weak ∞-categories were weaker than those of Penon. However, Batanin did not construct a specific ω-operad for each integer n ∈ ℕ whose algebras were models of weak (∞, n)-categories. It is therefore impossible in the present paper to compare our weak (∞, n)-categories with anything in Batanin’s work.

However it would be interesting to know whether our models of (∞, n)-categories do underlie a specific ω-operad for each integer n ∈ ℕ, of the kind Batanin produced for Penon’s weak ∞-categories. That project is quite difficult and deserves further work which is indeed in progress.

AN ALGEBRAIC DEFINITION OF (∞, N)-CATEGORIES  777

Also it is important to notice that in [15], Dominique Bourn and Jacques Penon developed an inductive procedure to categorify structures defined by a cartesian monad. Basically they start with a cartesian monad, say for instance the monad of monoids, and then their procedure allows the categorification of this monad to produce the monad of monoidal categories, and so on. Their procedure applies to the monad of monoids, because it is a cartesian monad. Unfortunately we cannot use their inductive procedure in this article, because our monads are not cartesian. For example, the monad for groupoids is not cartesian. So we cannot use their inductive procedure to obtain a monad for weak ∞-groupoids (case n = 0) of the kind we are able to construct in the present article. In this present article we prefer to build these higher structures directly, by using adapted stretchings (see 3.8), thereby avoiding any inductive process, similar to what Penon did in [34].

The plan of this article is as follows.

In Section 1 we introduce reversors, which are the operations algebraically describing equivalences. These operations plus the brilliant idea of categorical stretching developed by Jacques Penon (see [34]) are in the heart of our approach to weak (∞, n)-categories.

Section 2 introduces the reader to strict (∞, n)-categories, where we point out the important fact that reversors are “canonical” in the “strict world”. Reflexivity for strict (∞, n)-categories is seen as specific structure, using operations that we call reflexors, and we study in detail the relationships between reversors and reflexors (see 2.4). However most material of this section is well known.

Section 3 gives the steps in defining our algebraic approach to weak (∞, n)-categories. First we recall briefly the definition of weak ∞-categories in Penon’s sense. Then we define (∞, n)-magmas (see 3.5), which are the “(∞, n)-analogue” of the ∞-magmas of Penon. Then we define (∞, n)-categorical stretching (see 3.8), which is the “(∞, n)- analogue” of the categorical stretching of Penon. In [34], Jacques Penon used categorical stretching to weakened strict ∞-categories. Roughly speaking, the philosophy of Penon follows the idea that the “weak” must be controlled by the “strict”, and it is exactly what the (∞, n)-categorical stretchings do for the “(∞, n)-world”. Thirdly we give the definition of weak (∞, n)-categories (see 3.9) as algebras for specific monads 𝕋^{(∞,n)} on ∞-𝔾r. We show in 3.12 that each 𝕋^{(∞,n)}-algebra (G, v) puts on G a canonical (∞, n)- magma structure. Then, as we do for the strict case, we study the more subtle relationship between reversors and reflexors for weak (∞, n)-categories (see 3.13). Finally in 3.14, we make some computations for weak ∞-groupoids. We show that models of our weak ∞- groupoids in dimension 2 are bigroupoids.

The final section 4.2 explains how other choices of (∞, n)-structure could have been used to build other algebraic models of weak (∞, n)-categories, and among these choices, the maximal (∞, n)-structure is the one we use in this article, and the minimal (∞, n)- structure is another remarkable (∞, n)-structure.

The main ideas of this article were exposed for the first time in September 2011, in the Australian Category Seminar at Macquarie University [27].

Acknowledgement. First of all I am grateful to Michael Batanin and Ross Street, and the important fact that without their support and without their encouragement, this work could not have been done. I am also grateful to André Joyal for his invitation to Montréal, and for our many discussions, which helped me a lot to improve my point of view of higher category theory. I am also grateful to Denis Charles Cisinski who shared with me his point of view of many aspects of abstract homotopy theory. I am also grateful to Clemens Berger for many discussions with me, especially about the non trivial problem of “transfer” for Quillen model categories. I am also grateful to Paul-André Mèllies who gave me the chance to talk (in December 2011) about the monad for weak ∞-groupoids in his seminar in Paris, and also to René Guitart and Francois Métayer who proposed me to talk in their seminars. I am also grateful to Christian Lair who shared with me his point of view on sketch theory when I was living in Paris. I am also grateful to the categoricians and other mathematicians of our team at Macquarie University for their kindness, and their efforts for our seminar on Category Theory. Especially I want to mention Dominic Verity, Steve Lack, Richard Garner, Mark Weber, Tom Booker, Frank Valckenborgh, Rod Yager, Ross Moore, and Rishni Ratman. Also I have a thought for Brian Day who, unfortunately, left us too early. Finally I am grateful to Jacques Penon who taught me, many years ago, his point of view on weak ∞-categories.

I dedicate this work to Ross Street.

1. (∞, n)-graphs

Let 𝔾 be the globe category defined as follows. For each m ∈ ℕ, objects of 𝔾 are formal objects \overline{m}. Morphisms of 𝔾 are generated by the formal cosource and cotar-

get \xymatrix{\overline{m} \ar[r]^{s_{m}^{m+1}} \ar[r]_{t_{m}^{m+1}} & \overline{m + 1}} such that we have the following relations s_{m}^{m+1}s_{m−1}^{m} = s_{m}^{m+1}t_{m−1}^{m} and

t_{m}^{m+1}t_{m−1}^{m} = t_{m}^{m+1}s_{m−1}^{m}. An ∞-graph X is just a presheaf \xymatrix{𝔾^{op} \ar[r]^{X} & 𝕊et} . We denote by ∞-𝔾r := [𝔾^{op}, 𝕊et] the category of ∞-graphs where morphisms are just natural transformations. If X is an ∞-graph, sources and targets are still denoted by s_{m}^{m+1} and t_{m}^{m+1}. If 0 ≤ p < m we define s_{p}^{m} := s_{p}^{p+1} ∘ ... ∘ s_{m−1}^{m} and t_{p}^{m} := t_{p}^{p+1} ∘ ... ∘ t_{m−1}^{m}.

An (∞, n)-graph is given by a couple (X, (j_{p}^{m})_{0≤n≤p<m}) where X is an ∞-graph (see [34]) or “globular set” (see [7]), and j_{p}^{m} are maps (0 ≤ n ≤ p < m), called the reversors

\xymatrix{X_{m} \ar[r]^{j_{p}^{m}} & X_{m}} ,

such that for all integers n, m, and p such that 0 ≤ n ≤ p < m we have the following two diagrams in 𝕊et, each commuting serially.

\xymatrix{X_{m} \ar[rr]^{j_{p}^{m}} \ar[d]_{s_{m−1}^{m}} \ar[d]^{t_{m−1}^{m}} &  & X_{m} \ar[d]^{s_{m−1}^{m}} \ar[d]_{t_{m−1}^{m}} & X_{m} \ar[rr]^{j_{p}^{m}} \ar[d]_{s_{m−1}^{m}} \ar[d]^{t_{m−1}^{m}} &  & X_{m} \ar[d]^{s_{m−1}^{m}} \ar[d]_{t_{m−1}^{m}} \\ X_{m−1} \ar[rr]^{j_{p}^{m−1}} \ar[d]_{s_{m−2}^{m−1}} \ar[d]^{t_{m−1}^{m}} &  & X_{m−1} \ar[d]^{s_{m−2}^{m−1}} \ar[d]_{t_{m−1}^{m}} & X_{m−1} \ar[rr]^{j_{p}^{m−1}} \ar[d]_{s_{m−2}^{m−1}} \ar[d]^{t_{m−1}^{m}} &  & X_{m−1} \ar[d]^{s_{m−2}^{m−1}} \ar[d]_{t_{m−1}^{m}} \\ X_{m−2} \ar[rr]^{j_{p}^{m−2}} \ar@{.>}[d] \ar@{.>}[d] &  & X_{m−2} \ar@{.>}[d] \ar@{.>}[d] & X_{m−2} \ar[rr]^{j_{p}^{m−2}} \ar@{.>}[d] \ar@{.>}[d] &  & X_{m−2} \ar@{.>}[d] \ar@{.>}[d] \\ X_{p+2} \ar[rr]^{j_{p}^{p+2}} \ar[d]_{s_{p+1}^{p+2}} \ar[d]^{t_{m−1}^{m}} &  & X_{p+2} \ar[d]^{s_{p+1}^{p+2}} \ar[d]_{t_{m−1}^{m}} & X_{p+2} \ar[rr]^{j_{p}^{p+2}} \ar[d]_{s_{p+1}^{p+2}} \ar[d]^{t_{m−1}^{m}} &  & X_{p+2} \ar[d]^{s_{p+1}^{p+2}} \ar[d]_{t_{m−1}^{m}} \\ X_{p+1} \ar[rr]^{j_{p}^{p+1}} \ar[dr]_{t_{p}^{p+1}} &  & X_{p+1} \ar[dl]^{s_{p}^{p+1}} & X_{p+1} \ar[rr]^{j_{p}^{p+1}} \ar[dr]_{s_{p}^{p+1}} &  & X_{p+1} \ar[dl]^{t_{p}^{p+1}} \\  & X_{p} &  &  & X_{p}}

We shall say also that an (∞, n)-graph is an ∞-graph X equipped with an (∞, n)- structure.

1.1. Remark. These two diagrams looks equal, but it is their bottoms which are different, and are one of the key of our approach of reversibility. Also, to describe it we have preferred to use diagrams than equations, which we believe make it easier to be understood for the reader.

1.2. Remark. In the last section 4.2 we will see other interesting (∞, n)-structures on ∞-graphs, where the (∞, n)-structure just above shall be called the maximal (∞, n)- structure.

A morphism of (∞, n)-graphs

\xymatrix{(X, (j_{p}^{m})_{0≤n≤p<m}) \ar[r]^{φ} & (X′, (j_{p}^{′m})_{0≤n≤p<m})}

is given by a morphism of ∞-graphs \xymatrix{X \ar[r]^{φ} & X′} which is compatible with the reversors: this means that, for integers 0 ≤ n ≤ p < m, we have the following commutative square.

\xymatrix{X_{m} \ar[d]_{j_{p}^{m}} \ar[r]^{φ_{m}} & X′_{m} \ar[d]^{j_{p}^{′m}} \\ X_{m} \ar[r]_{φ_{m}} & X′_{m}}

The category of (∞, n)-graphs is denoted (∞, n)-𝔾r.

1.3. Remark. In [25] we defined the category (∞, n)-𝔾r of “∞-graphes n-cellulaires” (n-cellular ∞-graphs) which is a completely different category from this category (∞, n)- 𝔾r. The category (∞, n)-𝔾r was used to define an algebraic approach to “weak n-higher transformations”, still in the same spirit of the weak ∞-categories of Penon (see [34], but a quick review of this approach is in 3.1 below).

1.4. Remark. Throughout this paper the reversors are denoted by the symbols “j_{p}^{m}” except with weak (∞, n)-categories (see comment in Section 3.12) where they are denoted by the symbols “i_{p}^{m}”. Let us also make a little comment on reflexive ∞-graphs (see [34] for their definition). For us a reflexive ∞-graph (X, (1_{m}^{p})_{0≤p<m}) must be seen as a “structured ∞-graph” : that is, an ∞-graph X equipped with a structure (1_{m}^{p})_{0≤p<m},

where the maps \xymatrix{X(p) \ar[r]^{1_{m}^{p}} & X(m)} must be considered as specific operations that we call reflexors. Throughout this paper these operations are denoted by the symbols 1_{m}^{p} except for the underlying reflexive structure of weak (∞, n)-categories (see Section 3.12) where, instead, they are denoted by the symbols ι_{m}^{p} (with the Greek letter “iota”). Morphisms between reflexive ∞-graphs are morphisms of ∞-graphs which respect this structure. In [34] the category of reflexive ∞-graphs is denoted ∞-𝔾rr. The canonical forgetful functor \xymatrix{∞-𝔾rr \ar[r]^{𝕌} & ∞-𝔾r} is a right adjoint, and gives rise to the very important monad ℝ of reflexive ∞-graphs on ∞-graphs.

The reversors are built without using limits, and it is trivial to build the sketch^{1} G_{n} of (∞, n)-graphs. It has no cones and no cocones, thus (∞, n)-𝔾r is just a category of presheaves (∞, n)-𝔾r ≃ [G_{n}, 𝕊et]. Denote by G the sketch of ∞-graphs (it is the category 𝔾^{op} at the beginning of this section). We have the inclusions

\xymatrix{ & G \ar@{^{(}->}[dl] \ar@{^{(}->}[dr] \\ G_{n+1} \ar@{^{(}->}[rr] &  & G_{n}}

showing that the functor

\xymatrix{(∞, n)-𝔾r \ar[r]^{M_{n}} & (∞, n + 1)-𝔾r}

forgetting the reversors (j_{n}^{m})_{m≥n+2} of each (∞, n)-graph has a left and a right adjoint: L_{n} ⊣ M_{n} ⊣ R_{n}. The functor L_{n} is the “free (∞, n)-graphisation functor” on (∞, n + 1)- graphs, and the functor R_{n} is the “internal (∞, n)-graphisation functor” on (∞, n + 1)- graphs. The forgetful functor

\xymatrix{(∞, n)-𝔾r \ar[r]^{O_{n}} & ∞-𝔾r}

which forgets all the reversors, has a left and a right adjoint: G_{n} ⊣ O_{n} ⊣ D_{n}. The functor G_{n} is the “free (∞, n)-graphisation functor” on ∞-graphs, and the functor D_{n} is the “internal (∞, n)-graphisation functor” on ∞-graphs. Both M_{n} and O_{n} are monadic because they are conservative and, as well as left adjoints, have rights adjoints (and so preserve all coequalizers).

1see [18, 14] for good references on sketch theory.

2. Strict (∞, n)-categories (n ∈ ℕ)

2.1. Definition. The definition of the category ∞-ℂat of ∞-categories, in the form needed here, can be found in [34]. A strict ∞-category C has operations ∘_{p}^{m}, for all 0 ≤ p < m, which are maps

\xymatrix{∘_{p}^{m} : C(m) ×_{C(p)} C(m) \ar[r] & C(m)}

where C(m) ×_{C(p)} C(m) = {(y, x) ∈ C(m) × C(m) : s_{p}^{m}(y) = t_{p}^{m}(x)}.

Recall that the ∞-graph domains and codomains of these operations must satisfy the following conditions: If (y, x) ∈ C(m) ×_{C(p)} C(m), then

• for 0 ≤ p < q < m, s_{q}^{m}(y ∘_{p}^{m} x) = s_{q}^{m}(y) ∘_{p}^{q} s_{q}^{m}(x) and t_{q}^{m}(y ∘_{p}^{m} x) = t_{q}^{m}(y) ∘_{p}^{q} t_{q}^{m}(x)

• for 0 ≤ p = q < m, s_{q}^{m}(y ∘_{p}^{m} x) = s_{q}^{m}(x) and t_{q}^{m}(y ∘_{p}^{m} x) = t_{q}^{m}(x).

These are the positional axioms in the terminology of [34].

If we denote by (C, (1_{m}^{p})_{0≤p<m}) the underlying reflexivity structure on C, then the operations 1_{m}^{p} are just an abbreviation for 1_{m}^{m−1} ∘ ... ∘ 1_{p+1}^{p}. These reflexivity maps 1_{m}^{p} are called reflexors to emphasise that we see the reflexivity as specific structure.

Now let α ∈ C(m) be an m-cell of C. We say that α has an ∘_{p}^{m}-inverse (0 ≤ p < m) if there is an m-cell β ∈ C(m) such that α ∘_{p}^{m} β = 1_{m}^{p}(t_{p}^{m}(α)) and β ∘_{p}^{m} α = 1_{m}^{p}(s_{p}^{m}(α)).

A strict (∞, n)-category C is a strict ∞-category such that for all 0 ≤ n ≤ p < m, every m-cell α ∈ C(m) has an ∘_{p}^{m}-inverse. If such an inverse exists then it is unique, because it is an inverse for a morphism in a category. Thus every strict (∞, n)-category C has an underlying canonical (∞, n)-graph (C, (j_{p}^{m})_{0≤n≤p<m}) such that the maps j_{p}^{m} give the unique ∘_{p}^{m}-inverse for each m-cell of C. In other words, for each m-cell α of C such that 0 ≤ n ≤ p < m, we have α ∘_{p}^{m} j_{p}^{m}(α) = 1_{m}^{p}(t_{p}^{m}(α)) and j_{p}^{m}(α) ∘_{p}^{m} α = 1_{m}^{p}(s_{p}^{m}(α)). Strict ∞-functors respect the reversibility. As a matter of fact, consider two strict (∞, n)- categories C and C′ and a strict ∞-functor \xymatrix{C \ar[r]^{F} & C′} . If α is an m-cell of C, then for all 0 ≤ n ≤ p < m, we have

F (j_{p}^{m}(α) ∘_{p}^{m} α) = F (j_{p}^{m}(α)) ∘_{p}^{m} F (α)
= F (1_{m}^{p}(s_{p}^{m}(α)))
= 1_{m}^{p}(F (s_{p}^{m}(α)))1_{m}^{p}(s_{p}^{m}(F (α)))
= j_{p}^{m}(F (α)) ∘_{p}^{m} F (α)

which shows, by the unicity of j_{p}^{m}(F (α)), that F (j_{p}^{m}(α)) = j_{p}^{m}(F (α)). Thus morphisms between strict (∞, n)-categories are just strict ∞-functors. Thus the category of strict (∞, n)-categories, denoted by (∞, n)-ℂat, is a full subcategory of ∞-ℂat.

It is not difficult to see that there is a projective sketch C_{n} satisfying an equivalence of categories 𝕄od(C_{n}) ≃ (∞, n)-ℂat. Thus, for all n ∈ ℕ, the category (∞, n)-ℂat is locally presentable.

Furthermore, for each n ∈ ℕ, we have the following forgetful functor

\xymatrix{(∞, n)-ℂat \ar[r]^{U_{n}} & ∞-𝔾r} .

There is an inclusion G ⊂ C_{n}, and this inclusion of sketches produces, on passing to models, a functor C_{n} between the categories of models

\xymatrix{𝕄od(C_{n}) \ar[r]^{C_{n}} & 𝕄od(G)} ,

and the associated sheaf theorem for sketches of Foltz (see [21]) yields that C_{n} has a left adjoint. Thus the following commutative square induced by the previous equivalence of categories

\xymatrix{𝕄od(C_{n}) \ar[r]^{C_{n}} \ar[d]_{≀} & 𝕄od(G) \ar[d]^{≀} \\ (∞, n)-ℂat \ar[r]^{U_{n}} & ∞-𝔾r}

produces the required left adjoint \xymatrix{F_{n} ⊣ U_{n} : (∞, n)-ℂat \ar[r] & ∞-𝔾r} .

The unit and the counit of this adjunction are respectively denoted by λ_{s}^{(∞,n)} and ε_{s}^{(∞,n)}. Using Beck’s theorem of monadicity (see for instance [14]), we see that these functors U_{n} are monadic. This adjunction generates a monad 𝕋_{s}^{(∞,n)} = (T_{s}^{(∞,n)}, μ_{s}^{(∞,n)}, λ_{s}^{(∞,n)}) on ∞- 𝔾r. It is the monad for strict (∞, n)-categories on ∞-graphs.

2.2. Remark. For each n ∈ ℕ, when no confusion appears, we will simplify the notation of these monads by omitting the symbol ∞; so 𝕋_{s}^{n} = (T_{s}^{n}, μ_{s}^{n}, λ_{s}^{n}) is the same as 𝕋_{s}^{(∞,n)} = (T_{s}^{(∞,n)}, μ_{s}^{(∞,n)}, λ_{s}^{(∞,n)}).

As we did for (∞, n)-graphs (see Section 1) by building functors of “(∞, n)-graphisation”, we are going to build some functors of “strict (∞, n)-categorification” by using systematically the Dubuc adjoint triangle theorem (see theorem 1 page 72 in [20]).

For all n ∈ ℕ we have the following triangle in ℂAT

\xymatrix{(∞, n)-ℂat \ar[rr]^{V_{n}} \ar[dr]_{U_{n}} &  & (∞, n + 1)-ℂat \ar@{-->}[dl]^{U_{n+1}} \\  & ∞-𝔾r}

where the functor V_{n} forgets the reversors (j_{n}^{m})_{m≥n+2} for each strict (∞, n)-category, and we have the adjunctions F_{n} ⊣ U_{n} and F_{n+1} ⊣ U_{n+1}, where in particular U_{n+1}V_{n} = U_{n} and U_{n+1} is monadic. So we can apply the Dubuc adjoint triangle theorem which shows that the functor V_{n} has a left adjoint: L_{n} ⊣ V_{n}. For each strict (∞, n + 1)-category C, the left adjoint L_{n} of V_{n} assigns the free strict (∞, n)-category L_{n}(C) associated to C. The functor L_{n} is the “free strict (∞, n)-categorification functor” for strict (∞, n + 1)- categories. Notice that the functor V_{n} has an evident right adjoint R_{n}. For each strict (∞, n + 1)-category C, the right adjoint R_{n} of V_{n} assigns the maximal strict (∞, n)- category R_{n}(C) associated to C. This is simply because, if D is an object of (∞, n)-ℂat, then the unit map \xymatrix{D \ar[r]^{η_{n}} & R_{n}(V_{n}(D))} is just the identity 1_{D}, and its universality becomes straightforward.

We can apply the same argument to the following triangle in ℂAT (where here the functor V forgets all the reversors or can be seen as an inclusion)

\xymatrix{(∞, n)-ℂat \ar[rr]^{V} \ar[dr]_{U_{n}} &  & ∞-ℂat \ar[dl]^{U} \\  & ∞-𝔾r}

to prove that the functor V has a left adjoint: L ⊣ V. For each strict ∞-category C, the left adjoint L of V assigns the free strict (∞, n)-category L_{n}(C) associated to C. The functor L is the “free strict (∞, n)-categorification functor” for strict ∞-categories. Notice also that the functor V has an evident right adjoint R, by the same argument as before, for the adjunction V_{n} ⊣ R_{n}.

2.3. Remark. The previous functors V_{n} and V are, from our point of view, not only inclusions but also “trivial forgetful functors”. Indeed for instance, they occur in the paper [1] where they do not see strict ∞-groupoids (which are in our terminology (∞, 0)- categories) as strict ∞-categories equipped with canonical reversible structures. So from their point of view V is just an inclusion. We do not claim their point of view is incorrect but we believe that our point of view, which is more algebraic (the reversors j_{p}^{m} must be seen as unary operations), shows clearly that this inclusion is also a forgetful functor which forgets the canonical and unique reversible structures of some specific strict ∞- categories. Basically in our point of view, a strict (∞, n)-category (n ∈ ℕ) is a strict ∞-category equipped with some canonical specific structure.

2.4. (∞, n)-Involutive structures and (∞, n)-reflexivity structures. Involutive properties and reflexive structures (see below) are an important part of each strict (∞, n)-categories (n ∈ ℕ). We could have spoken about strict (∞, n)-categories without referring to these two specific structures. Yet we believe it is informative to especially point out that, while these two structures are canonical in the world of strict (∞, n)-categories, they are not canonical in the world of weak (∞, n)-categories (see Section 3.13). In particular we will show that they cannot be weakened for weak (∞, n)-categories, but only for some specific equalities which are part of these two kinds of structures (see Section 3).

This observation indicates that some properties or structures, true in the world of strict (∞, n)-categories, might or not be weakened in the world of weak (∞, n)-categories.

An involutive (∞, n)-graph is an (∞, n)-graph (X, (j_{p}^{m})_{0≤n≤p<m}) satisfying j_{p}^{m} ∘ j_{p}^{m} = 1_{X_{m}}. Involutive (∞, n)-graphs form a full reflexive subcategory i(∞, n)-𝔾r of (∞, n)- 𝔾r. For each n ∈ ℕ, each strict (∞, n)-category C has its underlying (∞, n)-graph (C, (j_{p}^{m})_{0≤n≤p<m}) an involutive (∞, n)-graph. Indeed, for each 0 ≤ n ≤ p < m and each m-cell α ∈ C(m), we have

j_{p}^{m}(j_{p}^{m}(α)) ∘_{p}^{m} j_{p}^{m}(α) = 1_{m}^{p}(s_{p}^{m}(j_{p}^{m}(α))) = 1_{m}^{p}(t_{p}^{m}(α)) ;

thus j_{p}^{m}(j_{p}^{m}(α)) is an ∘_{p}^{m}-inverse of j_{p}^{m}(α). By uniqueness, j_{p}^{m}(j_{p}^{m}(α)) = α.

A reflexive (∞, n)-graph is a triple (X, (1_{m}^{p})_{0≤p<m}, (j_{p}^{m})_{0≤n≤p<m}) where (X, (1_{m}^{p})_{0≤p<m}) is an ∞-graph equipped with a reflexivity structure (1_{m}^{p})_{0≤p<m}, where (X, (j_{p}^{m})_{0≤n≤p<m}) is an (∞, n)-graph, and such that we have the commutative diagram

\xymatrix{X_{n} \ar[rr]^{j_{p}^{n}} &  & X_{n} \\ X_{n−1} \ar[u]^{1_{n}^{n−1}} \ar[rr]^{j_{p}^{n−1}} &  & X_{n−1} \ar[u]_{1_{n}^{n−1}} \\ X_{n−2} \ar[u]^{1_{n−1}^{n−2}} \ar[rr]^{j_{p}^{n−2}} &  & X_{n−2} \ar[u]_{1_{n−1}^{n−2}} \\ X_{p+2} \ar@{.>}[u] \ar[rr]^{j_{p}^{p+2}} &  & X_{p+2} \ar@{.>}[u] \\ X_{p+1} \ar[u]^{1_{p+2}^{p+1}} \ar[rr]^{j_{p}^{p+1}} &  & X_{p+1} \ar[u]_{1_{p+2}^{p+1}} \\  & X_{p} \ar[ul]^{1_{p+1}^{p}} \ar@{-->}[ur]_{1_{p+1}^{p}}}

in 𝕊et, expressing the relation between the truncation at level n of the reflexors 1_{m}^{p} and the reversors j_{p}^{m} (0 ≤ n ≤ p < m). Thus, for all 0 ≤ n ≤ q < m and q ≥ p ≥ 0, we have j_{q}^{m}(1_{m}^{p}(α)) = 1_{m}^{p}(α) and, for all 0 ≤ n ≤ q < p < m, we have j_{q}^{m}(1_{m}^{p}(α)) = 1_{m}^{p}(j_{q}^{p}(α)). Morphisms between reflexive (∞, n)-graphs are those which are morphisms of reflexive ∞-graphs and morphisms of (∞, n)-graphs. The category of reflexive (∞, n)-graphs is denoted by (∞, n)-𝔾rr.

For each n ∈ ℕ, each strict (∞, n)-category C has its underlying (∞, n)-graph (C , (j_{p}^{m})_{0≤n≤p<m}) equipped with a reflexive (∞, n)-graph (C, (1_{m}^{p})_{0≤p<m}, (j_{p}^{m})_{0≤n≤p<m}) structure where (1_{m}^{p})_{0≤p<m} is the reflexive structure of the underlying strict ∞-category of C. As a matter of fact for all 0 ≤ n ≤ q < p < m, we have j_{q}^{m}(1_{m}^{p}(α)) ∘_{q}^{m} 1_{m}^{p}(α) = 1_{m}^{q}(s_{q}^{m}(1_{m}^{p}(α))) = 1_{m}^{q}(s_{q}^{p}(α)). Also we have the following axiom for strict ∞-categories: if q < p < m and s_{q}^{p}(y) = t_{q}^{p}(x) then 1_{m}^{p}(y ∘_{q}^{p} x) = 1_{m}^{p}(y) ∘_{q}^{m} 1_{m}^{p}(x) (see [34]). But here we have s_{q}^{p}(j_{q}^{p}(α)) = t_{q}^{q+1}s_{q+1}^{q+2}...s_{p−1}^{p}(α) = t_{q}^{p}(α), thus we can apply this axiom: 1_{m}^{p}(j_{q}^{p}(α)) ∘_{q}^{m} 1_{m}^{p}(α) = 1_{m}^{p}(j_{q}^{p}(α) ∘_{q}^{p} α) = 1_{m}^{p}(1_{p}^{q}(s_{q}^{p}(α))) = 1_{m}^{q}(s_{q}^{p}(α)) which shows that 1_{m}^{p}(j_{q}^{p}(α)) is the unique ∘_{q}^{m}-inverse of 1_{m}^{p}(α) and thus 1_{m}^{p}(j_{q}^{p}(α)) = j_{q}^{m}(1_{m}^{p}(α)). Also for all 0 ≤ n ≤ q < m and q ≥ p ≥ 0, we have j_{q}^{m}(1_{m}^{p}(α)) ∘_{q}^{m} 1_{m}^{p}(α) = 1_{m}^{q}(s_{q}^{m}(1_{m}^{p}(α))) = 1_{m}^{q}(1_{q}^{p}(α)) = 1_{m}^{p}(α) and 1_{m}^{p}(α) ∘_{q}^{m} 1_{m}^{p}(α) = 1_{m}^{p}(α) because q ≥ p, thus 1_{m}^{p}(α) is the unique ∘_{q}^{m}-inverse of 1_{m}^{p}(α) and thus 1_{m}^{p}(α) = j_{q}^{m}(1_{m}^{p}(α)).

As in Section 2, it is not difficult to show some similar results for the category i(∞, n)- 𝔾r and the category (∞, n)-𝔾rr (n ∈ ℕ):

• For each n ∈ ℕ, the categories i(∞, n)-𝔾r and (∞, n)-𝔾rr are both locally presentable.

• For each n ∈ ℕ, there is a monad 𝕀_{i}^{(∞,n)} = (I_{i}^{(∞,n)}, μ_{i}^{(∞,n)}, λ_{i}^{(∞,n)}) on ∞-𝔾r (i is for “involutive”) with 𝔸lg(𝕀_{i}^{(∞,n)}) ≃ i(∞, n)-𝔾r, and a monad ℝ_{r}^{(∞,n)} = (R_{r}^{(∞,n)}, μ_{r}^{(∞,n)}, λ_{r}^{(∞,n)}) on ∞-𝔾r (r is for “reflexive”) with 𝔸lg(ℝ_{r}^{(∞,n)}) ≃ (∞, n)-𝔾rr.

• We can also consider the category i(∞, n)-𝔾rr of involutive (∞, n)-graphs equipped with a specific reflexivity structure, whose morphisms are those of (∞, n)-𝔾r which respect the reflexivity structure. This category i(∞, n)-𝔾rr is also locally presentable. For each n ∈ ℕ, there is a monad 𝕂_{ir}^{(∞,n)} = (K_{ir}^{(∞,n)}, μ_{ir}^{(∞,n)}, λ_{ir}^{(∞,n)}) on ∞-𝔾r (ir is for “involutive-reflexive”) with 𝔸lg(𝕂_{ir}^{(∞,n)}) ≃ i(∞, n)-𝔾rr. There is a forgetful functor from the category i(∞, n)-𝔾rr to the category (∞, n)-𝔾rr which has a left adjoint, the functor “(∞, n)-involution” of any reflexive (∞, n)-graph. There is a forgetful functor from the category i(∞, n)-𝔾rr to the category i(∞, n)- 𝔾r which has a left adjoint, the functor “(∞, n)-reflexivisation” of any involutive (∞, n)-graph. These left adjoints are built using the Dubuc adjoint triangle theorem.

3. Weak (∞, n)-categories (n ∈ ℕ)

In this section we define our algebraic point of view of weak (∞, n)-categories for all n ∈ ℕ. As the reader will see, many kind of filtrations as in Section 2 could be studied here, because their filtered colimits do exist. But we have avoided that, because all the filtrations involved here are not built with “inclusion functors” but are all right adjunctions, and the author has not found a good description of their corresponding filtered colimits. We do hope to afford it in a future work because we believe that these filtered colimits have their own interest in abstract homotopy theory, and also in higher category theory. We start this section by recalling briefly^{2} the definition of the weak ∞-categories in Penon’s sense.

2For a self-contained text and the convenience of the reader.

3.1. Weak ∞-categories in Penon’s sense. For all n ∈ ℕ, each model of weak (∞, n)-category that we are going to define (see 3.9) is a weak ∞-categories in Penon’s sense.

3.2. Definition. An ∞-magma^{3} is a reflexive ∞-graph M equipped for all 0 ≤ p < m with operations ∘_{p}^{m}

\xymatrix{∘_{p}^{m} : M(m) ×_{M(p)} M(m) \ar[r] & M(m)}

where M(m) ×_{M(p)} M(m) = {(y, x) ∈ M(m) × M(m) : s_{p}^{m}(y) = t_{p}^{m}(x)}, and the operations

∘_{p}^{m} satisfy only positional axioms as in 2.1. Morphisms between ∞-magmas are morphisms of reflexive ∞-graphs which preserve these operations. We write ∞-𝕄ag for the category of ∞-magmas.

3.3. Definition. A categorical stretching is given by a quadruple

𝔼 = (M, C, π, ([−, −]_{m})_{m∈ℕ})

where M is an ∞-magma, C is a strict ∞-category, π is a morphism in ∞-𝕄ag, and ([−, −]_{m})_{m∈ℕ}) is an extra structure called the “bracketing structure”, and which is the key structure of the Penon approach to weakening the axioms of strict ∞-categories; let be more precise about it: If m ≥ 1, two m-cells c_{1}, c_{0} of M are parallel if t_{m−1}^{m}(c_{1}) = t_{m−1}^{m}(c_{0}) and if s_{m−1}^{m}(c_{1}) = s_{m−1}^{m}(c_{0}), and in that case we denote it c_{1}∥c_{0}.

([−, −]_{m} : \xymatrix{M̃_{m} \ar[r] & M_{m+1}} )_{m∈ℕ}

is a sequence of maps, where

M̃_{m} = {(c_{1}, c_{0}) ∈ M_{m} × M_{m} : c_{1}∥c_{0} and π_{m}(c_{1}) = π_{m}(c_{0})} ,

and such that

• ∀(c_{1}, c_{0}) ∈ M̃_{m}, t_{m−1}^{m}([c_{1}, c_{0}]_{m}) = c_{1}, s_{m−1}^{m}([c_{1}, c_{0}]_{m}) = c_{0},

• π_{m+1}([c_{1}, c_{0}]_{m}) = 1_{m}^{m−1}(π_{m}(c_{1})) = 1_{m}^{m−1}(π_{m}(c_{0})),

• ∀c ∈ M_{m}, [c, c]_{m} = 1_{m+1}^{m}(c).

A morphism of categorical stretchings,

\xymatrix{𝔼 \ar[r]^{(m,c)} & 𝔼′}

3In [26] ∞-magmas are defined without the reflexive structure. In [26] we can see that this approach of ∞-magmas have its own interest.

is given by the following commutative square in ∞-𝕄ag,

\xymatrix{M \ar[d]_{π} \ar[r]^{m} & M′ \ar[d]^{π′} \\ C \ar[r]_{c} & C′}

such that for all m ∈ ℕ, and for all (c_{1}, c_{0}) ∈ M̃_{m},

m_{m+1}([c_{1}, c_{0}]_{m}) = [m_{m}(c_{1}), m_{m}(c_{0})]_{m}.

Let ∞-𝔼tℂat denote the category of categorical stretchings.

Now consider the forgetful functor:

\xymatrix{∞-𝔼tℂat \ar[r]^{U} & ∞-𝔾r}

given by \xymatrix{(M, C, π, ([,])_{m∈ℕ}) \ar@{|->}[r] & M} . This functor has a left adjoint which produces a monad 𝕋^{P} = (T^{P} , μ^{P} , λ^{P}) on the category of ∞-graphs called the Penon’s monad.

3.4. Definition. Weak ∞-categories in the sense of Penon are algebras for the monad 𝕋^{P} above.

The original approach of the Penon’s monad is defined on the category ∞-𝔾rr of reflexive ∞-graphs, however Michael Batanin has proved in [8] that we obtain a better approach of this monad by considering ∞-graphs instead. See also the work in the article [16].

3.5. (∞, n)-Magmas. An (∞, n)-magma is an ∞-magma such that its underlying reflexive ∞-graph is equipped with a specific (∞, n)-structure in the sense of 1.

3.6. Remark. The reversibility part of an (∞, n)-magma has no relation with its reflexivity structure, neither with the involutive properties, contrary to strict (∞, n)-categories where their reversible structures, their involutive structures and their reflexivity structures are all related to one another (see 2.4). Instead we are going to see in this Section 3, that each underlying (∞, n)-categorical stretching of any weak (∞, n)-category (n ∈ ℕ) is especially going to be weakened, for the specific relation between the reversibility structure and the involutive structure, inside its underlying reflexive (∞, n)-magma the equalities j_{n}^{n+1} ∘ j_{n}^{n+1} = 1_{M_{n+1}}. Also we are going to see in 3.8, that each (∞, n)-categorical stretching is especially going to be weakened, for the specific relation between the reversibility structure and the reflexibility structure, inside its underlying reflexive (∞, n)-magma the equalities j_{q}^{m} ∘ 1_{m}^{m−1} = 1_{m}^{m−1} ∘ j_{q}^{m−1} and the equalities j_{m−1}^{m} ∘ 1_{m}^{m−1} = 1_{m}^{m−1}.

The basic examples of (∞, n)-magmas are strict (∞, n)-categories. Let us denote by 𝕄 = (M, (j_{p}^{m})_{0≤n≤p<m}) and 𝕄′ = (M′, (j_{p′}^{′m′})_{0≤n≤p′<m′}) two (∞, n)-magmas where M and M′ are respectively their underlying ∞-magmas, and (j_{p}^{m})_{0≤n≤p<m} and (j_{p′}^{′m′})_{0≤n≤p′<m′} are respectively their underlying reversors. A morphism between these (∞, n)-magmas

\xymatrix{𝕄 \ar[r]^{φ} & 𝕄′}

is given by its underlying morphism of ∞-magmas

\xymatrix{M \ar[r]^{φ} & M′}

such that φ preserves the (∞, n)-structure; this means that for integers 0 ≤ n ≤ p < m we have the following commutative square.

\xymatrix{M_{m} \ar[d]_{j_{p}^{m}} \ar[r]^{φ_{m}} & M′_{m} \ar[d]^{j_{p}^{′m}} \\ M_{m} \ar[r]_{φ_{m}} & M′_{m}}

The category of the (∞, n)-magmas is denoted by (∞, n)-𝕄ag; clearly it is not a full subcategory of ∞-𝕄ag.

As in Section 2, it is not difficult to show the following similar results for (∞, n)- magmas (n ∈ ℕ):

• For each n ∈ ℕ, the category (∞, n)-𝕄ag is locally presentable.

• For each n ∈ ℕ, there is a monad 𝕋_{m}^{(∞,n)} = (T_{m}^{(∞,n)}, μ_{m}^{(∞,n)}, λ_{m}^{(∞,n)}) on ∞-𝔾r (m is for “magmatic”) with 𝔸lg(𝕋_{m}^{(∞,n)}) ≃ (∞, n)-𝕄ag.

• By using Dubuc’s adjoint triangle theorem we can build functors of “(∞, n)-magma-fication” similar to those in Section 2.

3.7. Remark. Let us explain some informal intuition related to homotopy. The reader may notice that we can imagine many variations of “∞-magmas” similar to those of [34], or those that we propose in this paper (see above), but which still need to keep the presence of “higher equivalences”, encoded by the reversors (see the Section 1), or, in a less obvious way, by the reflexors plus some compositions ∘_{p}^{m} (see Section 2.4). For instance we can build kinds of “∞-magma”, their adapted “stretchings” (similar to those of Section 3.8), and their corresponding “weak ∞-structures” (similar to those of Section 3.9). All that just by using reversors, reflexors plus compositions. Such variations of “higher structures” must be all the time projectively sketchable (see Section 2). If we restrict to taking models of such sketches in 𝕊et, then these categories should be locally presentables and there are strong reasons to believe that there exists an interesting Quillen model structure on it. The Smith theorem could bring simplification to proving these intuitions. For instance, in [28], the authors have built a folk Quillen model structure on ω-ℂat, by using the Smith theorem, and ω-ℂat is such a “higher structure” where weak equivalences were build only with reflexors and compositions. So, even though the goal of this paper is to give an algebraic approach of weak (∞, n)-categories, we believe that such structures and variations should provide us many categories with interesting Quillen model structure. Our slogan is: “enough reversors, and (or) reflexors plus some higher compositions” captures enough equivalences for doing abstract homotopy theory, based on higher category theory.

3.8. (∞, n)-Categorical stretchings. Now we are going to define (∞, n)-categorical stretchings (n ∈ ℕ), which are for the weak (∞, n)-categories what categorical stretchings are for weak ∞-categories (see 3.1), and we are going to use these important tools to weaken the axioms of strict (∞, n)-categories.

An (∞, n)-categorical stretching is given by a categorical stretching 𝔼^{n} = (M^{n}, C^{n}, π^{n}, ([−, −]_{m})_{m∈ℕ}) where M^{n} is an (∞, n)-magma, C^{n} is a strict (∞, n)-category, π^{n} is a morphism in (∞, n)-𝕄ag. A morphism of (∞, n)-categorical stretchings

\xymatrix{𝔼 \ar[r]^{(m,c)} & 𝔼′}

is given by the following commutative square in (∞, n)-𝕄ag,

\xymatrix{M \ar[d]_{π} \ar[r]^{m} & M′ \ar[d]^{π′} \\ C \ar[r]_{c} & C′}

such that for all m ∈ ℕ, and for all (c_{1}, c_{0}) ∈ M̃_{m},

m_{m+1}([c_{1}, c_{0}]_{m}) = [m_{m}(c_{1}), m_{m}(c_{0})]_{m} .

Let (∞, n)-𝔼tℂat denote the category of (∞, n)-categorical stretchings.

As in Section 2, it is not difficult to show the following similar results for (∞, n)- categorical stretchings (n ∈ ℕ):

• For each n ∈ ℕ, the category (∞, n)-𝔼tℂat is locally presentable (see also 3.9).

• By using Dubuc’s adjoint triangle theorem we can build functors of “(∞, n)-categor-isation stretching” for any (∞, n + 1)-categorical stretching, and for any categorical stretching.

3.9. Definition. For each n ∈ ℕ consider the forgetful functors

\xymatrix{(∞, n)-𝔼tℂat \ar[r]^{U_{n}} & ∞-𝔾r}

given by \xymatrix{(M, C, π, ([,])_{m∈ℕ}) \ar@{|->}[r] & M}

Also, for each n ∈ ℕ the categories (∞, n)-𝔼tℂat and ∞-𝔾r are sketchable (in Section 2 we call G the sketch of ∞-graphs). Let us call E_{n} the sketch of (∞, n)-categorical stretchings. These sketches are both projective and there is an easy inclusion G ⊂ E_{n}. This inclusion of sketches produces, in passing to models, a functor W_{n}:

\xymatrix{𝕄od(E_{n}) \ar[r]^{W_{n}} & 𝕄od(G)}

and the associated sheaf theorem for sketches of Foltz ([21]) proves that W_{n} has a left adjoint. Furthermore, there is an equivalence of categories 𝕄od(E_{n}) ≃ (∞, n)-𝔼tℂat. Thus the following commutative square induced by these equivalences

\xymatrix{𝕄od(E_{n}) \ar[r]^{W_{n}} \ar[d]_{≀} & 𝕄od(G) \ar[d]^{≀} \\ (∞, n)-𝔼tℂat \ar[r]^{U_{n}} & ∞-𝔾r}

produces the required left adjoint F_{n} of U_{n}.

\xymatrix{(∞, n)-𝔼tℂat \ar[r]^{U_{n}} & ∞-𝔾r \ar[l]^{F_{n}}|{⊤}}

The unit and the counit of this adjunction are respectively denoted λ^{(∞,n)} and ε^{(∞,n)}. This adjunction generates a monad 𝕋^{(∞,n)} = (T^{(∞,n)}, μ^{(∞,n)}, λ^{(∞,n)}) on ∞-𝔾r.

3.10. Definition. For each n ∈ ℕ, a weak (∞, n)-category is an algebra for the monad 𝕋^{(∞,n)} = (T^{(∞,n)}, μ^{(∞,n)}, λ^{(∞,n)}) on ∞-𝔾r.

3.11. Remark. For each n ∈ ℕ, when no confusion occurs, we will simplify the notation of these monads: 𝕋^{n} = (T^{n}, μ^{n}, λ^{n}) = 𝕋^{(∞,n)} = (T^{(∞,n)}, μ^{(∞,n)}, λ^{(∞,n)}), by omitting the symbol ∞.

For each n ∈ ℕ, the category 𝔸lg(𝕋^{(∞,n)}) is locally presentable. As a matter of fact,

the adjunction \xymatrix{(∞, n)-𝔼tℂat \ar[r]^{U_{n}} & ∞-𝔾r \ar[l]^{F_{n}}|{⊤}} involves the categories (∞, n)-𝔼tℂat and ∞-

𝔾r which are both accessible (because they are both projectively sketchable thus locally presentable). But the forgetful functor U_{n} has a left adjoint, thus thanks to proposition 5.5.6 of [14], it preserves filtered colimits. Thus the monad 𝕋^{n} preserves filtered colimits in the locally presentable category ∞-𝔾r, and theorem 5.5.9 of [14] implies that the category 𝔸lg(𝕋^{(∞,n)}) is locally presentable as well.

Now we are going to build some functors of “weak (∞, n)-categorification” by using systematically Dubuc’s adjoint triangle theorem (see [20]). For all n ∈ ℕ we have the following triangle in ℂAT

\xymatrix{𝔸lg(𝕋^{n}) \ar[rr]^{V_{n}} \ar@{-->}[dr]_{U_{n}} &  & 𝔸lg(𝕋^{n+1}) \ar[dl]^{U_{n+1}} \\  & ∞-𝔾r}

The functors V_{n} can be thought of as forgetful functors which forget the reversors (i_{n}^{m})_{m≥n+2} for each weak (∞, n)-category (see 3.12 for the definition of the reversors produced by each weak (∞, n)-category). We have the adjunctions F_{n} ⊣ U_{n} and F_{n+1} ⊣ U_{n+1}, where in particular U_{n+1}V_{n} = U_{n} and U_{n+1} is monadic. So we can apply Dubuc’s adjoint triangle theorem (see [20]) to show that the functor V_{n} has a left adjoint: L_{n} ⊣ V_{n}. For each weak (∞, n+1)-category C, the left adjoint L_{n} of V_{n} yields the free weak (∞, n)-category L_{n}(C) associated to C. L_{n} can be seen as the “free weak (∞, n)-categorification functor” for weak (∞, n + 1)-categories.

We can apply the same argument to the following triangles in ℂAT (where here the functor V forgets all the reversors; see also 3.1)

\xymatrix{𝔸lg(𝕋^{n}) \ar[rr]^{V} \ar@{-->}[dr]_{U_{n}} &  & 𝔸lg(𝕋^{P}) \ar[dl]^{U} \\  & ∞-𝔾r}

to prove that the functor V has a left adjoint: L ⊣ V. For each weak ∞-category C, the left adjoint L of V builds the free weak (∞, n)-category L_{n}(C) associated to C. L is the “free weak (∞, n)-categorification functor” for weak ∞-categories.

In [8] Batanin has proved that Penon’s monad 𝕋^{P} is in fact a contractible ω-operad^{4} equipped with a composition system, and thus algebras for 𝕋^{P} are weak ∞-categories in Batanin’s sense. Thus for each n ∈ ℕ and thanks to the forgetful functor V above, our models of (∞, n)-categories are weak ∞-categories in Batanin’s sense.

3.12. Magmatic properties of weak (∞, n)-categories (n ∈ ℕ). If (G, v) is a

𝕋^{n}-algebra then \xymatrix{G \ar[r]^{λ_{G}^{n}} & 𝕋^{n}(G)} is the associated universal map and \xymatrix{M^{n}(G) \ar[r]^{π_{G}^{n}} & C^{n}(G)} is the free (∞, n)-categorical stretching associated to G, and we write (⋆_{p}^{m})_{0≤p<m} for the composition laws of M^{n}(G). Also let us define the following composition laws on G: If a, b ∈ G(m) are such that s_{p}^{m}(a) = t_{p}^{m}(b) then we put

a ∘_{p}^{m} b = v_{m}(λ_{G}^{n}(a) ⋆_{p}^{m} λ_{G}^{n}(b)) ,

if a ∈ G(p) then we put

ι_{m}^{p}(a) := v_{m}(1_{m}^{p}(λ_{G}^{n}(a))) ,

and if a ∈ G(m) and 0 ≤ n ≤ p < m then we put

i_{p}^{m}(a) := v_{m}(j_{p}^{m}(λ_{G}^{n}(a))) .

It is easy to show that with these definitions, the 𝕋^{n}-algebra (G, v) puts an (∞, n)-magma structure on G.

In [34] the author showed that if a, b are m-cells of 𝕋^{n}(G) such that s_{p}^{m}(a) = t_{p}^{m}(b) then v_{m}(a ⋆_{p}^{m} b) = v_{m}(a) ∘_{p}^{m} v_{m}(b). We are going to show that if a is a p-cell of 𝕋^{n}(G) such that 0 ≤ n ≤ p < m then v_{m}(1_{m}^{p}(a)) = ι_{m}^{p}(v_{p}(a)) and if a is an m-cell of 𝕋^{n}(G) such that 0 ≤ n ≤ p < m then v_{m}(j_{p}^{m}(a)) = i_{p}^{m}(v_{m}(a)). In other words, each underlying morphism \xymatrix{𝕋^{n}(G) \ar[r]^{v} & G} in ∞-𝔾r of a weak (∞, n)-category (G, v) is also a morphism of (∞, n)-𝕄ag when we consider them as equipped with the (∞, n)-magmatic structures defined above. Proofs of these magmatic properties becomes standard after the work in [25, 34], but for the comfort of the reader we shall give a complete proof.

4We use here the notation ω instead of ∞, in order to make clear that we are dealing with higher operads in Batanin’s sense and not with ∞-operads as defined in Jacob Lurie’s book (see [31])

All reversors for algebras are denoted “i_{p}^{m}” because there is no risk of confusion. The 𝕋^{n}-algebra (𝕋^{n}(G), μ_{G}^{n}) on 𝕋^{n}(G) is an (∞, n)-structure (i_{p}^{m})_{0≤n≤p<m} such that for all t in 𝕋^{n}(G)(m) we have j_{p}^{m}(t) = i_{p}^{m}(t). As a matter of fact

i_{p}^{m}(t) := μ_{G}^{n}(j_{p}^{m}(λ_{T^{n}(G)}^{n}(t))) = j_{p}^{m}(μ_{G}^{n}(λ_{T^{n}(G)}^{n}(t)))

because μ_{G}^{n} forgets that a morphism preserves the involutions, so i_{p}^{m}(t) = j_{p}^{m}(t). Further-

more a morphism of 𝕋^{n}-algebras \xymatrix{(G, v) \ar[r]^{f} & (G′, v′)} is such that for all t ∈ G(m) with 0 ≤ n ≤ p < m we have f(i_{p}^{m}(t)) = i_{p}^{m}(f(t)). Indeed

f(i_{p}^{m}(t)) = f(v_{m}(j_{p}^{m}(λ_{G}^{n}(t))))
= v′_{m}(T^{n}(f)(j_{p}^{m}(λ_{G}^{n}(t))))
= v′_{m}(j_{p}^{m}(T^{n}(f)(λ_{G}^{n}(t))))

because 𝕋^{n}(f) forgets that a morphism preserves the reversors. Thus

f(i_{p}^{m}(t)) = v′_{m}(j_{p}^{m}(T^{n}(f)(λ_{G}^{n}(t)))) = v′(j_{p}^{m}(λ_{G′}^{n}(f(t)) .

Because a 𝕋^{n}-algebra (G, v) determines a morphism of 𝕋^{n}-algebras

\xymatrix{(μ_{G}^{n}, 𝕋^{n}(G)) \ar[r]^{v} & (G, v)} ,

we deduce the useful formula v_{m}(j_{p}^{m}(t)) = i_{p}^{m}(v_{m}(t)).

All reflexors for algebras are denoted “ι_{m}^{p}” because there is no risk of confusion, and we use the symbols “1_{m}^{p}” for reflexors coming from the free (∞, n)-categorical stretchings. The 𝕋^{n}-algebra (𝕋^{n}(G), μ_{G}^{n}) puts a reflexive structure (ι_{m}^{p})_{0≤p<m} on 𝕋^{n}(G) such that for all t in 𝕋^{n}(G)(p) we have 1_{m}^{p}(t) = ι_{m}^{p}(t). As a matter of fact

ι_{m}^{p}(t) := μ_{G}^{n}(1_{m}^{p}(λ_{𝕋^{n}(G)}^{n}(t))) = 1_{m}^{p}(μ_{G}^{n}(λ_{𝕋^{n}(G)}^{n}(t)))

because μ_{G}^{n} forgets that a morphism preserves the reflexivities, so ι_{m}^{p}(t) = 1_{m}^{p}(t). Further-

more a morphism of 𝕋^{n}-algebras \xymatrix{(G, v) \ar[r]^{f} & (G′, v′)} is such that for all t ∈ G(p) with 0 ≤ p < m, we have f(ι_{m}^{p}(t)) = ι_{m}^{p}(f(t)). Indeed

f(ι_{m}^{p}(t)) = f(v_{m}(1_{m}^{p}(λ_{G}^{n}(t))))
= v′_{m}(𝕋^{n}(f)(1_{m}^{p}(λ_{G}^{n}(t))))
= v′_{m}(1_{m}^{p}(T^{n}(f)(λ_{G}^{n}(t))))

because 𝕋^{n}(f) forgets that a morphism preserves the reflexors, so

f(ι_{m}^{p}(t)) = v′(1_{m}^{p}(𝕋^{n}(f)(λ_{G}^{n}(t)))) = v′(1_{m}^{p}(λ_{G′}^{n}(f(t)))) = ι_{m}^{p}(f(t)) .

Thus, because a 𝕋^{n}-algebra (G, v) is also a morphism

\xymatrix{(μ_{G}^{n}, 𝕋^{n}(G)) \ar[r]^{v} & (G, v)}

of 𝕋^{n}-algebras, we have the useful formula v_{m}(1_{m}^{p}(t)) = ι_{m}^{p}(v_{m}(t)).

3.13. Interactions between reversibility structures, involutive structures, and reflexivity structures. The reversors for strict (∞, n)-categories and for (∞, n)-magmas are denoted by “j”, whereas the reversors for weak (∞, n)-categories are denoted by “i”. Let us fix an n ∈ ℕ and a strict (∞, n)-category C. We know that in C (see Section 2) we have for each 0 ≤ n ≤ p < m the involutive properties j_{p}^{m} ∘ j_{p}^{m} = 1_{X_{m}}. However for weak (∞, n)-categories this property does not hold even up to coherence cell; yet reversors of type i_{m}^{m+1} do permit a weakened version of the involutive property. As a matter of fact consider now a weak (∞, n)-category (G, v), the free (∞, n)-categorical

stretching \xymatrix{M^{n}(G) \ar[r]^{π_{G}^{n}} & C^{n}(G)} associated to G and the universal map \xymatrix{G \ar[r]^{λ_{G}^{n}} & 𝕋^{n}(G)} . For each α ∈ G(m + 1) we have

i_{m}^{m+1}(i_{m}^{m+1}(α)) = i_{m}^{m+1}(v(j_{m}^{m+1}(λ_{G}^{m}(α))))
= v(j_{m}^{m+1}(j_{m}^{m+1}(λ_{G}^{m}(α)))),

because the algebra (G, v) preserves the reversibility structure (see 3.14). This implies i_{m}^{m+1}(i_{m}^{m+1}(α))∥α because

s_{m}^{m+1}(i_{n}^{m+1}(i_{m}^{m+1}(α))) = s_{m}^{m+1}(v_{m+1}(j_{m}^{m+1}(j_{m}^{m+1}(λ_{G}^{m}(α)))))
= v_{m}(s_{m}^{m+1}(j_{m}^{m+1}(j_{m}^{m+1}(λ_{G}^{m}(α)))))
= v_{m}(t_{m}^{m+1}(j_{m}^{m+1}(λ_{G}^{m}(α))))
= v_{m}(s_{m}^{m+1}(λ_{G}^{m}(α)))
= v_{m}(λ_{G}^{m}(s_{m}^{m+1}(α)))
= s_{m}^{m+1}(α),

and similarly we show that t_{m}^{m+1}(i_{m}^{m+1}(i_{m}^{m+1}(α))) = t_{m}^{m+1}(α).

But also in the free (∞, n)-categorical stretching associated to G, which controls the algebraic nature of (G, v), creates between the (m+1)-cells j_{m}^{m+1}(j_{m}^{m+1}(λ_{G}^{m}(α))) and λ_{G}^{m}(α), an (m + 2)-cell of coherence:

[j_{m}^{m+1}(j_{m}^{m+1}(λ_{G}^{m}(α))); λ_{G}^{m}(α)]_{m+1}.

Thus at the level of algebras it shows that there is a coherence cell between i_{m}^{m+1}(i_{m}^{m+1}(α)) and α.

The other equalities j_{p}^{m} ∘ j_{p}^{m} = 1_{X_{m}} (0 ≤ n < p < m) which are valid in any strict (∞, n)-category have no reason to be weakened in any weak (∞, n)-category for the simple reason that the axioms of (∞, n)-structure do not imply parallelism between the m-cells i_{p}^{m}(i_{p}^{m}(α)) and α when p > n.

Now let us fix an n ∈ ℕ and a strict (∞, n)-category C. We know that for each p-cell α of C and for each 0 ≤ n ≤ q < p < m we have the equalities j_{q}^{m}(1_{m}^{p}(α)) = 1_{m}^{p}(j_{q}^{p}(α)) but also for each 0 ≤ n ≤ q < m and 0 ≤ p ≤ q we have the equalities j_{q}^{m}(1_{m}^{p}(α)) = 1_{m}^{p}(α) (see Section 2). However for a weak (∞, n)-category (G, v) and for any p-cell α in it, the (∞, n)-structure, for each 0 ≤ n ≤ q < p < m, does not ensure the parallelism between the m-cells i_{q}^{m}(ι_{m}^{p}(α)) and ι_{m}^{p}(i_{q}^{p}(α)), and for each 0 ≤ n ≤ q < m and 0 ≤ p ≤ q, does not ensure the parallelism between the m-cells i_{q}^{m}(ι_{m}^{p}(α)) and ι_{m}^{p}(α).

Thus these equalities which are true in the strict case are not necessarily weakened in the weak case. However there are certain situations where in the weak case these equalities are replaced by some coherence cells. As a matter of fact if now (G, v) is a weak (∞, n)-category then it is easy to check, thanks to the axioms for the (∞, n)-structure (see Section 1) that if p = m − 1 and 0 ≤ n ≤ q < m − 1, then for any (m − 1)-cell α of (G, v) we have i_{q}^{m}(ι_{m}^{m−1}(α))∥ι_{m}^{m−1}(i_{q}^{m−1}(α)). Indeed, we have

i_{q}^{m}(ι_{m}^{m−1}(α)) = i_{q}^{m}(v_{m}(1_{m}^{m−1}(λ_{G}^{n}(α))))
= v_{m}(j_{q}^{m}(1_{m}^{m−1}(λ_{G}^{n}(α)))).

Thus

s_{m−1}^{m}(i_{q}^{m}(ι_{m}^{m−1}(α))) = s_{m−1}^{m}(v_{m}(j_{q}^{m}(1_{m}^{m−1}(λ_{G}^{n}(α)))))
= v_{m−1}(s_{m−1}^{m}(j_{q}^{m}(1_{m}^{m−1}(λ_{G}^{n}(α)))))
= v_{m−1}(j_{q}^{m−1}(s_{m−1}^{m}(1_{m}^{m−1}(λ_{G}^{n}(α)))))
= v_{m−1}(j_{q}^{m−1}(λ_{G}^{n}(α)))
= i_{q}^{m−1}(α)
= v_{m−1}(λ_{G}^{n}(i_{q}^{m−1}(α)))
= v_{m−1}(s_{m−1}^{m}(1_{m}^{m−1}(λ_{G}^{n}(i_{q}^{m−1}(α)))))
= s_{m−1}^{m}(v_{m}(1_{m}^{m−1}(λ_{G}^{n}(i_{q}^{m−1}(α)))))
= s_{m−1}^{m}(ι_{m}^{m−1}(i_{q}^{m−1}(α))),

and similarly we see that t_{m−1}^{m}(i_{q}^{m}(ι_{m}^{m−1}(α))) = t_{m−1}^{m}(ι_{m}^{m−1}(i_{q}^{m−1}(α))). But also in the free (∞, n)-categorical stretching associated to G, which controls the algebraic nature of (G, v), between the m-cells j_{q}^{m}(1_{m}^{m−1}(λ_{G}^{n}(α))) and 1_{m}^{m−1}(j_{q}^{m−1}(λ_{G}^{n}(α))), an (m + 1)-cell of coherence is created:

[j_{q}^{m}(1_{m}^{m−1}(λ_{G}^{n}(α))); 1_{m}^{m−1}(j_{q}^{m−1}(λ_{G}^{n}(α)))]_{m}.

Thus at the level of algebras it shows that there is a coherence cell between i_{q}^{m}(ι_{m}^{m−1}(α)) and ι_{m}^{m−1}(i_{q}^{m−1}(α)).

Furthermore, thanks to the (∞, n)-structure (see Section 1) we easily prove that for p = q = m − 1 we have, for any (m − 1)-cells α of any weak (∞, n)-category (G, v), the parallelism i_{m−1}^{m}(ι_{m}^{m−1}(α))∥ι_{m}^{m−1}(α). As a matter of fact

i_{m−1}^{m}(ι_{m}^{m−1}(α)) = i_{m−1}^{m}(v_{m}(1_{m}^{m−1}(λ_{G}^{n}(α))))
= v_{m}(j_{m−1}^{m}(1_{m}^{m−1}(λ_{G}^{n}(α)))),

Thus

s_{m−1}^{m}(i_{m−1}^{m}(ι_{m}^{m−1}(α))) = s_{m−1}^{m}(v_{m}(j_{m−1}^{m}(1_{m}^{m−1}(λ_{G}^{n}(α)))))
= v_{m−1}(s_{m−1}^{m}(j_{m−1}^{m}(1_{m}^{m−1}(λ_{G}^{n}(α)))))
= v_{m−1}(t_{m−1}^{m}(1_{m}^{m−1}(λ_{G}^{n}(α))))
= v_{m−1}(λ_{G}^{n}(α))
= α
= s_{m−1}^{m}(ι_{m}^{m−1}(α)),

and similarly we show that t_{m−1}^{m}(i_{m−1}^{m}(1_{m−1}^{m}(α))) = t_{m−1}^{m}(1_{m−1}^{m}(α)). But also in the free (∞, n)-categorical stretching associated to G, which controls the algebricity of (G, v), between the m-cells j_{m−1}^{m}(1_{m}^{m−1}(λ_{G}^{n}(α))) and 1_{m}^{m−1}(λ_{G}^{n}(α)) creates an (m+1)-cell of coherence

[j_{m−1}^{m}(1_{m}^{m−1}(λ_{G}^{n}(α))); 1_{m}^{m−1}(λ_{G}^{n}(α))]_{m}.

Thus at the level of algebras it shows that there is a coherence cell between i_{m−1}^{m}(ι_{m}^{m−1}(α)) and ι_{m}^{m−1}(α).

For the other equalities j_{q}^{m}(1_{m}^{p}(α)) = 1_{m}^{p}(j_{q}^{p}(α)) (for 0 ≤ n ≤ q < p < m and p ≠ m−1) and j_{q}^{m}(1_{m}^{p}(α)) = 1_{m}^{p}(α) (for 0 ≤ n ≤ q < m, 0 ≤ p ≤ q and p, q ≠ m − 1), which are valid in any strict (∞, n)-category, they have no reason to be weakened in any weak (∞, n)- category for the simple reason that the axioms of the (∞, n)-structure does not imply the parallelism between these m-cells i_{q}^{m}(ι_{m}^{p}(α)) and ι_{m}^{p}(i_{q}^{p}(α)), or between the m-cells i_{q}^{m}(ι_{m}^{p}(α)) and ι_{m}^{p}(α).

3.14. Computations in dimensions 2. We are going to see that in dimension 2, algebras for the monad 𝕋^{0} (see Remark 3.11 for this notation) for weak (∞, 0)-categories (see Section 3), commonly called in the literature weak ∞-groupoids, are bigroupoids in the usual sense.

A bigroupoid is just a bicategory such that its 1-cells are equivalences and its 2-cells are ∘_{1}^{2}-isomorphism (see below). In [34] Penon has proved that weak ∞-categories in his sense are bicategories in dimension 2. However as we said in 3.1, this approach is slightly different from our approach because he used reflexive ∞-graphs whereas we use here ∞- graphs instead. It is important to recall that our models of weak (∞, 0)-categories are weak ∞-categories in Penon sense, where these weak ∞-categories are obtained with the monad 𝕋^{P} = (T^{P} , μ^{P} , λ^{P}) described in 3.1.

First let us recall some basic definitions that we can find in [25]. A reflexive ∞-graph has dimension p ∈ ℕ if all its q-cells for which q > p are identity cells and if there is at least one p-cell which is not an identity cell. Thus reflexive ∞-graphs of dimension 0 are just sets. An (∞, 0)-categorical stretching 𝔼^{0} = (M^{0}, C^{0}, π^{0}, ([,])_{m∈ℕ}) (that we should also call “groupoidal stretching” by analogy with the “categorical stretchings” of Penon) has dimension p ∈ ℕ if the underlying reflexive ∞-graph of M^{0} has dimension p. A 𝕋^{0}-algebra (G; v) has dimension p ∈ ℕ if G has dimension p when G is considered with its canonical reflexivity structure (see 3.12).

Because Penon has proved in [34] that algebras for the monad 𝕋^{P} on reflexive ∞- graphs are bicategories in dimension 2, we just need to prove the identities axiom for the bicategorical part : Basically, it says that given a 𝕋^{0}-algebra (G, v) of dimension 2, and given a diagram in G of the type

\xymatrix{a \ar[r]^{f} & b \ar[r]^{ι_{1}^{0}(b)} & b \ar[r]^{g} & c}

then the following commutative diagram in G holds

\xymatrix{(g ∘_{0}^{1} ι_{1}^{0}(b)) ∘_{0}^{1} f \ar[rr]^{a} \ar@{-->}[dr]_{r_{g}∘_{0}^{2}ι_{2}^{1}(f)} &  & g ∘_{0}^{1} (ι_{1}^{0}(b) ∘_{0}^{1} f) \ar[dl]^{ι_{2}^{1}(g)∘_{0}^{2}l_{f}} \\  & g ∘_{0}^{1} f}

where a is the coherence cell of associativity isomorphism described in [34], and

\xymatrix{g ∘_{0}^{1} 1_{1}^{0}(b) \ar[r]^{r_{g}} & g}, \xymatrix{1_{1}^{0}(b) ∘_{0}^{1} f \ar[r]^{l_{f}} & f} are respectively the right unit isomorphism cell and the left unit isomorphism cell.

Also we need to prove that 1-cells are equivalences and 2-cells are ∘_{1}^{2}-isomorphisms. We first start to prove these facts and then finish by proving the bicategorical nature of such algebras.

3.15. Proposition. Let (G; v) be a 𝕋^{0}-algebra of dimension 2 and let \xymatrix{a \ar[r]^{f} & b} be a 1-cell of G. Then f is an equivalence.

Proof. Actually we are going to exhibit a diagram in G of the following form

[figure: a; b; ι_{1}^{0}(a); i_{0}^{1}(f)∘_{1}^{0}f; v(α); ι_{1}^{0}(b); f∘_{1}^{0}i_{0}^{1}(f); v(β); f; i_{0}^{1}(f)]

and show that the 2-cells v(α) and v(β) are ∘_{1}^{2}-isomorphism. Let us denote by \xymatrix{G \ar[r]^{λ_{G}^{0}} & 𝕋^{0}(G)}

the universal map associated to G, and by \xymatrix{M^{0}(G) \ar[r]^{π_{G}^{0}} & C^{0}(G)} the free groupoidal stretching associated to G.

Consider the 2-cell

β = [λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1}

in M^{0}(G). We are going to show that a diagram of the type

[figure: [λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1} ⋆_{1}^{2} j_{1}^{2}([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1}); 1_{2}^{0}(λ_{G}^{0}(b)); λ_{f}^{0}]

lives in M^{0}(G). Because π_{G}^{0} is a morphism of (∞, 0)-magmas, we have

π_{G}^{0}([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b)))]_{1}
⋆_{1}^{2} j_{1}^{2}([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1}))
= π_{G}^{0}([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1})
∘_{1}^{2} π_{G}^{0}(j_{1}^{2}([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1}))
= π_{G}^{0}([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1})
∘_{1}^{2} j_{1}^{2}(π_{G}^{0}([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1}))
= 1_{2}^{1}(π_{G}^{0}(λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f))))
∘_{1}^{2} j_{1}^{2}(1_{2}^{1}(π_{G}^{0}(λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)))))
= 1_{2}^{1}(π_{G}^{0}(λ_{G}^{0}(f)) ∘_{0}^{1} j_{0}^{1}π_{G}((λ_{G}^{0}(f))))
∘_{1}^{2} j_{1}^{2}(1_{2}^{1}(π_{G}^{0}(λ_{G}^{0}(f)) ∘_{0}^{1} j_{0}^{1}π_{G}^{0}((λ_{G}^{0}(f)))))
= 1_{2}^{1}(1_{1}^{0}(π_{G}^{0}(λ_{G}^{0}(b)))) ∘_{1}^{2} j_{1}^{2}(1_{2}^{1}(1_{1}^{0}(π_{G}^{0}(λ_{G}^{0}(b)))))
= 1_{2}^{0}(π_{G}^{0}(λ_{G}^{0}(b))) ∘_{1}^{2} j_{1}^{2}(1_{2}^{0}(π_{G}^{0}(λ_{G}^{0}(b))))
= 1_{2}^{0}(π_{G}^{0}(λ_{G}^{0}(b))) ∘_{1}^{2} 1_{2}^{0}(π_{G}^{0}(λ_{G}^{0}(b)))
= 1_{2}^{0}(π_{G}^{0}(λ_{G}^{0}(b))) = π_{G}^{0}(1_{2}^{0}(λ_{G}^{0}(b))) .

The second equality holds because π_{G}^{0} respects the (∞, 0)-reversible structure and the third equality holds because of the definition of an (∞, 0)-stretching (see 3.8), the fourth equality is because π_{G}^{0} is a morphism of ∞-magmas and π_{G}^{0} preserves the reversible structure, whereas the fifth equality is because π_{G}^{0}(λ_{G}^{0}(f)) and j_{0}^{1}π_{G}^{0}((λ_{G}^{0}(f)) are ∘_{0}^{1}-inverse in the strict ∞-groupoid C^{0}(G). The seventh equality is because j_{1}^{2} ∘ 1_{2}^{0} = 1_{2}^{0}; see 2.4. Thus by the contractibility structure in M^{0}(G), we get the coherence 3-cell

[figure: [λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1} ⋆_{1}^{2} j_{1}^{2}([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1}); 1_{2}^{0}(λ_{G}^{0}(b)); λ_{f}^{0}]

in M^{0}(G), where λ_{f}^{0} is the 3-cell

[[λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)) ; 1_{1}^{0}(λ_{G}^{0}(b))]_{1} ⋆_{1}^{2} j_{1}^{2}([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)) ; 1_{1}^{0}(λ_{G}^{0}(b))]_{1})
; 1_{2}^{0}(λ_{G}^{0}(b))]_{2} .

By applying to it the 𝕋^{0}-algebra (G; v) we obtain the 3-cell in G

[figure: v([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1} ⋆_{1}^{2} j_{1}^{2}([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1})); v(1_{2}^{0}(λ_{G}^{0}(b))); v(λ_{f}^{0} )]

with

v([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1}
⋆_{1}^{2} j_{1}^{2}([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1})
= v([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1})
∘_{1}^{2} v(j_{1}^{2}([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1}))

because v is a morphism of (∞, 0)-magmas

= v([λ_{G}^{0}(f) ⋆_{0}^{1} j_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1})
∘_{1}^{2} i_{1}^{2}(v([λ_{G}^{0}(f) ⋆_{0}^{1} i_{0}^{1}(λ_{G}^{0}(f)); 1_{1}^{0}(λ_{G}^{0}(b))]_{1}))
= v(β) ∘_{1}^{2} i_{1}^{2}(v(β)) .

Thus we obtain the 3-cell

[figure: v(β) ∘_{1}^{2} i_{1}^{2}(v(β)); v(λ_{f}^{0} ); ι_{2}^{0}(b)]

in G. But the 𝕋^{0}-algebra (G; v) has dimension 2 forcing v(λ_{f}^{0}) to be an identity, proving v(β) ∘_{1}^{2} i_{1}^{2}(v(β)) = ι_{2}^{0}(b). By the same method we can prove that i_{1}^{2}(v(β)) ∘_{1}^{2} v(β) = ι_{2}^{1}(f ∘_{0}^{1} i_{0}^{1}(f)) which shows that v(β) is an ∘_{1}^{2}-isomorphism.

Furthermore with the 2-cell

α = [j_{0}^{1}(λ_{G}^{0}(f)) ⋆_{0}^{1} λ_{G}^{0}(f); 1_{1}^{0}(λ_{G}^{0}(a))]_{1}

in M^{0}(G), we can build a 3-cell in M^{0}(G) of the type

[figure: [j_{0}^{1}(λ_{G}^{0}(f)) ⋆_{0}^{1} λ_{G}^{0}(f); 1_{1}^{0}(λ_{G}^{0}(a))]_{1}] ⋆_{1}^{2} j_{1}^{2}([j_{0}^{1}(λ_{G}^{0}(f)) ⋆_{0}^{1} λ_{G}^{0}(f); 1_{1}^{0}(λ_{G}^{0}(a))]_{1}]); 1_{2}^{0}(λ_{G}^{0}(a)); ρ_{f}^{0}]

and with the same kind of arguments as above we can prove that in G we have the following 2-cell

\xymatrix{i_{0}^{1}(f) ∘_{0}^{1} f \ar@{=>}[r]^{v(α)} & ι_{0}^{1}(a)}

which is an ∘_{1}^{2}-isomorphism, that is we have v(α)∘_{1}^{2} i_{1}^{2}(v(α)) = ι_{2}^{0}(a) and i_{1}^{2}(v(α))∘_{1}^{2} v(α) = ι_{2}^{1}(i_{0}^{1}(f) ∘_{0}^{1} f), which finally shows that f is an equivalence.

3.16. Proposition. Let (G; v) be a 𝕋^{0}-algebra of dimension 2 and let \xymatrix{f \ar[r]^{α} & g} be a 2-cell of G. Then α is an ∘_{1}^{2}-isomorphism.

Proof. Let us denote by \xymatrix{G \ar[r]^{λ_{G}^{0}} & 𝕋^{0}(G)} the universal map associated to G, and by

\xymatrix{M^{0}(G) \ar[r]^{π_{G}^{0}} & C^{0}(G)} the free groupoidal stretching associated to G. Consider the following diagram in M^{0}(G)

\xymatrix{λ_{G}^{0}(g) \ar[r]^{j_{1}^{2}(λ_{G}^{0}(α))} & λ_{G}^{0}(f) \ar[r]^{λ_{G}^{0}(α)} & λ_{G}^{0}(g) \ar[r]^{j_{1}^{2}(λ_{G}^{0}(α))} & λ_{G}^{0}(f)}

The 2-cells λ_{G}^{0}(α)⋆_{1}^{2}j_{1}^{2}(λ_{G}^{0}(α)) and 1_{2}^{1}(λ_{G}^{0}(g)) are parallels and are connected by a coherence 3-cell A because

π_{G}^{0}(λ_{G}^{0}(α) ⋆_{1}^{2} j_{1}^{2}(λ_{G}^{0}(α))) = π_{G}^{0}(λ_{G}^{0}(α)) ∘_{1}^{2} π_{G}^{0}(j_{1}^{2}(λ_{G}^{0}(α)))
= π_{G}^{0}(λ_{G}^{0}(α)) ∘_{1}^{2} j_{1}^{2}(π_{G}^{0}(λ_{G}^{0}(α)))
= 1_{2}^{1}(π_{G}^{0}(λ_{G}^{0}(g)))
= π_{G}^{0}(1_{2}^{1}(λ_{G}^{0}(g)))) .

Also, the 2-cells j_{1}^{2}(λ_{G}^{0}(α)) ⋆_{1}^{2} λ_{G}^{0}(α) and 1_{2}^{1}(λ_{G}^{0}(f)) are parallels and are connected by a coherence 3-cell B because

π_{G}^{0}(j_{1}^{2}(λ_{G}^{0}(α)) ⋆_{1}^{2} λ_{G}^{0}(α)) = π_{G}^{0}(j_{1}^{2}(λ_{G}^{0}(α))) ∘_{1}^{2} π_{G}^{0}(λ_{G}^{0}(α))
= j_{1}^{2}(π_{G}^{0}(λ_{G}^{0}(α))) ∘_{1}^{2} π_{G}^{0}(λ_{G}^{0}(α))
= 1_{2}^{1}(π_{G}^{0}(λ_{G}^{0}(f)))
= π_{G}^{0}(1_{2}^{1}(λ_{G}^{0}(f))) .

These two equalities hold because π_{G}^{0} is a morphism of (∞, 0)-magmas.

But v is also a morphism of (∞, 0)-magmas, thus we have v(λ_{G}^{0}(α) ⋆_{1}^{2} j_{1}^{2}(λ_{G}^{0}(α))) = α ∘_{1}^{2} i_{1}^{2}(α) and v(j_{1}^{2}(λ_{G}^{0}(α)) ⋆_{1}^{2} λ_{G}^{0}(α)) = i_{1}^{2}(α) ∘_{1}^{2} α. Also (G; v) has dimension 2 thus it shows that v(A) and v(B) are identity cells, which brings equalities α ∘_{1}^{2} i_{1}^{2}(α) = ι_{2}^{1}(g) and i_{1}^{2}(α) ∘_{1}^{2} α = ι_{2}^{1}(f), which proves that α is indeed an ∘_{1}^{2}-isomorphism.

3.17. Proposition. If (G; v) is a 𝕋^{0}-algebra of dimension 2, then it puts on G a bicategorical structure.

Proof. The reflexive structure that (G; v) put on G has been defined in 3.12, and the associativity axiom for bicategories has been already proved in [34].

Consider the following diagram in G

\xymatrix{a \ar[r]^{f} & b \ar[r]^{ι_{1}^{0}(b)} & b \ar[r]^{g} & c}

We have to show that we have the following commutative diagram in G

\xymatrix{(g ∘_{0}^{1} ι_{1}^{0}(b)) ∘_{0}^{1} f \ar[rr]^{a} \ar@{-->}[dr]_{r_{g}∘_{0}^{2}ι_{2}^{1}(f)} &  & g ∘_{0}^{1} (ι_{1}^{0}(b) ∘_{0}^{1} f) \ar[dl]^{ι_{2}^{1}(g)∘_{0}^{2}l_{f}} \\  & g ∘_{0}^{1} f}

where a is the coherence cell of associativity isomorphism described in [34], and

\xymatrix{g ∘_{0}^{1} ι_{1}^{0}(b) \ar[r]^{r_{g}} & g}, \xymatrix{ι_{1}^{0}(b) ∘_{0}^{1} f \ar[r]^{l_{f}} & f} are respectively the right unit isomorphism cell and

the left unit isomorphism cell. First we define r_{g} and l_{f}. Let us denote by \xymatrix{G \ar[r]^{λ_{G}^{0}} & 𝕋^{0}(G)}

the universal map associated to G, and by \xymatrix{M^{0}(G) \ar[r]^{π_{G}^{0}} & C^{0}(G)} the free groupoidal stretching associated to G. Consider the coherence 2-cells

R_{g} = [λ_{G}^{0}(g) ⋆_{0}^{1} 1_{0}^{1}(λ_{G}^{0}(b)); λ_{G}^{0}(g)]_{1}

and

L_{f} = [1_{0}^{1}(λ_{G}^{0}(b)) ⋆_{0}^{1} λ_{G}^{0}(f) ; λ_{G}^{0}(f)]_{1}

in M^{0}(G). By definition we put r_{g} = v(R_{g}) and l_{f} = v(L_{f})

Now we have the following diagram in M^{0}(G)

\xymatrix{(λ_{G}^{0}(g) ⋆_{0}^{1} 1_{0}^{1}(λ_{G}^{0}(b))) ⋆_{0}^{1} λ_{G}^{0}(f) \ar[rr]|{A} \ar@{-->}[dr]_{R_{g}⋆_{0}^{2}1_{2}^{1}(λ_{G}^{0}(f))} &  & λ_{G}^{0}(g) ⋆_{0}^{1} (1_{0}^{1}(λ_{G}^{0}(b)) ⋆_{0}^{1} λ_{G}^{0}(f)) \ar@{-->}[dl]^{1_{2}^{1}(λ_{G}^{0}(g))⋆_{0}^{2}L_{f}} \\  & λ_{G}^{0}(g) ⋆_{0}^{1} λ_{G}^{0}(f)}

such that the v(A) = a and which is commutative up to a 3-coherence cell

[figure: (1_{2}^{1}(λ_{G}^{0}(g)) ⋆_{0}^{2} L_{f}) ⋆_{1}^{2} A; R_{g} ⋆_{0}^{2} 1_{2}^{1}(λ_{G}^{0}(f)); u]

This coherence 3-cell u exist because 1_{2}^{1}(λ_{G}^{0}(g))⋆_{0}^{2}L_{f})⋆_{1}^{2}A and R_{g}⋆_{0}^{2}1_{2}^{1}(λ_{G}^{0}(f)) are parallels; because π_{G}^{0} is a morphism of (∞, 0)-magmas, we have

π_{G}^{0}((1_{2}^{1}(λ_{G}^{0}(g)) ⋆_{0}^{2} L_{f}) ⋆_{1}^{2} A) = π_{G}^{0}(1_{2}^{1}(λ_{G}^{0}(g)) ⋆_{0}^{2} L_{f}) ∘_{1}^{2} π_{G}^{0}(A)
= (π_{G}^{0}(1_{2}^{1}(λ_{G}^{0}(g))) ∘_{0}^{2} π_{G}^{0}(L_{f})) ∘_{1}^{2} 1_{2}^{1}(f ∘_{0}^{1} g)
= 1_{2}^{1}(π_{G}^{0}(λ_{G}^{0}(g))) ∘_{0}^{2} π_{G}^{0}(L_{f})
= 1_{2}^{1}(π_{G}^{0}(λ_{G}^{0}(g))) ∘_{0}^{2} 1_{2}^{1}(π_{G}^{0}(λ_{G}^{0}(f)))
= π_{G}^{0}(R_{g}) ∘_{0}^{2} 1_{2}^{1}(π_{G}^{0}(λ_{G}^{0}(f)))
= π_{G}^{0}(R_{g}) ∘_{0}^{2} π_{G}^{0}(1_{2}^{1}(λ_{G}^{0}(f)))
= π_{G}^{0}(R_{g} ⋆_{0}^{2} 1_{2}^{1}(λ_{G}^{0}(f))) .

which shows existence of this coherence 3-cell u between the 2-cells (1_{2}^{1}(λ_{G}^{0}(g)) ⋆_{0}^{2} L_{f}) ⋆_{1}^{2} A and R_{g} ⋆_{0}^{2} 1_{2}^{1}(λ_{G}^{0}(f)).

Finally v is also a morphism of (∞, 0)-magmas (see 3.12), thus

• v(λ_{G}^{0}(g) ⋆_{0}^{1} 1_{0}^{1}(λ_{G}^{0}(b)))) ⋆_{0}^{1} λ_{G}^{0}(f) = (g ∘_{0}^{1} ι_{1}^{0}(b)) ∘_{0}^{1} f,

• v(λ_{G}^{0}(g) ⋆_{0}^{1} (1_{0}^{1}(λ_{G}^{0}(b)) ⋆_{0}^{1} λ_{G}^{0}(f))) = g ∘_{0}^{1} (ι_{1}^{0}(b) ∘_{0}^{1} f),

• v(λ_{G}^{0}(g) ⋆_{0}^{1} λ_{G}^{0}(f)) = g ∘_{0}^{1} f,

and

• v((1_{2}^{1}(λ_{G}^{0}(g)) ⋆_{0}^{2} L_{f}) ⋆_{1}^{2} A) = (ι_{2}^{1}(g) ∘_{0}^{2} l_{f}) ∘_{1}^{2} a,

• v(R_{g} ⋆_{0}^{2} 1_{2}^{1}(λ_{G}^{0}(f))) = r_{g} ∘_{0}^{2} ι_{2}^{1}(f).

If we apply v to this 3-cell u we obtain the required commutative diagram

\xymatrix{(g ∘_{0}^{1} ι_{1}^{0}(b)) ∘_{0}^{1} f \ar[rr]^{a} \ar@{-->}[dr]_{r_{g}∘_{0}^{2}ι_{2}^{1}(f)} &  & g ∘_{0}^{1} (ι_{1}^{0}(b) ∘_{0}^{1} f) \ar[dl]^{ι_{2}^{1}(g)∘_{0}^{2}l_{f}} \\  & g ∘_{0}^{1} f}

because v(u) is an identity cell which is a consequence of the dimension 2 of the algebra (G; v).

3.18. Proposition. If (G; v) is a 𝕋^{0}-algebra of dimension 2, then it puts on G a bigroupoidal structure.

Proof. It is a consequence of the propositions above

4. (∞, n)-structures

This last section was motivated by the suggestion of the anonymous referee who told us to include the proof that in dimension 2 our models of weak ∞-groupoids are bigroupoids.

Actually, we could have built all our article with many other combinatorics similar to those of the first section 1 and for each n ∈ ℕ it should give different flavours of what we suspect to be other models of weak (∞, n)-categories, where in dimension 2 corresponding models of weak ∞-groupoids are bigroupoids (see below).

These combinatorics are called regular (∞, n)-structure, and in our article we have used the maximal (∞, n)-structure (see below).

Surprisingly, the proposition 4.1 below related to strict ∞-groupoids, has helped us to understand better the key ingredients (see definitions 4.2 and 4.4) of all variations possible of models of weak ∞-groupoids which in dimension 2 produce bigroupoids.

Strict ∞-groupoids have been defined in 2.1 under the name of a strict (∞, 0)-categories. Also we have the following proposition 4.1. Proposition. A strict ∞-category is a strict ∞-groupoid if the following equivalent conditions hold

1. For each m ≥ 1, each m-cell has an ∘_{m−1}^{m}-inverse,

2. For each m ≥ 1, each m-cell has an ∘_{p}^{m}-inverse for one p such that 0 ≤ p < m,

3. For each m ≥ 1, each m-cell has an ∘_{p}^{m}-inverse for all p such that 0 ≤ p < m.

Proof of this proposition can be found in [1]. The underlying technology of stretchings that we used in this article is to use the “strict world” to build the “weak world”. This proposition informs us that other candidates of (∞, n)-structure (see below) could have been chosen for other approach of algebraic model of weak (∞, n)-categories. This proposition motivates the following definition

4.2. Definition. An (∞, 0)-structure on an ∞-graph X is the data of a family of diagrams constructed as follows : ∀m > 0, ∀s ∈ ℕ such that 1 ≤ s ≤ m + 1, we consider a finite sequence in ℕ

0 ≤ m_{1} < m_{2} < ... < m_{j} < ... < m_{s} ≤ m − 1

where for all j ∈ ℕ such that 1 ≤ j ≤ s, it corresponds two diagrams in 𝕊et, each commuting serially

\xymatrix{X_{m} \ar[rr]^{j_{m_{j}}^{m}} \ar[d]_{s_{m−1}^{m}} \ar[d]^{t_{m−1}^{m}} &  & X_{m} \ar[d]^{s_{m−1}^{m}} \ar[d]_{t_{m−1}^{m}} & X_{m} \ar[rr]^{j_{m_{j}}^{m}} \ar[d]_{s_{m−1}^{m}} \ar[d]^{t_{m−1}^{m}} &  & X_{m} \ar[d]^{s_{m−1}^{m}} \ar[d]_{t_{m−1}^{m}} \\ X_{m−1} \ar[rr]^{j_{m_{j}}^{m−1}} \ar[d]_{s_{m−2}^{m−1}} \ar[d]^{t_{m−1}^{m}} &  & X_{m−1} \ar[d]^{s_{m−2}^{m−1}} \ar[d]_{t_{m−1}^{m}} & X_{m−1} \ar[rr]^{j_{m_{j}}^{m−1}} \ar[d]_{s_{m−2}^{m−1}} \ar[d]^{t_{m−1}^{m}} &  & X_{m−1} \ar[d]^{s_{m−2}^{m−1}} \ar[d]_{t_{m−1}^{m}} \\ X_{m−2} \ar[rr]^{j_{m_{j}}^{m−2}} \ar@{.>}[d] \ar@{.>}[d] &  & X_{m−2} \ar@{.>}[d] \ar@{.>}[d] & X_{m−2} \ar[rr]^{j_{m_{j}}^{m−2}} \ar@{.>}[d] \ar@{.>}[d] &  & X_{m−2} \ar@{.>}[d] \ar@{.>}[d] \\ X_{m_{j}+2} \ar[rr]^{j_{m_{j}}^{m_{j} +2}} \ar[d]_{s_{m_{j} +1}^{m_{j} +2}} \ar[d]^{t_{m_{j} +1}^{m_{j} +2}} &  & X_{m_{j}+2} \ar[d]^{s_{m_{j} +1}^{m_{j} +2}} \ar[d]_{t_{m_{j} +1}^{m_{j} +2}} & X_{m_{j}+2} \ar[rr]^{j_{m_{j}}^{m_{j} +2}} \ar[d]_{s_{m_{j} +1}^{m_{j} +2}} \ar[d]^{t_{m_{j} +1}^{m_{j} +2}} &  & X_{m_{j}+2} \ar[d]^{s_{m_{j} +1}^{m_{j} +2}} \ar[d]_{t_{m_{j} +1}^{m_{j} +2}} \\ X_{m_{j}+1} \ar[rr]^{j_{m_{j}}^{m_{j} +1}} \ar[dr]_{t_{m_{j}}^{m_{j} +1}} &  & X_{m_{j}+1} \ar[dl]^{s_{m_{j}}^{m_{j} +1}} & X_{m_{j}+1} \ar[rr]^{j_{m_{j}}^{m_{j} +1}} \ar[dr]_{s_{m_{j}}^{m_{j} +1}} &  & X_{m_{j}+1} \ar[dl]^{t_{m_{j}}^{m_{j} +1}} \\  & X_{m_{j}} &  &  & X_{m_{j}}}

Now let us fix an integer n ∈ ℕ. An (∞, n)-structure on an ∞-graph X is the data of a family of diagrams that are obtained by n-truncation of an (∞, 0)-structure on an ∞- graph X′^{5}, that is we consider only diagrams of such (∞, 0)-structure for which m > n, and where X′ is replaced by X. It is equivalent to say that an (∞, n)-structure on an ∞-graph X is the realisation of the sketch obtained by n-truncation of the sketch of an (∞, 0)-structure on an ∞-graph X′. An (∞, n)-structure on an ∞-graph X is denoted by

R = (X, (j_{m_{j}}^{m})_{1≤j≤s;1≤s≤m−n+1;m>n})

4.3. Remark. Morphisms between two ∞-graphs equipped with the same kind of (∞, n)- structure, are build as in 1 : Such morphisms need just to preserve underlying reversors of such (∞, n)-structure.

4.4. Definition. Consider an (∞, n)-structure on an ∞-graph X as above. If for each m > n and for each 1 ≤ s ≤ m − n + 1 we have m_{s} = m − 1, then we say that this (∞, n)-structure is regular.

For each regular (∞, n)-structure R = (X, (j_{m_{j}}^{m})_{1≤j≤s;1≤s≤m−n+1;m>n}), the corresponding monad on ∞-𝔾r is denoted by

𝕋^{(∞,n,R)} = (T^{(∞,n,R)}, μ^{(∞,n,R)}, λ^{(∞,n,R)})

and its algebras are models of weak (∞, n)-categories build with this specific regular (∞, n)-structure. We believe that these regular (∞, n)-structures R produce categories 𝔸lg(𝕋^{(∞,n,R)}) which are Quillen equivalents.

The maximal (∞, n)-structure R_{max} that we used in our article, is to consider all kind of reversors j_{p}^{m} with 0 ≤ n ≤ p < m. It is a regular (∞, n)-structure as well, and we have shown in 3.18 that the corresponding models of weak ∞-groupoids in dimension 2 are bigroupoids. These are bigroupoids because the maximal (∞, n)-structure is regular, which forces the existence of the reversor j_{1}^{2}, and thus the proof of the bigroupoidal nature in dimension 2.

More generally, it is not difficult to check that with any regular (∞, n)-structure, the corresponding models of weak ∞-groupoids in dimension 2 are also bigroupoids, because regularity forces the existence of the reversor j_{1}^{2}. However it is not true for a non-regular (∞, n)-structure which doesn’t contain the reversor j_{1}^{2}, that the corresponding models of weak ∞-groupoids in dimension 2 are bigroupoids, just for the simple reason that each 2-cell in it have no reasons to be ∘_{1}^{2}-isomorphisms.

Also each regular (∞, n)-structure captures many kinds of higher equivalences, depending on the kinds of reversors we used. More precisely, consider for example the following definition build by decreasing induction :

4.5. Definition. Let us consider a strict ∞-category C and x ∈ C(m). We say that x is an (∘_{m−1}^{m}, q)-equivalence (q ∈ ℕ) if there exist a diagram

5X′ could be equal to X but not necessarily.

[figure: s_{m−1}^{m}(x); t_{m−1}^{m}(x); 1{m−1 \atop m}(s{m \atop m−1}(x)); j{m \atop m−1}(x)o{m \atop m−1}x; α; 1{m−1 \atop m}(t{m \atop m−1}(x)); xo{m \atop m−1}j{m \atop m−1}(x); β; x; j_{m−1}^{m}(x)]

such that α and β are (∘_{m}^{m+1}, q − 1)-equivalences. By convention, for each 0 ≤ s < r, (∘_{s}^{r}, 0)-equivalences are ∘_{s}^{r}-isomorphisms.

This definition gives also an intuition of what could be an (∘_{m−1}^{m}, q)-equivalence when q is taken to the infinite.

Also for any regular (∞, n)-structure R, for each n ∈ ℕ, and for each integer m such that 0 ≤ n < m, the m-cells of a 𝕋^{(∞,n,R)}-algebras are (∘_{m−1}^{m}, q)-equivalences where q could be taken to the infinite.

Let us finish by a fact that we believe is important : A remarkable regular (∞, n)- structure called the minimal (∞, n)-structure R_{min}, is to consider for each m > n, s = 1 and :

n ≤ m_{1} = m − 1.

and this (∞, n)-structure and the maximal (∞, n)-structure R_{max} are both extrema in the sense of the evident following proposition :

4.6. Proposition. For any regular (∞, n)-structure R we have the following functorial inclusions

\xymatrix{𝔸lg(𝕋^{(∞,n,R_{max})}) \ar@{^{(}->}[r] & 𝔸lg(𝕋^{(∞,n,R)} \ar@{^{(}->}[r] & 𝔸lg(𝕋^{(∞,n,R_{min})})}

This proposition raises the question that the collection of regular (∞, n)-structures might form a lattice. Other members of the collection, apart from the minimal and maximal, may provide useful models of weak (∞, n)-categories.

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Department of Mathematics, Macquarie University North Ryde, NSW 2109 Australia.

Email: camell.kachour@gmail.com
This article may be accessed at http://www.tac.mta.ca/tac/

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

Subscription information Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Information for authors The typesetting language of the journal is TEX, and L^{A}TEX2e strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files at http://www.tac.mta.ca/tac/.

Managing editor. Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca

TEXnical editor. Michael Barr, McGill University: barr@math.mcgill.ca

Assistant TEX editor. Gavin Seal, Ecole Polytechnique Fédérale de Lausanne:

gavin seal@fastmail.fm

Transmitting editors.

Clemens Berger, Université de Nice-Sophia Antipolis: cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown(at)btinternet.com

Valeria de Paiva: valeria.depaiva@gmail.com

Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne: kathryn.hess@epfl.ch Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, Macquarie University: steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Tom Leinster, University of Edinburgh: Tom.Leinster@ed.ac.uk Ieke Moerdijk, Radboud University Nijmegen: i.moerdijk@math.ru.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@disi.unige.it Alex Simpson, University of Edinburgh: Alex.Simpson@ed.ac.uk James Stasheff, University of North Carolina: jds@math.upenn.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca