# Groupoidal and truncated $n$-quasi-categories

Victor Brittes

(Date: 03 June 2022)

###### Abstract.

We define groupoidal and $(n+k)$-truncated $n$-quasi-categories, which are the
translation to the world of $n$-quasi-categories of groupoidal and truncated
$(\infty,n)$-$\Theta$-spaces defined by Rezk. We show that these objects are
the fibrant objects of model structures on the category of presheaves on
$\Theta_{n}$ obtained by localisation of Ara’s model structure for $n$-quasi-
categories. Furthermore, we prove that the inclusion $\Delta\to\Theta_{n}$
induces a Quillen equivalence between the model structure for groupoidal
(resp. and $n$-truncated) $n$-quasi-categories and the Kan-Quillen model
structure for spaces (resp. homotopy $n$-types) on simplicial sets. To get to
these results, we also construct a cylinder object for $n$-quasi-categories.

###### Key words and phrases:

n-quasi-categories, homotopy n-types, $\Theta_{n}$-sets

###### 2020 Mathematics Subject Classification:

18N20, 18N40, 18N55, 18N65, 55P15

###### Contents

  1. 1 Localisation of model category structures
  2. 2 Ara’s $n$-quasi-categories and Rezk’s $(\infty,n)$-$\Theta$-spaces
  3. 3 A cylinder for $n$-quasi-categories
  4. 4 Truncated objects
  5. 5 Groupoidal objects
  6. 6 $n$-quasi-groupoids vs spaces

## Introduction

There are several models for $(\infty,n)$-categories (see [Ber20] for a
survey). Two closely related models are $(\infty,n)$-$\Theta$-spaces and
$n$-quasi-categories, defined in [Rez10] and [Ara14], respectively. Ara
presents in [Ara14] two different Quillen equivalences

$p^{*}:\operatorname{\Theta_{n}Sp}\rightleftarrows\operatorname{n\mathchar
45\relax QCat}:i_{0}^{*}$

and

$t_{!}:\operatorname{n\mathchar 45\relax
QCat}\rightleftarrows\operatorname{\Theta_{n}Sp}:t^{!}$

between both models.

One could say that Rezk’s model is more homotopical or topological, since the
objects of the underlying category are simplicial presheaves, and many of the
constructions and definitions of this model use the Kan-Quillen model
structure on simplicial sets. Indeed, the family of theories of
$(\infty,n)$-$\Theta$-spaces starts at $n=0$, which corresponds exactly to the
classical homotopy theory of simplicial sets. On the other hand, Ara’s model
is more combinatorial, since it deals with presheaves of sets over a given
category. The family of theories of $n$-quasi-categories starts at $n=1$,
where we find the theory of (1-)quasi-categories. Quasi-categories are one of
the main models of $(\infty,1)$-categories, successfully and extensively
developed in [Lur09] and [Joy08b], for example. Thus, it is expected that
$n$-quasi-categories can provide a useful model for $(\infty,n)$-categories as
well.

The main goal of this work is to show how some ideas developed by Rezk in the
context of $(\infty,n)$-$\Theta$-spaces, such as that of truncated and
groupoidal objects, can be transferred to the world of $n$-quasi-categories.
The interest of having such definitions for $n$-quasi-categories is that, due
to its more set-theorical and minimal nature, it can be easier to construct
comparisons between them and more algebraic models of higher categories. For
example, we expect to be able to compare $n$-truncated $n$-quasi-categories
with some notion of semi-strict $n$-category, like what is done for $n=1$ and
$n=2$ in [CL20] and [Cam20], respectively, and later compare $n$-truncated
$n$-quasi-groupoids with semi-strict $n$-groupoids, like what is done in
[Bri22] in the case $n=2$. Of course, for $n=1,2$, semi-strict $n$-categories
are just strict $n$-categories, but for $n\geq 3$ it is known that the strict
versions are not enough.

Let us now explain more precisely what are Rezk’s results that we aim to
transfer to Ara’s model. Firstly, it is shown in [Rez10] that there is a
Bousfield localisation $\operatorname{\Theta_{n}Sp_{k}}$ of
$\operatorname{\Theta_{n}Sp}$ whose fibrant objects are
$(n+k,n)$-$\Theta$-spaces (a truncated version of
$(\infty,n)$-$\Theta$-spaces). We define $(n+k)$-truncated $n$-quasi-
categories, generalizing the inductive definition given in [Cam20] for the
case $n=2,k=0$ (the case $n=1$ was defined in [Joy08a] and further studied in
[CL20]). We show that $(n+k)$-truncated $n$-categories are fibrant objects for
a localisation of $\operatorname{n\mathchar 45\relax QCat}$, and that both
Ara’s Quillen equivalences presented above descend into the level of
$(n+k)$-truncated objects.

Also in [Rez10], there is another localisation $\operatorname{\Theta_{n}Gpd}$
of $\operatorname{\Theta_{n}Sp}$, whose homotopy category is equivalent to the
homotopy category of spaces. We define groupoidal $n$-quasi-categories,
generalising the definition given in [Bri22] for the case $n=2$. We show that
these are the fibrant objects of a localisation $\operatorname{n\mathchar
45\relax QGpd}$ of $\operatorname{n\mathchar 45\relax QCat}$, and that
$\operatorname{n\mathchar 45\relax QGpd}$ and $\operatorname{\Theta_{n}Gpd}$
are Quillen equivalent via both adjunctions above.

A last result of [Rez10] provides a direct equivalence comparing
$\operatorname{\Theta_{n}Gpd}$ and $\operatorname{Sp}$ (the Kan-Quillen model
category on simplicial sets), and similarly for their truncated versions.
Therefore, we could use the results of the previous paragraphs to provide
comparisons between $\operatorname{n\mathchar 45\relax QGpd}$ and
$\operatorname{Sp}$. However, the inclusion $i:\Delta\to\Theta_{n}$ allows us
to also provide a different, new Quillen equivalence between
$\operatorname{n\mathchar 45\relax QGpd}$ and $\operatorname{Sp}$, which does
not factor through $\operatorname{\Theta_{n}Gpd}$. Moreover, we show that this
Quillen equivalence induces another one between $\operatorname{n\mathchar
45\relax QGpd_{k}}$ and $\operatorname{Sp_{n+k}}$ (the model structure for
homotopy $(n+k)$-types).

We would like to highlight the main differences in the techniques used in this
work and in [Bri22], which addressed similar questions for the case $n=2$.
Both repeatedly use the theory of localisation of model structures. However,
some results for $n=2$, established in [Cam20] and used in [Bri22], do not
easily generalize to the general case. We give 3 examples.

The first one concerns the way we show that $n$-quasi-groupoids are equivalent
to spaces via an adjunction induced by the inclusion $i:\Delta\to\Theta_{n}$
(Theorem 6.10 of this article for general $n$ and [Bri22, Theorem 3.14] for
$n=2$) . In the case of 2-quasi-groupoids, we use a Quillen equivalence proven
by Campbell between quasi-categories and 2-quasi-categories which are locally
Kan complexes, and then proceed by localisation. Here, the strategy is to fit
the adjunction we want to show is a Quillen equivalence in a triangle of
Quillen adjunctions, where we use Rezk’s $(\infty,n)$-$\Theta$-groupoids to
construct one of its edges.

Secondly, the comparison between 2-truncated 2-quasi-groupoids and homotopy
2-types [Bri22, Corollary 6.3] uses Campbell’s nerve for 2-categories [Cam20]
and Moerdijk-Svensson’s equivalence between 2-groupoids and homotopy 2-types
[MS93]. To compare $(n+k)$-truncated $n$-quasi-groupoids and homotopy
$(n+k)$-types (Theorem 6.17), we show that two (a priori different) model
structures are the same: the one for $(n+k)$-truncated $n$-quasi-groupoids
($\operatorname{n\mathchar 45\relax QGpd_{k}}$) and a localisation of
$\operatorname{n\mathchar 45\relax QGpd}$, deliberately built to be equivalent
to the one for $(n+k)$-homotopy types. The key idea to do so is to compare the
boundaries of representable $\Theta_{n}$-sets, which can be interpreted in the
model structure $\operatorname{n\mathchar 45\relax QGpd}$ as model for
spheres. In the Kan-Quillen model structure on simplicial sets, for every
$k>0$, we can think of $\partial\Delta[k]$ as a model for the sphere
$\mathbb{S}^{k-1}$, which can be made precise by using the realisation functor
$\operatorname{\mathbf{PSh}(\Delta)}\to\operatorname{\mathbf{Top}}$, or by
giving an abstract definition of sphere as a given homotopy pushout, which is
the approach we use here. When working with presheaves on $\Theta_{n}$, there
can be more than one representable such that its boundary models a given
sphere. For example, in $\Theta_{2}$, the boundaries of both $[2]([0],[0])$
and $[1]([1])$ (respectively represented below) are models for
$\mathbb{S}^{1}$.

${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$

The third difference is in the proof that the suspension functor
$\Sigma:\operatorname{\mathbf{PSh}}(\Theta_{n})\to*\sqcup*/\operatorname{\mathbf{PSh}}(\Theta_{n+1})$
is a left Quillen functor. When $n=1$, this is proven in [Cam20], using a
simple recognition result for left Quillen functors from the model category
for quasi-categories. This recognition result generalises to $n$-quasi-
categories (Proposition 2.14), with the addition of one extra condition to be
verified, involving a certain cylinder object. The problem is that we do not
know a simple combinatorial description for the "obvious" choice of cylinder
for $n$-quasi-categories (the product with an interval object). The solution
we found is to build a smaller, more combinatorial cylinder, in order to use
such recognition principle. This construction, therefore fundamental to the
proof of this result, is also of independent interest for other applications
in the theory of $n$-quasi-categories. With this new cylinder, we are able to
get a refined version of Proposition 2.14, stated in Theorem 3.19.

The following table summarizes all the model categories we work with in this
article, the respective fibrant-cofibrant objects, and how each one of them
fits in the general framework of $(r,s)$-categories: higher (weak) categories
where all $k$-morphisms are (weakly) invertible for $k>s$ and (weakly) trivial
for $k>r$.

Category | Model structure | Fibrant objects |  $(r,s)$-category
---|---|---|---
$\operatorname{\mathbf{PSh}(\Delta)}$ | $\operatorname{Sp}$ | Kan complexes | $(\infty,0)$
$\operatorname{Sp_{m}}$ | Homotopy $m$-types | $(m,0)$
$\operatorname{\mathbf{PSh}(\Theta_{n})}$ | $\operatorname{n\mathchar 45\relax QCat}$ |  $n$-quasi-categories | $(\infty,n)$
$\operatorname{n\mathchar 45\relax QGpd}$ |  $n$-quasi-groupoids | $(\infty,0)$
$\operatorname{n\mathchar 45\relax QCat_{k}}$ |  $(n+k)$-truncated $n$-quasi-categories | $(n+k,n)$
| $\operatorname{n\mathchar 45\relax QGpd_{k}}$ |  $(n+k)$-truncated $n$-quasi-groupoids | $(n+k,0)$
$\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}$ | $\operatorname{\Theta_{n}Sp}$ |  $(\infty,n)$-$\Theta$-categories | $(\infty,n)$
$\operatorname{\Theta_{n}Gpd}$ |  $(\infty,n)$-$\Theta$-groupoids | $(\infty,0)$
$\operatorname{\Theta_{n}Sp_{k}}$ |  $(n+k,n)$-$\Theta$-categories | $(n+k,n)$
| $\operatorname{\Theta_{n}Gpd_{k}}$ |  $(n+k,n)$-$\Theta$-groupoids | $(n+k,0)$

Organisation of the paper. In the first preliminary section, we recall some
aspects of the theory of localisation of model structures, our main technical
tool. In section 2, we briefly recall the definition of the category
$\Theta_{n}$, Ara’s and Rezk’s model structures for $(\infty,n)$-categories,
and Ara’s Quillen equivalences between them. The third is devoted to the
construction of an alternative cylinder object for $n$-quasi-categories, that
will be used in the next section to show that the suspension functor between
$\Theta_{n}$-sets and bipointed $\Theta_{n+1}$-sets is left Quillen. Its right
adjoint, the functor $\operatorname{Hom}$, plays a fundamental role in the
definitions of truncated $n$-quasi-categories, presented in section 4, and of
groupoidal $n$-quasi-categories, in section 5. Besides the definitions, these
two sections also contain theorems stating the existence, for each version of
$n$-quasi-category (truncated and groupoidal), of model structures which are
Quillen equivalent to Rezk’s ones. The last section provides a third, direct,
comparison between $n$-quasi-groupoids and spaces, and its truncated version,
which uses the sphere argument mentioned above

Acknowledgement. We would like to thank Muriel Livernet and Clemens Berger for
the helpful and insightful conversations. We would also like to thank Felix
Loubaton for the idea for the construction of the cylinder of Section 3.

Notation. If $\mathcal{A}$ is a small category, we denote by
$\widehat{\mathcal{A}}$ the category of presheaves of sets on $\mathcal{A}$,
i.e., of functors
$\mathcal{A}^{\operatorname{op}}\to\operatorname{\mathbf{Set}}$. If $X$ is an
object of $\widehat{\mathcal{A}}$ and $a$ is an object of $\mathcal{A}$, we
write $X_{a}$ for the set $X(a)$. The Yoneda embedding
$a\mapsto\mathcal{A}[a]:=\operatorname{Hom}_{\mathcal{A}}(-,a)$ defines a
fully faithful functor $\mathcal{A}\to\widehat{\mathcal{A}}$. If $f:a\to b$ is
a morphism in $\mathcal{A}$, we still denote by
$f:\mathcal{A}[a]\to\mathcal{A}[b]$ its image under the Yoneda embedding.
Sometimes, when there is no risk of confusion, we also denote by $a$ the image
of an object $a\in\mathcal{A}$ by the Yoneda embedding.

When representing an adjunction by
$F:\mathcal{C}\rightleftarrows\mathcal{D}:G$, the functor $F$ is left adjoint
to the functor $G$.

If $\mathcal{A}$ and $\mathcal{B}$ are two small categories and
$\varphi:\mathcal{A}\to\mathcal{B}$ is a functor, we denote by
$\varphi^{*}:\widehat{\mathcal{B}}\to\widehat{\mathcal{A}}$ the restriction
functor given by precomposition with (the opposite of) $\varphi$. Given an
adjunction $\varphi:\mathcal{A}\rightleftarrows\mathcal{B}:\psi$, considering
restriction functors induces an adjunction
$\varphi^{*}:\widehat{\mathcal{B}}\rightleftarrows\widehat{\mathcal{A}}:\psi^{*}$
between presheaf categories.

If $\mathcal{A}$ is a small category and $\mathcal{C}$ is a cocomplete
category, a functor $f:\mathcal{A}\to\mathcal{C}$ induces an adjunction that
we will denote by
$f_{!}:\widehat{\mathcal{A}}\rightleftarrows\mathcal{C}:f^{!}$, where $f_{!}$
is the left Kan extension of $f$ along the Yoneda embedding, and $f^{!}$ is
the nerve functor defined by
$f^{!}(X)_{a}=\operatorname{Hom}_{\mathcal{C}}(f(a),X)$ for $X\in\mathcal{C}$
and $a\in\mathcal{A}$. In the case where $\mathcal{C}=\widehat{\mathcal{B}}$
for some small category $\mathcal{B}$ and
$f:\mathcal{A}\to\widehat{\mathcal{B}}$ is the composite of a functor
$\varphi:\mathcal{A}\to\mathcal{B}$ with the Yoneda embedding, we abuse
notation and denote by
$\varphi_{!}:\widehat{\mathcal{A}}\rightleftarrows\widehat{\mathcal{B}}:\varphi^{!}$
the induced adjunction. Note that the Yoneda lemma implies that $\varphi^{!}$
is isomorphic to the restriction functor $\varphi^{*}$.

##  1\. Localisation of model category structures

We recall some notions and results about the localisation of model categories.
A complete reference is [Hir03].

######  1.1.

Let $(\mathcal{M},\mathsf{Cof},\mathsf{W},\mathsf{Fib})$ be a model category
structure on a category $\mathcal{M}$. A model category structure
$(\mathcal{M},\mathsf{Cof}_{\operatorname{loc}},\mathsf{W}_{\operatorname{loc}},\mathsf{Fib}_{\operatorname{loc}})$
on $\mathcal{M}$ is a (left) Bousfield localisation of
$(\mathcal{M},\mathsf{Cof},\mathsf{W},\mathsf{Fib})$ if
$\mathsf{Cof}_{\operatorname{loc}}=\mathsf{Cof}$ and
$\mathsf{W}\subset\mathsf{W}_{\operatorname{loc}}$. When studying a model
category structure and a given localisation, we shall write $\mathcal{M}$ and
$\mathcal{M}_{\operatorname{loc}}$ to refer to the original model structure
and to its localisation, respectively. We shall also call local fibration
(resp. local fibrant object, resp. local weak equivalence) a fibration (resp.
fibrant object, resp. weak equivalence) of $\mathcal{M}_{\operatorname{loc}}$.
Note that we have a Quillen adjunction

$\operatorname{id}:\mathcal{M}\rightleftarrows\mathcal{M}_{\operatorname{loc}}:\operatorname{id}$

We see that a Bousfield localisation is completely determined by its fibrant
objects, i.e., the local fibrant objects. It is useful to know that a morphism
between local fibrant objects is a weak equivalence (resp. fibration) in
$\mathcal{M}$ if and only if it is a local weak equivalence (resp. local
fibration).

######  1.2.

Given a model category $\mathcal{M}$ and two objects $X,Y$ of $\mathcal{M}$,
we can consider the homotopy mapping space
$\underline{\operatorname{Ho}\mathcal{M}}(X,Y)$, which is the image of the
pair $(X,Y)$ by the functor
$\underline{\operatorname{Ho}\mathcal{M}}:\operatorname{Ho}(\mathcal{M})^{\operatorname{op}}\times\operatorname{Ho}(\mathcal{M})\to\operatorname{Ho}(\widehat{\Delta})$111When
writing $\operatorname{Ho}(\widehat{\Delta})$, we always consider the Kan-
Quillen model structure on the category of simplicial sets induced by
$\operatorname{Hom}_{\mathcal{M}}:\mathcal{M}^{\operatorname{op}}\times\mathcal{M}\to\operatorname{\mathbf{Set}}$
(see [Ara14, A.2] or [Hir03] for the details).

Let $S$ be a class of morphisms of $\mathcal{M}$. An object $X$ of
$\mathcal{M}$ is $S$-local (or local with respect to $S$) if for every
morphism $f:A\to B$ of $S$, the induced map

$\underline{\operatorname{Ho}\mathcal{M}}(f,X):\underline{\operatorname{Ho}\mathcal{M}}(B,X)\to\underline{\operatorname{Ho}\mathcal{M}}(A,X)$

is an isomorphism (in the homotopy category
$\operatorname{Ho}(\widehat{\Delta})$).

A morphism $f:A\to B$ of $\mathcal{M}$ is an $S$-equivalence if for every
$S$-local object $X$, the induced map

$\underline{\operatorname{Ho}\mathcal{M}}(f,X):\underline{\operatorname{Ho}\mathcal{M}}(B,X)\to\underline{\operatorname{Ho}\mathcal{M}}(A,X)$

is an isomorphism (in the homotopy category
$\operatorname{Ho}(\widehat{\Delta})$).

If there is a Bousfield localisation $\mathcal{M}_{\operatorname{loc}}$ of
$\mathcal{M}$ whose local fibrant objects are the $S$-local objects and whose
weak equivalences are the $S$-equivalences, we say that
$\mathcal{M}_{\operatorname{loc}}$ is a (Bousfield) localisation of
$\mathcal{M}$ with respect to $S$, and we denote it by $L_{S}\mathcal{M}$.

######  Example 1.3.

The Kan-Quillen model category structure on simplicial sets is a localisation
of Joyal’s model structure with respect to the morphism
$\Delta[1]\to\Delta[0]$, cf. [CL20, Proposition 3.30].

We will state a result, due to Smith, about the existence of the localisation
of a model category with respect to a certain set of morphisms. Before, let us
recall some definitions. A model category is left proper if the pushout of
every weak equivalence along a cofibration is a weak equivalence. A model
category where all objects are cofibrant is left proper (see [Hir03, Corollary
13.1.3]). A model category is combinatorial if it is cofibrantly generated and
locally presentable.

######  Theorem 1.4.

Let $\mathcal{M}$ be a left proper and combinatorial model category. Let $S$
be a set of morphisms of $\mathcal{M}$. Then the localisation
$L_{S}\mathcal{M}$ of $\mathcal{M}$ with respect to $S$ exists and is left
proper and combinatorial.

###### Proof.

See [Bar10, Theorem 4.7]. ∎

######  Remark 1.5.

If $F:\mathcal{M}\rightleftarrows\mathcal{N}:G$ is a Quillen adjunction
between model categories, the induced adjunction between the homotopy
categories is usually denoted by
$\mathbb{L}F:\operatorname{Ho}(\mathcal{M})\rightleftarrows\operatorname{Ho}(\mathcal{N}):\mathbb{R}G$.
In what follows, we will abuse language and also denote by $\mathbb{L}F$ the
functor $\mathbb{L}F:=FQ:\mathcal{M}\to\mathcal{N}$, where $Q$ is a fixed
functorial cofibrant replacement in the model category $\mathcal{M}$. In all
the applications presented in this paper, all objects of $\mathcal{M}$ will be
cofibrant, and so we shall take $Q=\operatorname{id}_{\mathcal{M}}$.

We can transfer localisations of model structures along Quillen adjunctions.

######  Proposition 1.6.

Let $F:\mathcal{M}\rightleftarrows\mathcal{N}:G$ be a Quillen adjunction
between model categories $\mathcal{M}$ and $\mathcal{N}$. Let $S$ be a class
of morphisms of $\mathcal{M}$. A fibrant object $Y$ of $\mathcal{N}$ is
$\mathbb{L}F(S)$-local if and only if $G(Y)$ is $S$-local.

###### Proof.

See [Hir03, Proposition 3.1.12]. ∎

######  Theorem 1.7.

Let $F:\mathcal{M}\rightleftarrows\mathcal{N}:G$ be a Quillen adjunction
between model categories $\mathcal{M}$ and $\mathcal{N}$. Let $S$ be a class
of morphisms of $\mathcal{M}$. If the localisations $L_{S}\mathcal{M}$ and
$L_{\mathbb{L}F(S)}\mathcal{N}$ exist, then

$F:L_{S}\mathcal{M}\rightleftarrows L_{\mathbb{L}F(S)}\mathcal{N}:G$

is a Quillen adjunction between the localised model categories.

Moreover, if $F:\mathcal{M}\rightleftarrows\mathcal{N}:G$ is a Quillen
equivalence, then so is $F:L_{S}\mathcal{M}\rightleftarrows
L_{\mathbb{L}F(S)}\mathcal{N}:G$.

###### Proof.

See [Hir03, Proposition 3.3.20] ∎

The two following theorems (recalled from the appendix of [CL20]) present ways
to know when Quillen adjunctions and equivalences are preserved after
localisation.

######  Theorem 1.8.

Let $F:\mathcal{M}\rightleftarrows\mathcal{N}:G$ be a Quillen adjunction
between model categories $\mathcal{M}$ and $\mathcal{N}$. Let
$\mathcal{M}_{\operatorname{loc}}$ be a localisation of $\mathcal{M}$. The
adjunction $F:\mathcal{M}_{\operatorname{loc}}\rightleftarrows\mathcal{N}:G$
is Quillen if and only if for every fibrant every object $Y$ of $\mathcal{N}$,
its image $G(Y)$ is fibrant in $\mathcal{M}_{\operatorname{loc}}$.

###### Proof.

See [CL20, Proposition A.13]. ∎

######  Theorem 1.9.

Let $F:\mathcal{M}\rightleftarrows\mathcal{N}:G$ be a Quillen equivalence
between model categories $\mathcal{M}$ and $\mathcal{N}$. Let
$\mathcal{M}_{\operatorname{loc}}$ and $\mathcal{N}_{\operatorname{loc}}$ be
localisations of these model structures. We have a Quillen equivalence
$F:\mathcal{M}_{\operatorname{loc}}\rightleftarrows\mathcal{N}_{\operatorname{loc}}:G$
if and only if a fibrant object $Y$ of $\mathcal{N}$ is fibrant in
$\mathcal{N}_{\operatorname{loc}}$ precisely when $G(Y)$ is fibrant in
$\mathcal{M}_{\operatorname{loc}}$.

###### Proof.

See [CL20, Theorem A.15]. ∎

Next, we state a proposition which will allow us to understand successive
localisations of a model category.

######  Lemma 1.10.

Let $\mathcal{M}$ be a model category and $\mathcal{M}_{\operatorname{loc}}$
be a Bousfield localisation of $\mathcal{M}$. For every object $X$ and every
local fibrant object $Y$ of $\mathcal{M}$, the homotopy mapping spaces
$\underline{\operatorname{Ho}\mathcal{M}}(X,Y)$ and
$\underline{\operatorname{Ho}\mathcal{M}_{\operatorname{loc}}}(X,Y)$ are
naturally isomorphic in $\operatorname{Ho}(\widehat{\Delta})$.

###### Proof.

See [Ara14, Lemma A.4] ∎

######  Proposition 1.11.

Let $\mathcal{M}$ be a model category and $S,T$ be two classes of morphisms of
$\mathcal{M}$. Suppose that the localisations $L_{S}\mathcal{M}$,
$L_{T}\mathcal{M}$, $L_{T}L_{S}\mathcal{M}$, $L_{S}L_{T}\mathcal{M}$ and
$L_{S\cup T}\mathcal{M}$ exist. Then the model categories
$L_{T}L_{S}\mathcal{M}$, $L_{S}L_{T}\mathcal{M}$ and $L_{S\cup T}\mathcal{M}$
are the same.

###### Proof.

Since a model structure is completely determined by its cofibrations and
fibrant objects (cf. [Joy08b, Proposition E.1.10]) and the 3 considered model
structures have the same cofibrations, it is sufficient to show that they have
the same fibrant objects. Let $X$ be an object of $\mathcal{M}$. We claim that
the following assertions are equivalent:

  1. (1)

$X$ is a $T$-local object of $L_{S}\mathcal{M}$

  2. (2)

$X$ is an $S$-local object of $L_{T}\mathcal{M}$

  3. (3)

$X$ is an $(S\cup T)$-local objects of $\mathcal{M}$

We will show that $(1)\Leftrightarrow(3)$. The equivalence
$(2)\Leftrightarrow(3)$ follows by exchanging the roles of $S$ and $T$.

$(1)\Rightarrow(3)$ Suppose that $X$ is a $T$-local object of
$L_{S}\mathcal{M}$. We have to show that, for every $f\in S\cup T$, $f:A\to
B$, the map

$\underline{\operatorname{Ho}\mathcal{M}}(f,X):\underline{\operatorname{Ho}\mathcal{M}}(B,X)\to\underline{\operatorname{Ho}\mathcal{M}}(A,X)$

is an isomorphism. This is true if $f\in S$, since $X$ is fibrant in
$L_{T}L_{S}\mathcal{M}$, so it is in particular fibrant in $L_{S}\mathcal{M}$,
which means it is $S$-local in $\mathcal{M}$. If $f\in T$, we consider the
commutative square

${{\underline{\text{Ho}\mathcal{M}}(B,X)}}$${{\underline{\text{Ho}\mathcal{M}}(A,X)}}$${{\underline{\text{Ho}L_{S}\mathcal{M}}(B,X)}}$${{\underline{\text{Ho}L_{S}\mathcal{M}}(A,X)}}$$\scriptstyle{\underline{\text{Ho}\mathcal{M}}(f,X)}$$\scriptstyle{\underline{\text{Ho}L_{S}\mathcal{M}}(f,X)}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

where the isomorphisms are given by Lemma 1.10. The bottom arrow is an
isomorphism, since $X$ is $T$-local in $L_{S}\mathcal{M}$ by assumption, and
thus the top arrow is also an isomorphism, as desired.

$(3)\Rightarrow(1)$ Let $X$ be a $(S\cup T)$-local object of $\mathcal{M}$.
Let $f:A\to B$ be a morphism in the class $T$. We have to show that the map

$\underline{\operatorname{Ho}L_{S}\mathcal{M}}(f,X):\underline{\operatorname{Ho}L_{S}\mathcal{M}}(B,X)\to\underline{\operatorname{Ho}L_{S}\mathcal{M}}(A,X)$

is an isomorphism. Since $X$ is $(S\cup T)$-local in $\mathcal{M}$, it is in
particular $S$-local in $\mathcal{M}$, and so it is a fibrant object of
$L_{S}\mathcal{M}$. Therefore, we can use Lemma 1.10 to consider a commutative
square as above. The top arrow is an isomorphism since $X$ is $T$-local in
$\mathcal{M}$, which implies that the bottom arrow is also an isomorphism. ∎

We end this section recalling some results about model category structures on
slice categories.

######  1.12.

If $\mathcal{M}$ is a category and $C$ is an object of $\mathcal{M}$, we
denote by $C/\mathcal{M}$ the slice category of objects under $C$. The objects
of $C/\mathcal{M}$ are pairs $(A,a)$, where $A$ is an object of $\mathcal{M}$
and $a:C\to A$ is a morphism of $\mathcal{M}$. A morphism $f:(A,a)\to(B,b)$ in
$C/\mathcal{M}$ is a morphism $f:A\to B$ of $\mathcal{M}$ such that $fa=b$.
There is an obvious forgetful functor $C/\mathcal{M}\to\mathcal{M}$ taking
$(A,a)$ to $A$.

When $\mathcal{M}$ is endowed with a model category structure, there is an
induced model structure on $C/\mathcal{M}$ where a morphism is a weak
equivalence (resp. cofibration, resp. fibration) when its image by the
forgetful functor is a weak equivalence (resp. cofibration, resp. fibration)
of $\mathcal{M}$ [Hir03, Theorem 7.6.5].

######  Lemma 1.13.

Let $\mathcal{M}$ be a model category, $C$ be a cofibrant object of
$\mathcal{M}$ and $f:(A,a)\to(B,b)$ be a morphism in $C/\mathcal{M}$. A
fibrant object $X$ of $\mathcal{M}$ is local with respect to $f$ in
$\mathcal{M}$ if and only if for every $x:C\to X$, the object $(X,x)$ of
$C/\mathcal{M}$ is local with respect to $f$ in $C/\mathcal{M}$.

###### Proof.

Exercise in [Cam20, §8.1]. ∎

##  2\. Ara’s $n$-quasi-categories and Rezk’s $(\infty,n)$-$\Theta$-spaces

Let $n\geq 0$. We recall the recursive definition of the category
$\Theta_{n}$, using the wreath product introduced in [Ber07].

######  2.1.

Let $\mathcal{C}$ be a category. The wreath product $\Delta\wr\mathcal{C}$ is
the category described as follows. The objects are lists
$[p](x_{1},\ldots,x_{p})$, where $p\geq 0$ and $x_{1},\ldots,x_{p}$ are
objects of $\mathcal{C}$. A morphism
$[f](\alpha):[p](x_{1},\ldots,x_{p})\to[q](y_{1},\ldots,y_{q})$ is the data of
a morphism $f:[p]\to[q]$ in $\Delta$ and of morphisms $\alpha^{i}_{j}:x_{i}\to
y_{j}$ in $\mathcal{C}$ for every $i,j$ such that $f(i-1)<j\leq f(i)$.

######  2.2.

There is a suspension functor $\sigma:\mathcal{C}\to\Delta\wr\mathcal{C}$,
which sends an object $x$ to $[1](x)$ and a morphism $f:x\to y$ to
$[\operatorname{id}_{1}](f)$. If the category $\mathcal{C}$ has a terminal
object $t$, there is an inclusion functor $i:\Delta\to\Delta\wr\mathcal{C}$,
defined on objects by $[p]\mapsto[p](t,\ldots,t)$. The inclusion functor $i$
has a left adjoint, given by the truncation
$\pi:\Delta\wr\mathcal{C}\to\Delta$ sending $[p](x_{1},\ldots,x_{p})$ to
$[p]$.

######  2.3.

Let $\Theta_{0}$ be the terminal category. We recursively define $\Theta_{n}$
for $n>0$ by letting

$\Theta_{n}:=\Delta\wr\Theta_{n-1}$

We note that $\Theta_{1}$ is exactly $\Delta$. Moreover, by the previous
paragraph we have a suspension

$\sigma:\Theta_{n}\to\Theta_{n+1}$

and an inclusion (since $\Theta_{n-1}$ has a terminal object $[0]$)

$i:\Delta\to\Theta_{n}$

for every $n\geq 0$.

######  2.4.

For every $n\geq 0$, there is fully faithful inclusion
$\Theta_{n}\to\operatorname{n\mathbf{\mathchar 45\relax Cat}}$. Conceptually,
we can see $\Theta_{0}$ as the full subcategory of $0\mathchar
45\relax\operatorname{\mathbf{Cat}}:=\operatorname{\mathbf{Set}}$ formed the
singleton. Reasoning inductively and using the fact that the wreath product
preserves fully faithfulness, we can see $\Theta_{n}=\Delta\wr\Theta_{n-1}$ as
a full subcategory of $\Delta\wr(n-1)\mathchar
45\relax\operatorname{\mathbf{Cat}}$, which is a full subcategory of
$\operatorname{n\mathbf{\mathchar 45\relax Cat}}$. From now on, we may
identify $\Theta_{n}$ with a full subcategory of
$\operatorname{n\mathbf{\mathchar 45\relax Cat}}$.

More explicitly, we picture an object $[p](\theta_{1},\ldots,\theta_{p})$ of
$\Theta_{n}$ as the $n$-category freely generated by the ($(n-1)\mathchar
45\relax\operatorname{\mathbf{Cat}}$)-graph with objects $0,1,\ldots p$ and
hom-$(n-1)$-categories $\operatorname{Hom}(i-1,i)=\theta_{i}$ for $1\leq i\leq
p$.

If $\mathcal{C}$ is an $n$-category, the suspension $\sigma(\mathcal{C})$ is
the $(n+1)$-category with two objects $0$ and $1$, and with hom-$n$-categories
given by $\operatorname{Hom}(0,1)=\mathcal{C}$,
$\operatorname{Hom}(1,0)=\emptyset$,
$\operatorname{Hom}(0,0)=\operatorname{Hom}(1,1)=\\{*\\}$. The restriction of
the suspension functor $\sigma:\operatorname{n\mathbf{\mathchar 45\relax
Cat}}\to(n+1)\mathchar 45\relax\operatorname{\mathbf{Cat}}$ to $\Theta_{n}$
factors through $\Theta_{n+1}$ and is exactly the suspension defined in §2.3.

######  Example 2.5.

The object $\theta=[3](2,0,1)$ of $\Theta_{2}$ is the free 2-category
generated by the 2-graph

${0}$${1}$${2}$${3}$

Its suspension $\sigma(\theta)\in\Theta_{3}$ is the free 3-category generated
by the 3-graph (where the small horizontal arrows should be triple)

${0}$${1}$

######  2.6.

Of special interest among the objects of $\Theta_{n}$ are the globes
$D_{0},\ldots,D_{n}$. For $n=0$, the globe $D_{0}$ is the only object of
$\Theta_{0}$. For $n>0$, we have $D_{0}=[0]$ and $D_{k}=\sigma(D_{k-1})$ for
$1\leq k\leq n$.

For $1\leq k\leq n$, there are two morphisms $s,t:D_{k-1}\to D_{k}$ in
$\Theta_{n}$ corresponding to sending the non-identity $(k-1)$-cell of
$D_{k-1}$ to the source and target of the non-identity $k$-cell of $D_{k}$,
respectively.

On the other hand, the unique morphism $\tau^{1}_{1}:[1]\to[0]$ of
$\Delta=\Theta_{1}$ induces by suspensions morphisms $\tau^{n}_{k}:D_{k}\to
D_{k-1}$ in $\Theta_{n}$ for $2\leq k\leq n$. The unique morphism $D_{1}\to
D_{0}$ in $\Theta_{n}$ is denoted by $\tau^{n}_{1}$. In practice, the
$n$-functor $\tau^{n}_{k}:D_{k}\to D_{k-1}$ sends the two non-identity
$(k-1)$-cells of $D_{k}$ to the only non-identity $(k-1)$-cell of $D_{k-1}$.

######  Example 2.7.

The globes $D_{0},\ldots,D_{3}$ of $\Theta_{3}$ are pictured below

${{D_{0}}}$${{D_{1}}}$${{D_{2}}}$${{D_{3}}}$${0}$${0}$${1}$${0}$${1}$${0}$${1}$

######  2.8.

We denote by $\operatorname{\mathbf{PSh}(\Theta_{n})}$ the category of
presheaves on $\Theta_{n}$, i.e., of functors
$\Theta_{n}^{\operatorname{op}}\to\operatorname{\mathbf{Set}}$ and natural
transformations between them. The representable presheaves are denoted by
$\Theta_{n}[\theta]$, and by definition we have

$\Theta_{n}[\theta]_{\theta^{\prime}}=\operatorname{Hom}_{\Theta_{n}}(\theta^{\prime},\theta)$

for every $\theta,\theta^{\prime}\in\Theta_{n}$.

The boundary $\partial\Theta_{n}[\theta]$ is the presheaf generated by the
monomorphisms $\theta^{\prime}\to\theta$ which are not the identity. We denote
by $\delta_{\theta}:\partial\Theta_{n}[\theta]\to\Theta_{n}[\theta]$ the
boundary inclusion monomorphism.

######  2.9.

The inclusion $\Theta_{n}\to\operatorname{n\mathbf{\mathchar 45\relax Cat}}$
induces a fully faithful (cf. [Ber02, Theorem 1.12]) nerve functor

$N_{n}:\operatorname{n\mathbf{\mathchar 45\relax
Cat}}\to\operatorname{\mathbf{PSh}(\Theta_{n})}$

Explicitly, if $\mathcal{C}$ is an $n$-category and $\theta$ is an object of
$\Theta_{n}$, we have

$N_{n}(\mathcal{C})_{\theta}=\operatorname{Hom}_{\operatorname{n\mathbf{\mathchar
45\relax Cat}}}(\theta,\mathcal{C})$

######  2.10.

Ara defines in [Ara14] a model category structure $\operatorname{n\mathchar
45\relax QCat}$ on the category $\operatorname{\mathbf{PSh}(\Theta_{n})}$
which provides a model for $(\infty,n)$-categories. The fibrant objects of
$\operatorname{n\mathchar 45\relax QCat}$ are called $n$-quasi-categories.

We will state a recognition principle for left Quillen functors from
$\operatorname{n\mathchar 45\relax QCat}$, generalizing the one for the case
$n=2$ given by Campbell [Cam20, Proposition 4.13]. To this end, we introduce
three classes of morphisms of presheaves on $\Theta_{n}$.

######  2.11.

The first class, denoted by $\mathcal{I}_{n}$, is the class of spine
inclusions. For $n=1$, $[p]\in\Theta_{1}=\Delta$, the spine $I[p]$ is the
colimit of the image in $\operatorname{\mathbf{PSh}(\Delta)}$ of the following
diagram $\mathcal{D}([p])$ in $\Delta$

${{D_{1}}}$${\ldots}$${{D_{1}}}$${{D_{0}}}$${{D_{0}}}$$\scriptstyle{t}$$\scriptstyle{s}$$\scriptstyle{t}$$\scriptstyle{s}$

where there are $p$ copies of $D_{1}$. There is an obvious inclusions
$i_{p}:I[p]\to\Delta[p]$ induced by the universal property of the colimit.

For $n>1$, define the spines inductively. Consider an object
$\theta=[p](\theta_{1},\ldots,\theta_{p})$ of $\Theta_{n}$. The spine
$I[\theta_{i}]$ associated to $\theta_{i}$ is defined as the colimit of (the
image by the Yoneda embedding of) a diagram $\mathcal{D}(\theta_{i})$ in
$\Theta_{n-1}$. The spine $I[\theta]$ is then the colimit in
$\operatorname{\mathbf{PSh}(\Theta_{n})}$ of the diagram

${{\sigma(\mathcal{D}(\theta_{1}))}}$${\ldots}$${{\sigma(\mathcal{D}(\theta_{p}))}}$${{D_{0}}}$${{D_{0}}}$$\scriptstyle{t}$$\scriptstyle{s}$$\scriptstyle{t}$$\scriptstyle{s}$

where the target of the $t$’s (resp. $s$’s) are the rightmost (resp. leftmost)
globe appearing in $\mathcal{D}(\theta_{i})$. Once again, there are inclusions

$i_{\theta}:I[\theta]\to\Theta_{n}[\theta]$

The class $\mathcal{I}_{n}$ is formed by all the spine inclusions, i.e.,

$\mathcal{I}_{n}:=\\{i_{\theta}:I[\theta]\to\Theta_{n}[\theta],\theta\in\Theta_{n}\\}$

######  2.12.

The second class is the one of generating equivalences. Let $J=J_{1}$ be the
free groupoid generated by one arrow $0\to 1$. We define the $k$-category
$J_{k}$ as the suspension $\sigma(J_{k-1})$ for $k>1$. The functor $j_{1}:J\to
D_{0}$ induces functors

$j_{k}:J_{k}\to D_{k-1}$

for every $k\geq 1$. For $n\geq 1$, the class $\mathcal{J}_{n}$ is given by

$\mathcal{J}_{n}:=\\{N_{n}(j_{k}):N_{n}(J_{k})\to N_{n}(D_{k-1}),1\leq k\leq
n\\}$

######  2.13.

The third class is given by the following projections:

$\mathcal{P}_{n}:=\\{p_{2}:N_{n}(J)\times\Theta_{n}[\theta]\to\Theta_{n}[\theta],\theta\in\Theta_{n}\\}$

######  Proposition 2.14.

Let $\mathcal{M}$ be a model category. Let
$F:\operatorname{\mathbf{PSh}(\Theta_{n})}\to\mathcal{M}$ be a cocontinuous
functors sending monomorphisms to cofibrations. Then $F$ sends weak
equivalences of $\operatorname{n\mathchar 45\relax QCat}$ to weak equivalences
of $\mathcal{M}$ if and only if $F$ sends the morphisms in
$\mathcal{I}_{n}\cup\mathcal{J}_{n}\cup\mathcal{P}_{n}$ to weak equivalences.

###### Proof.

Suppose that $F$ sends weak equivalences to weak equivalences. By definition,
the morphisms of $\mathcal{I}_{n}$ and $\mathcal{J}_{n}$ are weak equivalences
of $\operatorname{n\mathchar 45\relax QCat}$, and thus they are sent to weak
equivalences. Since $N_{n}(J)\to N_{n}(D_{0})$ is a trivial fibration [Ara14,
Corollary 6.7], then for every $\theta\in\Theta_{n}$, the projection
$N_{n}(J)\times\Theta_{n}[\theta]\to\Theta_{n}[\theta]$ is also a trivial
fibration (as a pullback of a trivial fibration), and hence a weak
equivalence. Therefore, the morphisms of $\mathcal{P}_{n}$ are also sent to
weak equivalences.

To show the converse implication, we use the theory of localizers developed by
Cisinski in [Cis06]. Let $\mathsf{W}$ be the class of morphisms of
$\widehat{\Theta_{n}}$ which are sent to weak equivalences by $F$. We want to
show that $\mathsf{W}(\mathcal{I}_{n}\cup\mathcal{J}_{n})\subset\mathsf{W}$,
where $\mathsf{W}(\mathcal{I}_{n}\cup\mathcal{J}_{n})$ is the localizer
generated by $\mathcal{I}_{n}\cup\mathcal{J}_{n}$, which is by definition the
class of weak equivalences of $\operatorname{n\mathchar 45\relax QCat}$. For
that, it is sufficient to show that $\mathsf{W}$ is a localizer, since it
contains $\mathcal{I}_{n}\cup\mathcal{J}_{n}$ by hypothesis. This is done by
applying [Cis06, Proposition 8.2.15] to the pre-localizer [Cis06, Def. 8.2.10]
$\mathsf{W}$ and the catégorie squelettique régulière [Cis06, Definition
8.2.3] $\Theta_{n}$. The hypothesis of the cited proposition is precisely the
fact that $\mathcal{P}_{n}\subset\mathsf{W}$, since $N_{n}(J)\times\theta$ is
a donnée homotopique élémentaire (see Section 3 for more details on the
definition). ∎

######  2.15.

A simplicial presheaf on a small category $\mathcal{A}$ is a functor
$\mathcal{A}^{\operatorname{op}}\to\operatorname{\mathbf{PSh}(\Delta)}$. A
simplicial presheaf
$X:\mathcal{A}^{\operatorname{op}}\to\operatorname{\mathbf{PSh}(\Delta)}$
corresponds to a presheaf $\hat{X}$ on $\mathcal{A}\times\Delta$ by putting
$\hat{X}_{(a,[n])}=X(a)_{n}$, for every $a\in\mathcal{A}$ and $n\geq 0$. From
now on, we identify the categories of simplicial presheaves on $\mathcal{A}$
and of presheaves on $\mathcal{A}\times\Delta$, which we denote by
$\operatorname{\mathbf{PSh}_{\Delta}}(\mathcal{A})$.

A simplicial presheaf $X$ on $\mathcal{A}$ is said to be discrete if for every
$a\in\mathcal{A}$, the simplicial set $X(a)$ is discrete. If $a$ is an object
of $\mathcal{A}$, we denote by $F(a)$ the discrete simplicial presheaf whose
value in $b\in\mathcal{A}$ is the constant simplicial set at
$\operatorname{Hom}_{\mathcal{A}}(b,a)$.

######  2.16.

In [Rez10], Rezk defines a model structure $\operatorname{\Theta_{n}Sp}$ on
the category of simplicial presheaves on $\Theta_{n}$ (or presheaves on
$\Theta_{n}\times\Delta$), which provides another model for
$(\infty,n)$-categories. The fibrant objects of $\operatorname{\Theta_{n}Sp}$
are called $(\infty,n)$-$\Theta$-spaces.

Let us recall the two equivalences proven in [Ara14] between $n$-quasi-
categories and $(\infty,n)$-$\Theta$-spaces. We also state some lemmas
relating the introduced functors, which will be useful in the following
sections.

######  2.17.

Let $p:\Theta_{n}\times\Delta\to\Theta_{n}$ be the projection functor and
$i_{0}:\Theta_{n}\to\Theta_{n}\times\Delta$ be the functor that sends
$\theta\in\Theta_{n}$ to $(\theta,[0])\in\Theta_{n}\times\Delta$. Since $[0]$
is the terminal object of $\Delta$, there is an adjunction:

$p:\Theta_{n}\times\Delta\rightleftarrows\Theta_{n}:i_{0}$

which induces an adjunction between that presheaf categories

$p^{*}:\operatorname{\mathbf{PSh}(\Theta_{n})}\rightleftarrows\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}:i_{0}^{*}$
(2.18)

######  Theorem 2.19.

The adjunction (2.18) is a Quillen equivalence

$p^{*}:\operatorname{n\mathchar 45\relax
QCat}\rightleftarrows\operatorname{\Theta_{n}Sp}:i_{0}^{*}$

###### Proof.

See [Ara14, Theorem 8.4.(1)]. ∎

######  Lemma 2.20.

The functor $F:\Theta_{n}\to\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}$
defined in §2.15 can be factored in two ways displayed in the following
commuting (up to isomorphism) diagram

${{\Theta_{n}}}$${{\Theta_{n}\times\Delta}}$${\operatorname{\mathbf{PSh}(\Theta_{n})}}$${\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}}$$\scriptstyle{F}$$\scriptstyle{p^{*}}$$\scriptstyle{i_{0}}$

where the vertical arrows are Yoneda embeddings.

###### Proof.

Straightforward computation from the definitions. ∎

######  2.21.

Let $G:\Delta\to\operatorname{\mathbf{PSh}(\Theta_{n})}$ be the composite of
the functors

$\Delta\xrightarrow[]{}\operatorname{\mathbf{Cat}}\xrightarrow[]{\Pi}\operatorname{\mathbf{Gpd}}\to\operatorname{n\mathbf{\mathchar
45\relax Cat}}\xrightarrow[]{N_{n}}\operatorname{\mathbf{PSh}(\Theta_{n})}$

where the unlabeled arrows are inclusions, $\Pi$ is the free-groupoid functor
and $N_{n}$ is the strict $n$-nerve of §2.9. We define
$t:\Theta_{n}\times\Delta\to\operatorname{\mathbf{PSh}(\Theta_{n})}$ on
objects by $t(\theta,[n])=\Theta_{n}[\theta]\times G([n])$. The functor $t$
induces an adjunction

$t_{!}:\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}\rightleftarrows\operatorname{\mathbf{PSh}(\Theta_{n})}:t^{*}$
(2.22)

######  Theorem 2.23.

The adjunction (2.22) is a Quillen equivalence

$t_{!}:\operatorname{\Theta_{n}Sp}\rightleftarrows\operatorname{n\mathchar
45\relax QCat}:t^{*}$

###### Proof.

See [Ara14, Corollary 8.8]. ∎

######  Lemma 2.24.

The following triangle commutes up to isomorphism of functors

${{\Theta_{n}}}$${\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}}$${\operatorname{\mathbf{PSh}(\Theta_{n})}}$$\scriptstyle{F}$$\scriptstyle{t_{!}}$

where the right diagonal functor is the Yoneda embedding.

###### Proof.

We want to show that the composite $t_{!}F$ is isomorphic to the Yoneda
embedding of $\Theta_{n}$. Using the upper commutative triangle of Lemma 2.20,
we replace $F$ by $\mathcal{Y}i_{0}$, where $\mathcal{Y}$ is the Yoneda
embedding of $\Theta_{n}\times\Delta$. So $t_{!}F\cong
t_{!}\mathcal{Y}i_{0}=ti_{0}$, the last equality coming from the definition of
$t_{!}$ as the left Kan extension of $t$ along $\mathcal{Y}$. For every
$\theta\in\Theta_{n}$, we have

$ti_{0}(\theta)=t(\theta,[0])=\Theta_{n}[\theta]\times
G([0])\cong\Theta_{n}[\theta]$

since $G(0)=\Theta_{n}[0]$ is the terminal object of
$\operatorname{\mathbf{PSh}(\Theta_{n})}$. ∎

##  3\. A cylinder for $n$-quasi-categories

In this section, we recall the definition of a functorial cylinder in a
presheaf category and of an elementary homotopical datum (donnée homotopique
élémentaire), following [Cis06]. We then specialize to the case of presheaves
on a wreath product $\mathcal{C}=\Delta\wr\mathcal{A}$, where $\mathcal{A}$
has a terminal object (denoted by $*$). We give a description, for every
object $c$ of $\mathcal{C}$, of the product $c\times[1](*)$ in
$\operatorname{\mathbf{PSh}}(\mathcal{C})$ as a colimit of representables,
which generalizes the description of products $[p]\times[1]$ of simplicial
sets. Finally, we look at the case
$\mathcal{C}=\Theta_{n}(=\Delta\wr\Theta_{n-1})$, and modify this cylinder to
obtain another one, better suited to the model structure for $n$-quasi-
categories, while still having a simple combinatorial description.

Let $\mathcal{C}$ be a small category.

######  Definition 3.1.

A cylinder for a presheaf $X$ on $\mathcal{C}$ is the data of a presheaf
$\mathcal{I}X$ and of morphisms $\partial_{X}^{\varepsilon}:X\to\mathcal{I}X$,
$\varepsilon=0,1$ and $\sigma_{X}:\mathcal{I}X\to X$ such that the following
diagram commutes

${X}$${\mathcal{I}X}$${X}$${X}$$\scriptstyle{\sigma_{X}}$$\scriptstyle{\partial_{X}^{0}}$$\scriptstyle{\partial_{X}^{1}}$$\scriptstyle{\operatorname{id}_{X}}$$\scriptstyle{\operatorname{id}_{X}}$

and that $\langle\partial_{X}^{0},\partial_{X}^{1}\rangle:X\sqcup
X\to\mathcal{I}X$ is a monomorphism.

######  Definition 3.2.

A functorial cylinder on $\mathcal{C}$ is the data of a functor
$\mathcal{I}:\operatorname{\mathbf{PSh}}(\mathcal{C})\to\operatorname{\mathbf{PSh}}(\mathcal{C})$
and of natural transformations
$\partial^{\varepsilon}:\operatorname{id}\to\mathcal{I}$, $\varepsilon=0,1$,
and $\sigma:\mathcal{I}\to\operatorname{id}$, such that for every presheaf $X$
on $\mathcal{C}$,
$(\mathcal{I}(X),\partial^{0}_{X},\partial^{1}_{X},\sigma_{X})$ is a cylinder
for $X$.

######  Definition 3.3.

An elementary homotopical datum (EHD) on $\mathcal{C}$ is a functorial
cylinder $(\mathcal{I},\partial^{0},\partial^{1},\sigma)$ verifying the two
following axioms:

(HD1) The functor $\mathcal{I}$ commutes with small colimits and preserves
monomorphisms.

(HD2) For every monomorphism $j:K\to L$ in
$\operatorname{\mathbf{PSh}}(\mathcal{C})$ and $\varepsilon=0,1$, the
following square is a pullback

${K}$${L}$${{\mathcal{I}(K)}}$${{\mathcal{I}(L)}}$$\scriptstyle{j}$$\scriptstyle{\partial^{\varepsilon}_{K}}$$\scriptstyle{\partial^{\varepsilon}_{L}}$$\scriptstyle{\mathcal{I}(j)}$

######  Definition 3.4.

An interval of $\operatorname{\mathbf{PSh}}(\mathcal{C})$ is a presheaf $I$ on
$\mathcal{C}$ equipped with two morphisms $\\{\varepsilon\\}:\\{*\\}\to I$,
$\varepsilon=0,1$. An interval is separating if
$\\{0\\}\cap\\{1\\}=\emptyset$.

######  3.5.

If $I$ is a separating interval of $\operatorname{\mathbf{PSh}}(\mathcal{C})$,
the functor
$I\times(-):\operatorname{\mathbf{PSh}}(\mathcal{C})\to\operatorname{\mathbf{PSh}}(\mathcal{C})$
defines a functorial cylinder on $\mathcal{C}$. The components at a presheaf
$X$ of the natural transformations $\partial^{\varepsilon}$ and $\sigma$ are
given by $\partial^{\varepsilon}_{X}=X\times\\{\varepsilon\\}:X\cong
X\times\\{*\\}\to X\times I$ and $\sigma_{X}=p_{1}:X\times I\to X$. This
functorial cylinder is an EHD [Cis06, Example 1.3.8].

######  3.6.

If $(I,\\{\varepsilon_{I}\\})$ is a separating interval and $j:I\to J$ is a
monomorphism, then $J$ can be endowed with the structure of a separating
interval by taking $\\{\varepsilon_{J}\\}=j\\{\varepsilon_{I}\\}$.

######  Example 3.7.

The nerves $N_{n}(D_{1})$ and $N_{n}(J)$ are separating intervals of
$\operatorname{\mathbf{PSh}(\Theta_{n})}$.

Suppose that the category $\mathcal{C}$ has a terminal object $[0]$. If $X$ a
presheaf on $\mathcal{C}$, there is a monomorphism $i_{X}:X_{0}\to X$, where
$X_{0}$ is seen as the constant presheaf at the set $X_{0}$ (simpler notation
for $X_{[0]}$).

Our first goal is to show that, given a separating interval $I$ and a
monomorphism $j:I\to J$, the presheaf

$\mathcal{J}(X)=X\times I\bigsqcup_{X_{0}\times I}X_{0}\times J$

has the structure of an EHD.

######  3.8.

Since both morphisms $i_{X}:X_{0}\to X$ and $j:I\to J$ are monomorphisms, all
the morphisms of the following pushout square are monomorphisms.

${{X_{0}\times I}}$${{X\times I}}$${{X_{0}\times J}}$${{\mathcal{J}(X)}}$

Since the square is also a pullback, the induced arrow $\mathcal{J}(X)\to
X\times J$ is also a monomorphism, and we write $\mathcal{J}(X)=X\times I\cup
X_{0}\times J\subset X\times J$.

######  Proposition 3.9.

The construction $X\mapsto\mathcal{J}(X)$ above defines a functorial cylinder
on $\mathcal{C}$, which is moreover an EHD.

###### Proof.

The functoriality of $\mathcal{J}$ comes from the functoriality of $X\mapsto
X_{0}$, the functoriality of the product and the universal property of the
pushout.

The morphisms $\partial^{\varepsilon}_{X}:X\to\mathcal{J}(X)$ are the
composites

$X\xrightarrow[]{X\times\\{\varepsilon\\}}X\times I\to\mathcal{J}(X)$

Since $I$ is a separating interval, $X\times I$ is a cylinder and $X\sqcup
X\to X\times I$ is a monomorphism. Therefore,
$\langle\partial^{0}_{X},\partial^{1}_{X}\rangle:X\sqcup X\to\mathcal{J}(X)$
is a monomorphism, as the composite of the monomorphisms $X\sqcup X\to X\times
I$ and $X\times I\to\mathcal{J}(X)$.

The morphism $\sigma_{X}:X\to\mathcal{J}(X)$ is the composite of the inclusion
$\mathcal{J}(X)\subset X\times J$ with the projection $X\times J\to X$. For
$\varepsilon=0,1$, we have
$\sigma_{X}\partial^{\varepsilon}_{X}=\operatorname{id}_{X}$, as can be seen
in the commuting diagram

${X}$${{X\times I}}$${{\mathcal{J}(X)}}$${{X\times
J}}$${X}$$\scriptstyle{X\times\\{\varepsilon\\}}$$\scriptstyle{p_{1}}$$\scriptstyle{\operatorname{id}_{X}}$$\scriptstyle{p_{1}}$$\scriptstyle{\partial^{\varepsilon}_{X}}$$\scriptstyle{\sigma_{X}}$

where all the horizontal arrows are inclusions. Thus $\mathcal{J}$ is a
functorial cylinder.

Let us show that $(\mathcal{J},\partial^{0},\partial^{1},\sigma)$ is an EHD.
The functor $\mathcal{J}$ commutes with small colimits, since products and
pushouts of presheaves commute with small colimits. Now let $X\to Y$ be a
monomorphism between presheaves on $\mathcal{C}$. In particular, the map
$X_{0}\to Y_{0}$ is a monomorphism. The induced arrow
$\mathcal{J}(X)\to\mathcal{J}(Y)$ is simply the inclusion $X\times I\cup
X_{0}\times J\subset Y\times I\cup Y_{0}\times J$, so $\mathcal{J}$ preserves
monomorphisms. It remains to show (HD2), that is, that given a monomorphism
$X\to Y$ and $\varepsilon=0,1$, the following commuting square is a pullback

${X}$${Y}$${{\mathcal{J}(X)}}$${{\mathcal{J}(Y)}}$$\scriptstyle{\partial^{\varepsilon}_{X}}$$\scriptstyle{\partial^{\varepsilon}_{Y}}$

This is clear since all arrows are monomorphisms of presheaves of sets. ∎

Now we work in the case $\mathcal{C}=\Delta\wr\mathcal{A}$, where
$\mathcal{A}$ is a small category with a terminal object $*$. Let
$a\in\mathcal{A}$ and $I=[1](*)$. We will give a presentation of $a\times I$
as a colimit of representables (see Proposition 3.11).

######  3.10.

Let $c\in\mathcal{C}$. By definition, $c$ is of the form
$[p](a_{1},\ldots,a_{p})$, where $p\geq 0$ and
$a_{1},\ldots,a_{p}\in\mathcal{A}$. Let $0\leq i\leq p$. We define
$c^{i}\in\mathcal{C}$ as

$c^{i}=[p+1](a_{1},\ldots,a_{i},*,a_{i+1},\ldots,a_{p})$

for $1\leq i<p$ and

$c^{0}=[p+1](*,a_{1},\ldots,a_{p})$ $c^{p}=[p+1](a_{1},\ldots,a_{p},*)$

We introduce morphisms $\alpha^{i}:c\to c^{i}$, for $i=0,\ldots,p-1$, induced
by $(0,\ldots,i,i+2,\ldots,p+1):[p]\to[p+1]$, and morphisms $\beta^{i}:c\to
c^{i}$, for $i=1,\ldots,p$, induced by
$(0,\ldots,i-1,i+1,\ldots,p+1):[p]\to[p+1]$. We can picture the morphisms
$\alpha^{i}$ and $\beta^{i}$ by the following plain and dotted arrows,
respectively.

${c}$${{i-1}}$${i}$${{i+1}}$${{c^{i}}}$${{i-1}}$${i}$${{i+1}}$${{i+2}}$$\scriptstyle{a_{i}}$$\scriptstyle{a_{i+1}}$$\scriptstyle{a_{i}}$$\scriptstyle{a_{i+1}}$$\scriptstyle{*}$$\scriptstyle{\alpha^{i}}$$\scriptstyle{\beta^{i}}$

Let $\mathcal{I}(c)$ be the colimit of the following diagram in
$\operatorname{\mathbf{PSh}(\Theta_{n})}$

${c}$${c}$${\ldots}$${c}$${{c^{0}}}$${{c^{1}}}$${{c^{p}}}$$\scriptstyle{\alpha^{0}}$$\scriptstyle{\beta^{1}}$$\scriptstyle{\alpha^{1}}$$\scriptstyle{\beta^{p}}$

######  Proposition 3.11.

Let $\mathcal{A}$ be a small category with a terminal object $*$. Let
$\mathcal{C}$ be the wreath product $\Delta\wr\mathcal{A}$ let $I$ be the
object $[1](*)$ of $\mathcal{C}$. For every $c\in\mathcal{C}$, there is an
isomorphism

$\mathcal{I}(c)\xrightarrow[]{\cong}c\times I$

in $\operatorname{\mathbf{PSh}}(\mathcal{C})$.

###### Proof.

A morphism $\mathcal{I}(c)\to c\times I$ corresponds to a morphism
$\mathcal{I}(c)\to c$ and a morphism $\mathcal{I}(c)\to I$. For $0\leq i\leq
p$, let $\sigma^{i}:c^{i}\to c$ be the morphism induced by
$\sigma^{i}:[p+1]\to[p]$ sending $i$ and $i+1$ to $i$. Since for every
$i=1,\ldots p$ the diagrams

${{c^{i-1}}}$${c}$${c}$${{c^{i}}}$$\scriptstyle{\alpha^{i-1}}$$\scriptstyle{\beta^{i}}$$\scriptstyle{\sigma^{i-1}}$$\scriptstyle{\sigma^{i}}$

commute (both composites equal $\operatorname{id}_{c}$), the morphisms
$\sigma^{i}$ induce a morphism $\sigma:\mathcal{I}(c)\to c$.

For $0\leq i\leq p$, let $\gamma^{i}:c^{i}\to[1](*)=I$ be the morphism sending
$i$ to $0$ and $i+1$ to $1$. Since for every $i=1,\ldots p$ the diagrams

${{c^{i-1}}}$${c}$${I}$${{c^{i}}}$$\scriptstyle{\alpha^{i-1}}$$\scriptstyle{\beta^{i}}$$\scriptstyle{\gamma^{i-1}}$$\scriptstyle{\gamma^{i}}$

commute, the morphisms $\gamma^{i}$ induce a morphism
$\gamma:\mathcal{I}(c)\to I$.

It remains to show that the morphism
$\langle\sigma,\gamma\rangle:\mathcal{I}(c)\to c\times I$ is an isomorphism.
We will show that, for every $d=[q](b_{1},\ldots,b_{q})\in\mathcal{C}$, the
induced morphism

$\operatorname{Hom}_{\operatorname{\mathbf{PSh}}(\mathcal{C})}(d,\mathcal{I}(c))\to\operatorname{Hom}_{\operatorname{\mathbf{PSh}}(\mathcal{C})}(d,c\times
I)$

is a bijection.

Let $f:d\to c\times I$, $g=p_{1}f:d\to c$ and $h=p_{2}f:d\to I$. Since
$I=[1](*)$, the map $h$ corresponds to a morphism $[q]\to[1]$ in $\Delta$,
which we still denote by $h$. If $h$ sends all objects of $[q]$ to $0$ (resp.
1), consider $f^{\prime}:d\to\mathcal{I}(c)$ defined as the composite
$d\xrightarrow[]{g}c\xrightarrow[]{[\delta^{p+1}]}c^{p}\to\mathcal{I}(c)$
(resp.
$d\xrightarrow[]{g}c\xrightarrow[]{[\delta^{0}]}c^{0}\to\mathcal{I}(c)$). It
is clear that $\langle\sigma,\gamma\rangle f^{\prime}=f$. Note that the
construction of $f^{\prime}$ is the unique possible one.

If not all objects of $[q]$ are sent to the same object of $[1]$, then there
exists $1\leq j_{h}\leq q$ such that $h(i)=0$ for $i<j_{h}$ and $h(i)=1$ for
$i\geq j_{h}$. Let $k=g(j_{h}-1)$. Consider $f^{\prime}:d\to\mathcal{I}(c)$
defined as the composite
$d\xrightarrow{f^{\prime\prime}}c^{k}\to\mathcal{I}(c)$ where
$f^{\prime\prime}$ is induced by $g$ in the following way:
$f^{\prime\prime}(i)=g(i)$ for $i<j_{h}$ and $f^{\prime\prime}(i)=g(i)+1$ for
$i\geq j_{h}$ (the maps between $b_{j}$’s and $a_{i}$’s are those of $g$). We
have $\sigma^{k}f^{\prime\prime}=g$ and $\gamma^{k}f^{\prime\prime}=h$, so
$\langle\sigma,\gamma\rangle f^{\prime}=f$, as desired. This time, the only
choice we made was that of $k$. Indeed, we could have chosen any $k$ from
$g(j_{h}-1)$ to $g(j_{h})-1$, and produced $f^{\prime}_{k}:d\to\mathcal{I}(c)$
factoring through $f^{\prime\prime}_{k}:d\to c^{k}$. Suppose that there are
two possible consecutive choices $k$ and $k+1$. Following the constructions,
we see that $f^{\prime\prime}_{k}=\alpha^{k}g$ and
$f^{\prime\prime}_{k+1}=\beta^{k+1}g$, and therefore $f^{\prime\prime}_{k}$
and $f^{\prime\prime}_{k+1}$ define the same morphism
$f^{\prime}:d\to\mathcal{I}(c)$.

∎

In the last part of this section, we restrict our attention to the case
$\mathcal{C}=\Theta_{n}$, for $n>0$ (or $\mathcal{A}=\Theta_{n-1}$, since
$\Theta_{n}=\Delta\wr\Theta_{n-1}$). As before, let $I=[1]([0])$ and
$J=N_{n}(J)$. Recall that $\mathcal{J}(\theta)=\theta\times
I\cup\theta_{0}\times J$. Let us provide an alternative construction
$\mathcal{J}^{\prime}(\theta)$ for $\mathcal{J}(\theta)$.

######  3.12.

Let $\theta=[p](\tau_{1},\ldots,\tau_{p})\in\Theta_{n}$ and $0\leq i\leq p$.
Let $\theta^{i}_{J}$ be the presheaf on $\Theta_{n}$ defined as the following
pushout

${I}$${{\theta^{i}}}$${J}$${{\theta^{i}_{J}}}$$\scriptstyle{\\{i,i+1\\}}$$\scriptstyle{\varphi^{i}}$$\scriptstyle{\psi^{i}}$$\scriptstyle{\lrcorner}$

where the vertical arrow is the inclusion $I\to J$ and the top horizontal
arrow sends the object $0$ (resp. $1$) of $I$ to the object $i$ (resp. $i+1$)
of $\theta^{i}$. Since the arrows $I\to\theta^{i}$ and $I\to J$ are
monomorphisms, the arrows $\varphi^{i}$ and $\psi^{i}$ are also monomorphisms,
as pushouts of monomorphisms of presheaves.

Let $\mathcal{J}^{\prime}(\theta)$ be the colimit of the following diagram in
$\operatorname{\mathbf{PSh}(\Theta_{n})}$

${\theta}$${\theta}$${\ldots}$${\theta}$${{\theta^{0}}}$${{\theta^{1}}}$${{\theta^{p}}}$${{\theta^{0}_{J}}}$${{\theta^{1}_{J}}}$${{\theta^{p}_{J}}}$$\scriptstyle{\alpha^{0}}$$\scriptstyle{\beta^{1}}$$\scriptstyle{\alpha^{1}}$$\scriptstyle{\beta^{p}}$$\scriptstyle{\varphi^{0}}$$\scriptstyle{\varphi^{1}}$$\scriptstyle{\varphi^{p}}$

######  Proposition 3.13.

For every $\theta\in\Theta_{n}$, there is an isomorphism

$\mathcal{J}(\theta)\cong\mathcal{J}^{\prime}(\theta)$

in $\operatorname{\mathbf{PSh}(\Theta_{n})}$.

###### Proof.

It follows from the fact that colimits commute with colimits. Let us explicit
all the diagrams and functors. If $\theta=[p](\tau_{1},\ldots,\tau_{p})$, let
$W_{p}$ be the category

${{1^{\prime}}}$${{2^{\prime}}}$${\ldots}$${{p^{\prime}}}$${0}$${1}$${p}$

and $P$ be the category

${a}$${b}$${c}$

Recall that $\mathcal{J}(\theta)$ is defined as the pushout

${{\coprod_{0\leq i\leq p}I}}$${{\mathcal{I}(\theta)}}$${{\coprod_{0\leq i\leq
p}J}}$${{\mathcal{J}(\theta)}}$$\scriptstyle{\lrcorner}$

We observe that the 3 objects defining this pushout can be seen as colimits
indexed by $W_{p}$. Indeed, the coproduct $\coprod_{0\leq i\leq p}I$ (resp.
$\coprod_{0\leq i\leq p}J$) is the colimit of
$F^{I}:W_{p}\to\operatorname{\mathbf{PSh}(\Theta_{n})}$ (resp. $F^{J}$)
sending $j^{\prime}$ to $\emptyset$ for every $1\leq j\leq p$ and $i$ to $I$
(resp. $J$), for every $0\leq i\leq p$. Moreover, $\mathcal{I}(\theta)$ is
defined as the colimit of a functor
$F^{I}_{\theta}:W_{p}\to\operatorname{\mathbf{PSh}(\Theta_{n})}$. Therefore,
consider the functor

$F:W_{p}\times P\to\operatorname{\mathbf{PSh}(\Theta_{n})}$

whose adjoint

$F^{\prime}:P\to[W_{p},\operatorname{\mathbf{PSh}(\Theta_{n})}]$

sends $a$ to $F^{I}$, $b$ to $F^{I}_{\theta}$ and $c$ to $F^{J}$. We have that

$\mathcal{J}(\theta)=\operatorname{colim}_{W_{p}}(\operatorname{colim}_{P}F^{\prime})$

The adjoint

$F^{\prime\prime}:W_{p}\to[P,\operatorname{\mathbf{PSh}(\Theta_{n})}]$

is the functor that sends $j^{\prime}$ to the diagram

${\emptyset}$${\theta}$${\emptyset}$

for every $1\leq j\leq p$ and $i$ to the diagram

${I}$${{\theta^{i}}}$${J}$$\scriptstyle{\\{i,i+1\\}}$

for every $0\leq i\leq p$. The images by $F^{\prime\prime}$ of the arrows of
$W_{p}$ are given by the unique morphisms $\emptyset\to I$ and $\emptyset\to
J$ and by the morphisms $\alpha^{i},\beta^{i}:\theta\to\theta^{i}$. By
definition, we have

$\mathcal{J}^{\prime}(\theta)=\operatorname{colim}_{P}(\operatorname{colim}_{W_{p}}F^{\prime\prime})$

and hence

$\mathcal{J}(\theta)\cong\mathcal{J}^{\prime}(\theta)$

∎

######  3.14.

Let us construct a morphism $\mu:\mathcal{J}^{\prime}(\theta)\to\theta$. Let
$0\leq i\leq p$. Recall from the proof of Proposition 3.11 the morphism
$\sigma^{i}:\theta^{i}\to\theta$. It fits in the commutative diagram below,
inducing an arrow $\mu^{i}:\theta^{i}_{J}\to\theta$.

${I}$${{\theta^{i}}}$${J}$${{\theta^{i}_{J}}}$${{\\{*\\}}}$${\theta}$$\scriptstyle{\sigma^{i}}$$\scriptstyle{\\{i\\}}$$\scriptstyle{\\{i,i+1\\}}$$\scriptstyle{\lrcorner}$$\scriptstyle{\mu^{i}}$

The morphisms $\mu^{i}$ assemble to form a morphism
$\mu:\mathcal{J}^{\prime}(\theta)\to\theta$. It follows from the constructions
that $\mu$ fits in a commutative triangle

${{\mathcal{J}(\theta)}}$${{\mathcal{J}^{\prime}(\theta)}}$${\theta}$$\scriptstyle{\varphi}$$\scriptstyle{\sigma}$$\scriptstyle{\mu}$

where $\varphi$ is the isomorphism of Proposition 3.13 and $\sigma$ is
described in the proof of Proposition 3.9.

We now study these cylinder constructions considering the model structure for
$n$-quasi-categories on $\operatorname{\mathbf{PSh}(\Theta_{n})}$.

######  Proposition 3.15.

For $0\leq i\leq p$, the morphism $\mu^{i}:\theta^{i}_{J}\to\theta$ is a weak
equivalence of $\operatorname{n\mathchar 45\relax QCat}$.

###### Proof.

Consider the commutative diagram below

${I}$${{I[\theta^{i}]}}$${{\theta^{i}}}$${J}$${A}$${{\theta^{i}_{J}}}$${{\\{*\\}}}$${{I[\theta]}}$${B}$${\theta}$$\scriptstyle{\sim}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$$\scriptstyle{\sim}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$$\scriptstyle{\sigma^{i}}$$\scriptstyle{\\{i\\}}$$\scriptstyle{\mu^{i}}$$\scriptstyle{\sim}$

where the upper rectangle

${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$

is the pushout defining $\theta^{i}_{J}$, where we factorised the arrow
$I\to\theta^{i}$ through the spine inclusion $I[\theta^{i}]\to\theta^{i}$,
which is a weak equivalence of $\operatorname{n\mathchar 45\relax QCat}$ by
definition. The morphisms $J\to\\{*\\}$ and $I[\theta]\to\theta$ are also weak
equivalences by definition of $\operatorname{n\mathchar 45\relax QCat}$. The
objects $A$ and $B$ are defined as the respective pushouts, and we will not
need to compute them explicitly.

Since $I\to I[\theta^{i}]$ and $I\to J$ are monomorphisms (equivalently,
cofibrations of $\operatorname{n\mathchar 45\relax QCat}$), the arrows
$I[\theta^{i}]\to A$ and $J\to A$ are also cofibrations, since cofibrations
are stable by pushout. Moreover, the arrow $A\to\theta^{i}_{J}$ is a trivial
cofibration, as a pushout of the trivial cofibration
$I[\theta^{i}]\to\theta^{i}$.

Since $\operatorname{n\mathchar 45\relax QCat}$ is left proper, we have that
that $A\to I[\theta]$ and $\theta^{i}_{J}\to B$ are weak equivalences. The
2-out-of-3 property of weak equivalences then implies that $I[\theta]\to B$
and $B\to\theta$ are weak equivalences, and thus that
$\mu^{i}:\theta^{i}_{J}\to\theta$ is a weak equivalence, as desired. ∎

######  Theorem 3.16.

For every $\theta\in\Theta_{n}$, the morphism
$\mu:\mathcal{J}^{\prime}(\theta)\to\theta$ is a weak equivalence of
$\operatorname{n\mathchar 45\relax QCat}$.

###### Proof.

The morphism $\mu:\mathcal{J}^{\prime}(\theta)\to\theta$ is induced by a
morphism of diagrams
$\mathsf{W}_{p}\to\operatorname{\mathbf{PSh}(\Theta_{n})}$. Indeed,
$\mathcal{J}^{\prime}(\theta)$ is a colimit of a functor
$F^{J}_{\theta}:W_{p}\to\operatorname{\mathbf{PSh}(\Theta_{n})}$ by
definition. We can see $\theta$ as such a colimit by considering
$F_{\theta}:W_{p}\to\operatorname{\mathbf{PSh}(\Theta_{n})}$ sending all
objects of $W_{p}$ to $\theta$ and all arrows to the identity of $\theta$. The
morphism of diagrams $F^{J}_{\theta}\to F_{\theta}$ is induced by the
identities $\theta\to\theta$ (in particular, weak equivalences) and by the
arrows $\mu^{i}:\theta^{i}_{J}\to\theta$, which are weak equivalences by
Proposition 3.15.

In the diagrams given by $F^{J}_{\theta}$ and by $F_{\theta}$, all objects are
cofibrant and all arrows are cofibrations, so their colimits are in fact
homotopy colimits (a proof of this fact for diagrams of shape $W_{p}$ can be
found in the proof of [CLS22, Proposition B.2.1], under the first diagram
appearing in the proof). Since homotopy colimits preserve weak equivalences of
diagrams, the induced arrow $\mu:\mathcal{J}^{\prime}(\theta)\to\theta$ is a
weak equivalence. ∎

######  Corollary 3.17.

For every $\theta\in\Theta_{n}$, the morphism
$\sigma:\mathcal{J}(\theta)\to\theta$ is a weak equivalence in
$\operatorname{n\mathchar 45\relax QCat}$.

###### Proof.

Apply the 2-out-of-3 property of weak equivalences to the commutative triangle
of §3.14. ∎

######  Corollary 3.18.

For every $X\in\operatorname{\mathbf{PSh}(\Theta_{n})}$, the morphism
$\sigma_{X}:\mathcal{J}(X)\to X$ is a weak equivalence in
$\operatorname{n\mathchar 45\relax QCat}$.

###### Proof.

We apply [Cis06, Proposition 8.2.15], using the fact that $\Theta_{n}$ is a
catégorie squelettique régulière and that $\mathcal{J}(X)$ is an EHD (cf.
Proposition 3.9). We take $\mathsf{W}$ as the class of weak equivalences of
$\operatorname{n\mathchar 45\relax QCat}$, which is a (pre-)localizer, and
Corollary 3.17 is exactly the hypothesis that
$\sigma:\mathcal{J}(\theta)\to\theta$ is in $\mathsf{W}$ for every
$\theta\in\Theta_{n}$. ∎

Recall from Paragraphs 2.11 and 2.12 the definitions of the classes
$\mathcal{I}_{n}$ and $\mathcal{J}_{n}$. The following theorem is a refinement
of the recognition principle given in Proposition 2.14.

######  Theorem 3.19.

Let $\mathcal{M}$ be a model category. Let
$F:\operatorname{\mathbf{PSh}(\Theta_{n})}\to\mathcal{M}$ be a cocontinuous
functor sending monomorphisms to cofibrations. Then $F$ sends weak
equivalences of $\operatorname{n\mathchar 45\relax QCat}$ to weak equivalences
of $\mathcal{M}$ if and only if $F$ sends the morphisms in
$\mathcal{I}_{n}\cup\mathcal{J}_{n}$ to weak equivalences.

###### Proof.

The direct implication is clear, since the morphisms in $\mathcal{I}_{n}$ and
$\mathcal{J}_{n}$ are weak equivalences by definition.

For the converse implication, we use once again [Cis06, Proposition 8.2.15].
The proof follows exactly as the second paragraph of the proof of Proposition
2.14, the only difference is that we exchange the EHD $J\times\theta$ by the
EHD $\mathcal{J}(\theta)$.

Therefore, it suffices to show that $F$ sends the morphims
$\sigma:\mathcal{J}(\theta)\to\theta$ to weak equivalences of $\mathcal{M}$.
Since, by hypothesis, the functor $F$ preserves pushouts, and sends spine
inclusions and the projection $J\to\\{*\\}$ to weak equivalences of
$\mathcal{M}$, we can apply the same reasoning of the proof of Proposition
3.15 to show that $F(\mu^{i}):F(\theta^{i}_{J})\to F(\theta)$ is a weak
equivalence of $\mathcal{M}$. Thus, the equivalence of diagrams
$F^{J}_{\theta}\to F_{\theta}$ which induces the weak equivalence
$\mu:\mathcal{J}^{\prime}(\theta)\to\theta$ of Theorem 3.16 provides an
equivalence of diagrams $F(F^{J}_{\theta})\to F(F_{\theta})$. Since $F$
preserves homotopy colimits, the same argument of the proof Theorem 3.16
applies, showing that $F(\mu):F(\mathcal{J}^{\prime}(\theta))\to F(\theta)$ is
a weak equivalence, and hence that $F(\sigma)$ is a weak equivalence. ∎

##  4\. Truncated objects

In this section, we recall Rezk’s definition of $(n+k,n)$-$\Theta$-spaces and
we define $(n+k)$-truncated $n$-quasi-categories, generalizing the definitions
of [Joy08a] for $n=1$ and [Cam20] for $(n=2,k=0)$. It is shown in [CL20] and
[Cam20] that the adjunctions (2.18) and (2.22) induce Quillen equivalences
between $(n+k,n)$-$\Theta$-spaces and $(n+k)$-truncated $n$-quasi-categories,
for $(n=1,k\geq 0)$ and $(n=2,k=0)$, respectively. Using the same techniques,
we extend this results to all $n\geq 1$, $k\geq 0$, constructing a model
structure for $(n+k)$-truncated $n$-quasi-categories and showing that it is
Quillen equivalent to the one for $(n+k,n)$-$\Theta$-spaces via the same two
adjunctions.

######  4.1.

Let $m\geq 0$. A Kan complex $X$ is a homotopy $k$-type if it has the right
lifting property with respect to the boundary inclusions
$\partial\Delta[m]\to\Delta[m]$, for all $m\geq k+2$. Equivalently, a homotopy
$k$-type is a Kan complex such that for all $x\in X_{0}$, the homotopy groups
$\pi_{m}(X,x)$ are trivial for every $m>k$.

######  4.2.

If $X$ is a quasi-category, and $x,y\in X_{0}$, the Kan complex
$\operatorname{Hom}_{X}(x,y)$ is defined as the pullback

${{\operatorname{Hom}_{X}(x,y)}}$${{X^{\Delta[1]}}}$${{\\{(x,y)\\}}}$${{X^{\partial\Delta[1]}}}$$\scriptstyle{\lrcorner}$

where the right vertical arrow is induced by the boundary inclusion
$\partial\Delta[1]\to\Delta[1]$.

######  4.3.

Let $k\geq 0$. Recall from [Joy08a, §26] that a (1-)quasi-category $X$ is
$(1+k)$-truncated if for every $x,y\in X_{0}$, the Kan complex
$\operatorname{Hom}_{X}(x,y)$ is a homotopy $k$-type.

######  4.4.

For every $n\geq 1$, the suspension functor $\sigma:\Theta_{n}\to\Theta_{n+1}$
induces an adjunction

$\Sigma:\operatorname{\mathbf{PSh}}(\Theta_{n})\rightleftarrows*\sqcup*/\operatorname{\mathbf{PSh}}(\Theta_{n+1}):\operatorname{Hom}$
(4.5)

between presheaves on $\Theta_{n}$ and bipointed presheaves on $\Theta_{n+1}$.
Indeed, composing $\sigma$ with the Yoneda embedding
$\Theta_{n+1}\to\operatorname{\mathbf{PSh}}(\Theta_{n+1})$, we get a functor
$\Theta_{n}\to\operatorname{\mathbf{PSh}}(\Theta_{n+1})$ which factorises via
the forgetful functor
$*\sqcup*/\operatorname{\mathbf{PSh}}(\Theta_{n+1})\to\operatorname{\mathbf{PSh}}(\Theta_{n+1})$.
The functor
$\hat{\sigma}:\Theta_{n}\to*\sqcup*/\operatorname{\mathbf{PSh}}(\Theta_{n+1})$
sends $\theta\in\Theta_{n}$ to $(\Theta_{n+1}[\sigma(\theta)],0,1)$. The
adjunction (4.5) is the Kan extension-nerve adjunction induced by this
functor, i.e., $\Sigma:=\hat{\sigma}_{!}$ and
$\operatorname{Hom}:=\hat{\sigma}^{!}$.

Explicitly, for $X\in\operatorname{\mathbf{PSh}(\Theta_{n})}$ and
$\theta\in\Theta_{n-1}$, the value of
$\operatorname{Hom}_{X}(x,y):=\operatorname{Hom}(X,x,y)$ at $\theta$ is given
by the pullback

${{\operatorname{Hom}_{X}(x,y)_{\theta}}}$${{X_{\sigma(\theta)}}}$${{\\{(x,y)\\}}}$${{X_{0}\times
X_{0}}}$$\scriptstyle{\lrcorner}$

where the right vertical arrow is induced by the two morphisms
$[0]\to\sigma(\theta)$ in $\Theta_{n}$.

######  Proposition 4.6.

The adjunction (4.5) is a Quillen adjunction

$\Sigma:\operatorname{n\mathchar 45\relax
QCat}\rightleftarrows*\sqcup*/\operatorname{(n+1)\mathchar 45\relax
QCat}:\operatorname{Hom}$

###### Proof.

The functor $\Sigma$ is cocontinuous (as a left adjoint) and preserves
monomorphisms. Therefore, it suffices by Theorem 3.19 to show that it sends
the morphisms of the classes $\mathcal{I}_{n}$ and $\mathcal{J}_{n}$ to weak
equivalences. The spine inclusions $i_{\theta}$ are sent to the spine
inclusions $i_{\sigma(\theta)}$, and the morphisms $j_{k}$ of
$\mathcal{J}_{n}$ are sent to the morphisms $j_{k+1}$ of $\mathcal{J}_{n+1}$,
so we are done. ∎

######  Definition 4.7.

Let $n\geq 2$, $k\geq 0$. An $n$-quasi-category $X$ is $(n+k)$-truncated if
for every $x,y\in X_{0}$ the $(n-1)$-quasi-category
$\operatorname{Hom}_{X}(x,y)$ is $(n-1+k)$-truncated.

######  4.8.

Let $Z\in\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}$, and $x,y$ be two
points of the space $Z(0)$. Rezk defines the mapping object $M_{Z}(x,y)$ to be
the simplicial presheaf on $\Theta_{n-1}$ whose space at
$\theta\in\Theta_{n-1}$ is given by the pullback

${{M_{Z}(x,y)(\theta)}}$${{Z(\sigma(\theta))}}$${{\\{(x,y)\\}}}$${{Z(0)\times
Z(0)}}$$\scriptstyle{\lrcorner}$

If $Z$ is an $(\infty,n)$-$\Theta$-space, then $M_{Z}(x,y)$ is an
$(\infty,n-1)$-$\Theta$-space [Rez10, Proposition 8.3].

Mapping objects for set-valued and space-valued presheaves on $\Theta_{n}$ can
be related via the functor
$i_{0}^{*}:\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}\to\operatorname{\mathbf{PSh}(\Theta_{n})}$.

######  Lemma 4.9.

Let $Z\in\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}$, and $x,y$ be two
points of the space $Z(0)$. There is a natural isomorphism

$i_{0}^{*}(M_{Z}(x,y))\cong\operatorname{Hom}_{i_{0}^{*}(Z)}(x,y)$

###### Proof.

This follows directly from the definitions of the mapping objects and from the
fact that $i_{0}^{*}$ preserves pullbacks, as a right adjoint (or as a
restriction functor between presheaf categories). ∎

######  4.10.

Let $k\geq 0$. A $(k,0)$-$\Theta$-space is the same as a homotopy $k$-type. An
$(n+k,n)$-$\Theta$-space $Z$ is an $(\infty,n)$-$\Theta$-space such that for
every pair $(x,y)$ of points of $Z(0)$ the mapping object $M_{Z}(x,y)$ is an
$(n-1+k,n-1)$-$\Theta$-space.

######  Proposition 4.11 (Rezk).

Let $k\geq 0$. There is a localisation $\operatorname{\Theta_{n}Sp_{k}}$ of
the model structure $\operatorname{\Theta_{n}Sp}$ of
$\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}$ such that the fibrant
objects are exactly $(n+k,n)$-$\Theta$-spaces.

###### Proof.

The model structure is defined in [Rez10, §11.4] and the characterisation of
the fibrant objects is given in [Rez10, Proposition 11.20]. ∎

######  Theorem 4.12.

Let $k\geq 0$. There is a model structure $\operatorname{n\mathchar 45\relax
QCat_{k}}$ on $\operatorname{\mathbf{PSh}(\Theta_{n})}$ whose fibrant objects
are $(n+k)$-truncated $n$-quasi-categories. This model structure is the
localization of $\operatorname{n\mathchar 45\relax QCat}$ with respect to the
boundary inclusion
$\partial\Theta_{n}[\sigma^{n-1}[k+3]]\to\Theta_{n}[\sigma^{n-1}[k+3]]$.
Moreover, the adjunctions (2.18) and (2.22) are Quillen equivalences

$p^{*}:\operatorname{n\mathchar 45\relax
QCat_{k}}\rightleftarrows\operatorname{\Theta_{n}Sp_{k}}:i_{0}^{*}$

and

$t_{!}:\operatorname{\Theta_{n}Sp_{k}}\rightleftarrows\operatorname{n\mathchar
45\relax QCat_{k}}:t^{*}$

###### Proof.

First, let us show that $(n+k)$-truncated $n$-quasi-categories are exactly the
objects of $\operatorname{\mathbf{PSh}(\Theta_{n})}$ which are local with
respect to
$\partial\Theta_{n}[\sigma^{n-1}[k+3]]\to\Theta_{n}[\sigma^{n-1}[k+3]]$ in
$\operatorname{n\mathchar 45\relax QCat}$. We reason by induction on $n$. For
$n=1$, this is [CL20, Proposition 3.23]. Now let $X$ be an $(n+1)$-quasi-
category. The following assertions are equivalent:

  * •

$X$ is $(n+1+k)$-truncated

  * •

For every $x,y\in X_{0}$, the $n$-quasi-category $\operatorname{Hom}_{X}(x,y)$
is $(n+k)$-truncated (by Definition 4.7)

  * •

For every $x,y\in X_{0}$, the $n$-quasi-category $\operatorname{Hom}_{X}(x,y)$
is local with respect
$\partial\Theta_{n}[\sigma^{n-1}[k+3]]\to\Theta_{n}[\sigma^{n-1}[k+3]]$ in
$\operatorname{n\mathchar 45\relax QCat}$ (by induction on $n$)

  * •

For every $x,y\in X_{0}$ the bipointed $(n+1)$-quasi-category $(X,x,y)$ is
local with respect to
$\Sigma(\partial\Theta_{n}[\sigma^{n-1}[k+3]]\to\Theta_{n}[\sigma^{n-1}[k+3]])=(\partial\Theta_{n+1}[\sigma^{n}[k+3]]\to\Theta_{n+1}[\sigma^{n}[k+3]])$
in $*\sqcup*/\operatorname{(n+1)\mathchar 45\relax QCat}$ (by Proposition 1.6)

  * •

$X$ is local with respect to
$\partial\Theta_{n+1}[\sigma^{n}[k+3]]\to\Theta_{n+1}[\sigma^{n}[k+3]]$ in
$\operatorname{(n+1)\mathchar 45\relax QCat}$ (by Lemma 1.13)

Given the characterisation of truncated objects as local objects with respect
to a single morphism, the existence of the model structure comes from Theorem
1.4.

We know that the adjunctions in question are Quillen equivalences between
$\operatorname{n\mathchar 45\relax QCat}$ and $\operatorname{\Theta_{n}Sp}$
(by Theorems 2.19 and 2.23). To show that they remain Quillen equivalences
after localisation to their truncated versions, we use Theorem 1.9.

For the first adjunction, we have to show that an $(\infty,n)$-$\Theta$-space
$Z$ is an $(n+k,n)$-$\Theta$-space if and only if $i_{0}^{*}(Z)$ is an
$(n+k)$-truncated $n$-quasi-category. Once again, we proceed by induction on
$n$. For $n=1$, this is [CL20, Proposition 5.8.(2)]. The induction step
follows from applying the definitions of truncated objects and using the
isomorphism of Lemma 4.9.

For the second one, we have to show that an $n$-quasi-category $X$ is
$(n+k)$-truncated if and only if $t_{!}(X)$ is an $(n+k,n)$-$\Theta$-space.
Note that, for $X\in\operatorname{\mathbf{PSh}(\Theta_{n})}$ and
$\theta\in\Theta_{n}$, we have isomorphisms

$i_{0}^{*}(t^{!}(X))_{\theta}=t^{!}(X)_{(\theta,0)}=\operatorname{Hom}_{\operatorname{\mathbf{PSh}(\Theta_{n})}}(\Theta_{n}[\theta]\times
G(0),X)\cong X_{\theta}$

natural in $\theta$, and hence an isomorphism $X\cong i_{0}^{*}(t^{!}(X))$. So
$X$ is $(n+k)$-truncated if and only if $i_{0}^{*}(t^{!}(X))$ is
$(n+k)$-truncated, which by the last paragraph is the same as saying that
$t^{!}(X)$ is an $(n+k,n)$-$\Theta$-space. ∎

##  5\. Groupoidal objects

In this section, we define groupoidal $n$-quasi-categories, generalizing the
definitions of Kan complexes and of groupoidal 2-quasi-categories [Bri22]. We
show that these are the fibrant objects of a localisation of
$\operatorname{n\mathchar 45\relax QCat}$, and that both Quillen equivalences
between $\operatorname{n\mathchar 45\relax QCat}$ and
$\operatorname{\Theta_{n}Sp}$ still work in the level of groupoidal objects.

######  5.1.

The inclusion $i:\Delta\to\Theta_{n}$ induces an adjunction between the
presheaf categories:

$i_{!}:\operatorname{\mathbf{PSh}(\Delta)}\rightleftarrows\operatorname{\mathbf{PSh}(\Theta_{n})}:i^{*}$
(5.2)

######  Proposition 5.3.

The adjunction (5.2) is a Quillen adjunction

$i_{!}:\operatorname{QCat}\rightleftarrows\operatorname{n\mathchar 45\relax
QCat}:i^{*}$

###### Proof.

We use the recognition criterion for left Quillen functors (Theorem 3.19). The
functor $i_{!}$ preserves colimits and monomorphisms. By definition, it sends
$\Delta[p]$ to $\Theta_{n}[i[p]]$. Since it preserves colimits, it preserves
spine inclusions.

Recall that the functor $i$ admits a left adjoint $\pi:\Theta_{n}\to\Delta$,
sending $\theta=[p](\theta_{1},\ldots,\theta_{p})$ to $[p]$. Thus, the left
adjoint $i_{!}$ is isomorphic to the restriction functor $\pi^{*}$. A direct
computation shows that we have an isomorphism of functors $N_{n}i\cong
i_{!}N:\operatorname{\mathbf{Cat}}\to\operatorname{\mathbf{PSh}(\Theta_{n})}$,
and so $i_{!}(N(J)\to D_{0})=(N_{n}(J)\to D_{0})$.

We have shown that $i_{!}$ sends the classes $\mathcal{I}_{1}$ and
$\mathcal{J}_{1}$ to the classes $\mathcal{I}_{n}$ and $\mathcal{J}_{n}$,
which are in particular weak equivalences. ∎

######  Definition 5.4.

A quasi-category is groupoidal if it is a Kan complex. Let $n\geq 2$. A
$n$-quasi-category $X$ is groupoidal if

  1. (1)

$i^{*}(X)$ is a Kan complex

  2. (2)

For every $x,y\in X_{0}$, the $(n-1)$-quasi-category
$\operatorname{Hom}_{X}(x,y)$ is groupoidal

For short, we will call a groupoidal $n$-quasi-category an $n$-quasi-groupoid.

Recall from §2.15 the functor
$F:\Theta_{n}\to\operatorname{\mathbf{PSh}_{\Delta}}(\Theta_{n})$.

######  Proposition 5.5 (Rezk).

There exists a model structure $\operatorname{\Theta_{n}Gpd}$ on
$\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}$ given by localizing the
model structure $\operatorname{\Theta_{n}Sp}$ with respect to the set

$\\{\\{F(\tau^{n}_{k}):F(D_{k})\to F(D_{k-1}),1\leq k\leq n\\}\\}$

###### Proof.

The argument amounts to the 2-out-of-3 property of weak equivalences, used to
relate the morphisms presented here with the ones used in Rezk’s original
definition.

The set of morphisms used by Rezk in his definition [Rez10, §11.25] of the
model structure $\operatorname{\Theta_{n}Gpd}$ is another one, namely
$\\{V[1]^{k}(T_{\\#}q),0\leq k<n\\}$, where $T_{\\#}q:T_{\\#}F[1]\to T_{\\#}E$
in the notation of [Rez10]. Since the morphisms $V[1]^{k}(T_{\\#}p)$ for
$0\leq k<n$ (where $T_{\\#}p:T_{\\#}E\to T_{\\#}F[0]$) are already weak
equivalences in $\operatorname{\Theta_{n}Sp}$ [Rez10, §11.6], we can replace
(using 2-out-of-3) the first set by $\\{V[1]^{k}(T_{\\#}F(\tau^{1}_{1})),0\leq
k<n\\}$, where $\tau^{1}_{1}:[1]\to[0]$ is the unique morphism $[1]\to[0]$ in
$\Delta$. This is precisely the set $\\{F(\tau^{n}_{k}):F(D_{k})\to
F(D_{k-1}),1\leq k\leq n\\}\\}$, which follows from a straightforward
calculation using [Rez10, Proposition 4.2, Lemma 11.10]. ∎

######  Theorem 5.6.

There is a model structure $\operatorname{n\mathchar 45\relax QGpd}$ on
$\operatorname{\mathbf{PSh}(\Theta_{n})}$ whose fibrant objects are $n$-quasi-
groupoids. This model structure is the localization of
$\operatorname{n\mathchar 45\relax QCat}$ with respect to the morphisms
$\tau^{n}_{k}:\Theta_{n}[D_{k}]\to\Theta_{n}[D_{k-1}]$, for $1\leq k\leq n$.
Moreover, the adjunctions (2.18) and (2.22) are Quillen equivalences

$p^{*}:\operatorname{n\mathchar 45\relax
QGpd}\rightleftarrows\operatorname{\Theta_{n}Gpd}:i_{0}^{*}$

and

$t_{!}:\operatorname{\Theta_{n}Gpd}\rightleftarrows\operatorname{n\mathchar
45\relax QGpd}:t^{*}$

###### Proof.

First, let us show that an $n$-quasi-category $X$ is local with respect to
$\\{\tau^{n}_{k},1\leq k\leq n\\}$ if and only if $X$ is groupoidal. For $n=1$
the result holds, since Kan complexes are exactly quasi-categories which are
local with respcet to $\Delta[1]\to\Delta[0]$ in $\operatorname{QCat}$ (cf.
[CL20, Proposition 3.30]).

Suppose it holds for $n=m$, and let us show it is true for $n=m+1$. Let $X$ be
an $(m+1)$-quasi-category. The underlying quasi-category $i^{*}(X)$ is a Kan
complex if and only if it is local with respect to $\tau^{1}_{1}$. By
Propositions 5.1 and 1.6, it amounts to say that $X$ is local with respect to
$i_{!}(\tau^{1}_{1})=\tau^{m+1}_{1}$.

Now let $x,y\in X_{0}$. By the induction hypothesis,
$\operatorname{Hom}_{X}(x,y)$ is groupoidal if and only if it is local with
respect to $\\{\tau^{m}_{k},1\leq k\leq m\\}$. By Propositions 4.6 and 1.6, it
is the case exactly when $(X,x,y)$ is local with respect to the set of
bipointed morphisms

$\Sigma(\\{\tau^{m}_{k},1\leq k\leq m\ \\})=\\{\tau^{m+1}_{k},2\leq k\leq m+1\
\\}$

We conclude using Lemma 1.13 that $\operatorname{Hom}_{X}(x,y)$ is groupoidal
for all $x,y\in X_{0}$ if and only if $X$ is local with respect to
$\\{\tau^{m+1}_{k},2\leq k\leq m+1\ \\}$ in $(m+1)$-$\operatorname{QCat}$.

Combining conditions (1) and (2) of Definition 5.4, we have the desired
result.

The existence of the model structure follows directly from Theorem 1.4.

Finally, we show that the adjunctions are indeed Quillen equivalences. Since
the two adjunctions are already Quillen equivalences between
$\operatorname{n\mathchar 45\relax QCat}$ and $\operatorname{\Theta_{n}Sp}$ by
Theorems 2.19 and 2.23, it is sufficient by Theorem 1.7 to show that the class
of morphisms by which we localise the right-hand side model structures is the
image by the left Quillen functors of the class by which we localise the left-
hand side model structures. This is the case thanks to Lemmas 2.20 and 2.24,
as the morphisms used in the localisation $\operatorname{n\mathchar 45\relax
QGpd}$ (resp. $\operatorname{\Theta_{n}Gpd}$) are the image by the Yoneda
embedding (resp. by the functor
$F:\Theta_{n}\to\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}$) of the
morphisms $\tau^{n}_{k}$ of $\Theta_{n}$. ∎

In the Kan-Quillen model structure on simplicial sets, every representable is
contractible. The same is true in the model structure
$\operatorname{n\mathchar 45\relax QGpd}$ on
$\operatorname{\mathbf{PSh}(\Theta_{n})}$.

######  Proposition 5.7.

Let $\theta\in\Theta_{n}$. The morphism $\Theta_{n}[\theta]\to\Theta_{n}[0]$
is a weak equivalence in $\operatorname{n\mathchar 45\relax QGpd}$.

###### Proof.

We omit the notation $\Theta_{n}[-]$ for representables in this proof.

Let $\theta\in\Theta_{n}$. The spine $I[\theta]$ is a colimit of the form

${{D_{i_{1}}}}$${{D_{i_{2}}}}$${\ldots}$${{D_{i_{m}}}}$${{D_{j_{1}}}}$${{D_{j_{2}}}}$${{D_{j_{m-1}}}}$

for some $i_{1},\ldots,i_{m},j_{1},\ldots,j_{m-1}$ with $i_{k}>j_{k}<i_{k+1}$,
$1\leq k<m$ (cf. §2.11).

By definition of the model structure $\operatorname{n\mathchar 45\relax
QGpd}$, the morphisms

$D_{n}\to D_{n-1}\to\ldots\to D_{1}\to D_{0}$

are all weak equivalences, so the diagram above is equivalent in
$\operatorname{n\mathchar 45\relax QGpd}$ to the diagram of same shape and
only $D_{0}$’s and identities. The colimit of both diagram is actually a
homotopy colimit (see proof of Theorem 3.16), so the equivalence of diagrams
induces a weak equivalence $I[\theta]\to D_{0}$.

Since the spine inclusion $I[\theta]\to\theta$ is a weak equivalence of
$\operatorname{n\mathchar 45\relax QCat}$ (and hence of
$\operatorname{n\mathchar 45\relax QGpd}$), the 2-out-of-3 property of weak
equivalences implies that $\theta\to D_{0}$ is a weak equivalence of
$\operatorname{n\mathchar 45\relax QGpd}$. ∎

##  6\. $n$-quasi-groupoids vs spaces

In [Rez10], Rezk shows that there is a Quillen equivalence between the model
structure for groupoidal objects in
$\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}$ and the model structure for
Kan complexes in $\operatorname{\mathbf{PSh}(\Delta)}$. Combining this
equivalence with both equivalences of Theorem 5.6, we get respectively a
zigzag of Quillen equivalences and a direct Quillen equivalence between the
model structure for groupoidal $n$-quasi-categories and the one for Kan
complexes. The aim of this section is to provide another direct Quillen
equivalence between these model structures, induced by the inclusion
$i:\Delta\to\Theta_{n}$, which does not factors through
$\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}$.

######  6.1.

Let
$\operatorname{ev}_{0}:\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}\to\operatorname{\mathbf{PSh}(\Delta)}$
be the functor defined on objects by $X\mapsto X([0],-)$. It is the right
adjoint of an adjoint pair

$\operatorname{ct}:\operatorname{\mathbf{PSh}(\Delta)}\rightleftarrows\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}:\operatorname{ev}_{0}$
(6.2)

where $\operatorname{ct}$ is the constant functor, i.e., for
$K\in\operatorname{\mathbf{PSh}(\Delta)}$, $\operatorname{ct}(K)_{\theta}=K$.

######  6.3.

Let $k:\Delta\to\operatorname{\mathbf{PSh}(\Delta)}$ be the composite

$\Delta\hookrightarrow\operatorname{\mathbf{Cat}}\xrightarrow{\Pi}\operatorname{\mathbf{Gpd}}\hookrightarrow\operatorname{\mathbf{Cat}}\xrightarrow{N}\operatorname{\mathbf{PSh}(\Delta)}$

This functor induces a Kan extension-nerve adjunction

$k_{!}:\operatorname{\mathbf{PSh}(\Delta)}\rightleftarrows\operatorname{\mathbf{PSh}(\Delta)}:k^{!}$
(6.4)

######  Remark 6.5.

The functor $k^{!}$ is related to the maximal sub-Kan complex functor $J$,
whose definition we recall. Let $X$ be a quasi-category,
$\operatorname{ho}(X)$ be its homotopy category and
$J:\operatorname{\mathbf{Cat}}\to\operatorname{\mathbf{Gpd}}$ be the maximal
sub-groupoid functor, which is the right adjoint of the inclusion
$\operatorname{\mathbf{Gpd}}\to\operatorname{\mathbf{Cat}}$. The simplicial
set $J(X)$ is the pullback

${{J(X)}}$${X}$${{N(J\operatorname{ho}(X))}}$${{N(\operatorname{ho}(X))}}$$\scriptstyle{\lrcorner}$

The simplicial set $J(X)$ is a Kan complex for every quasi-category $X$.
Indeed, it is the maximal Kan complex included in $X$, since $J$ defines a
functor $J:\mathbf{QCat}\to\mathbf{KanCx}$, right adjoint to the inclusion
$\mathbf{KanCx}\to\mathbf{QCat}$, where $\mathbf{KanCx}$ (resp.
$\mathbf{Qcat}$) is the full subcategory of
$\operatorname{\mathbf{PSh}(\Delta)}$ formed by Kan complexes (resp. quasi-
categories) [Joy08b, Theorem 4.19].

If $X$ is a quasi-category, the Quillen adjunction
$k_{!}:\operatorname{Sp}\rightleftarrows\operatorname{QCat}:k^{!}$ [Joy08b,
Theorem 6.22] implies that $k^{!}(X)$ is a Kan complex. Moreover, there is a
morphism $k^{!}(X)\to X$ induced by the inclusion $k([p])\to\Delta[p]$ for
every $[p]\in\Delta$. Therefore, the arrow $k^{!}(X)\to X$ factors through
$J(X)\to X$. The morphism $k^{!}(X)\to J(X)$ is in fact a trivial fibration
[Joy08b, Proposition 6.26].

######  Lemma 6.6.

The following square commutes up to isomorphism.

${{\operatorname{\mathbf{PSh}(\Theta_{n})}}}$${{\operatorname{\mathbf{PSh}_{\Delta}(\Theta_{n})}}}$${{\operatorname{\mathbf{PSh}(\Delta)}}}$${{\operatorname{\mathbf{PSh}(\Delta)}}}$$\scriptstyle{i^{*}}$$\scriptstyle{t^{!}}$$\scriptstyle{\operatorname{ev}_{0}}$$\scriptstyle{k^{!}}$

###### Proof.

This is a direct computation using the definitions of the functors. Indeed,
let $X\in\operatorname{\mathbf{PSh}(\Theta_{n})}$ and $[m]\in\Delta$. We have

$\operatorname{ev}_{0}t^{!}(X)_{m}=t^{!}(X)_{([0],[m])}=\operatorname{Hom}_{\operatorname{\mathbf{PSh}(\Theta_{n})}}(\Theta_{n}[0]\times
G([m]),X)\cong\operatorname{Hom}_{\operatorname{\mathbf{PSh}(\Theta_{n})}}(N_{n}\Pi([m]),X)$

and

$k^{!}i^{*}(X)_{m}=\operatorname{Hom}_{\operatorname{\mathbf{PSh}(\Delta)}}(k([m]),i^{*}(X))\cong\operatorname{Hom}_{\operatorname{\mathbf{PSh}(\Theta_{n})}}(i_{!}N\Pi([m]),X)\cong\operatorname{Hom}_{\operatorname{\mathbf{PSh}(\Theta_{n})}}(N_{n}\Pi([m]),X)$

∎

######  Proposition 6.7 (Rezk).

The adjunction (6.2) is a Quillen equivalence

$\operatorname{ct}:\operatorname{Sp}\rightleftarrows\operatorname{\Theta_{n}Gpd}:\operatorname{ev}_{0}$

###### Proof.

See [Rez10, Proposition 11.27.(1)]. ∎

######  Proposition 6.8 (Joyal).

The adjunction (6.4) is a Quillen equivalence

$k_{!}:\operatorname{Sp}\rightleftarrows\operatorname{Sp}:k^{!}$

###### Proof.

The adjunction is Quillen as a composite of two Quillen adjunctions:

$k_{!}:\operatorname{Sp}\rightleftarrows\operatorname{QCat}:k^{!}$

which is a Quillen adjunction by [Joy08b, Theorem 6.22], and

$\operatorname{id}:\operatorname{QCat}\rightleftarrows\operatorname{Sp}:\operatorname{id}$

given by the fact that $\operatorname{Sp}$ is a left Bousfield localisation of
$\operatorname{QCat}$.

Therefore, we have to show that the derived adjunction

$\mathbb{L}k_{!}:\operatorname{Ho}(\operatorname{Sp})\rightleftarrows\operatorname{Ho}(\operatorname{Sp}):\mathbb{R}k^{!}$

is an equivalence of categories. This is true since for every (cofibrant)
object $X$ of $\operatorname{\mathbf{PSh}(\Delta)}$, there is a natural weak
homotopy equivalence $X\to k_{!}X$ [Joy08b, Theorem 6.22], and then a natural
isomorphism of functors
$\operatorname{id}\cong\mathbb{L}k_{!}:\operatorname{Ho}(\operatorname{Sp})\to\operatorname{Ho}(\operatorname{Sp})$.
∎

######  Proposition 6.9.

The adjunction (5.2) is a Quillen adjunction

$i_{!}:\operatorname{Sp}\rightleftarrows\operatorname{n\mathchar 45\relax
QGpd}:i^{*}$

###### Proof.

The adjunction
$i_{!}:\operatorname{QCat}\rightleftarrows\operatorname{n\mathchar 45\relax
QGpd}:i^{*}$ is Quillen as a composite of the Quillen adjunctions

$i_{!}:\operatorname{QCat}\rightleftarrows\operatorname{n\mathchar 45\relax
QCat}:i^{*}$

of Proposition 5.3 and

$\operatorname{id}:\operatorname{n\mathchar 45\relax
QCat}\rightleftarrows\operatorname{n\mathchar 45\relax
QGpd}:\operatorname{id}$

given by left Bousfield localisation.

Since for every $n$-quasi-groupoid $X$, the simplicial set $i^{*}(X)$ is a Kan
complex (i.e., a fibrant object in $\operatorname{Sp}$) by definition, and
$\operatorname{Sp}$ is a localisation of $\operatorname{QCat}$, the adjunction

$i_{!}:\operatorname{Sp}\rightleftarrows\operatorname{n\mathchar 45\relax
QGpd}:i^{*}$

is Quillen by Theorem 1.8. ∎

######  Theorem 6.10.

The adjunction (5.2) is a Quillen equivalence

$i_{!}:\operatorname{Sp}\rightleftarrows\operatorname{n\mathchar 45\relax
QGpd}:i^{*}$

###### Proof.

We use the 2-out-of-3 property of Quillen equivalences, applied to the diagram
of Lemma 6.6 with the following model structures

${\operatorname{n\mathchar 45\relax
QGpd}}$${\operatorname{\Theta_{n}Gpd}}$${\operatorname{Sp}}$${\operatorname{Sp}}$$\scriptstyle{i^{*}}$$\scriptstyle{t^{!}}$$\scriptstyle{\operatorname{ev}_{0}}$$\scriptstyle{k^{!}}$

The horizontal arrows are the right Quillen functors of Quillen equivalences
by Theorem 5.6 and Proposition 6.8. The right vertical arrow is the right
Quillen functor of the Quillen equivalence of Proposition 6.7. The left
vertical arrow is the right Quillen functor of a Quillen adjunction by
Proposition 6.9. ∎

######  6.11.

We say that two objects of a model category are weakly equivalent if there is
a zigzag of weak equivalences between them, that is, if they are isomorphic in
the homotopy category.

We introduce some terminology to clarify the nature of the next lemma.

######  6.12.

Let $\mathcal{M}$ be a model category with a terminal object $*$. We will
define what we mean by a sphere in $\mathcal{M}$. The 0-sphere
$\mathbb{S}^{0}$ is the object $*\sqcup*$. For $k>0$, the $k$-sphere
$\mathbb{S}^{k}$ is the homotopy colimit of the diagram

${{\mathbb{S}^{k-1}}}$${{*}}$${{*}}$

in $\mathcal{M}$. This definition is unique up to isomorphism in the homotopy
category $\operatorname{Ho}(\mathcal{M})$, i.e., all $k$-spheres are weakly
equivalent.

######  Example 6.13.

Consider the Kan-Quillen model structure on the category of simplicial sets.
For every $k>0$, the boundary $\partial\Delta[k]$ is a model for
$\mathbb{S}^{k-1}$. Indeed, $\partial\Delta[1]=\Delta[0]\sqcup\Delta[0]$, and
$\Delta[0]$ is the terminal object of $\operatorname{\mathbf{PSh}(\Delta)}$.
By induction, suppose that $\partial\Delta[k]$ is a model for
$\mathbb{S}^{k-1}$. Consider the following pushout diagram

${{\partial\Delta[k]}}$${{\Lambda^{0}[k+1]}}$${{\Delta[k]}}$

where the horizontal arrow sends $\partial\Delta[k]$ to the boundary of the
missing face, and the vertical arrow is the boundary inclusion. It can be used
to compute the homotopy pushout in the definition above (cf. [Lur09,
Proposition A.2.4.4]), since by induction $\partial\Delta[k]$ models
$\mathbb{S}^{k-1}$, and we know that all objects are cofibrant, both arrows
are cofibrations, and the objects $\Delta[k]$ and $\Lambda^{0}[k+1]$ are
contractible in $\operatorname{Sp}$. The pushout, which equals
$\partial\Delta[k+1]$, is then a model of $\mathbb{S}^{k}$.

In the following statement and proof, we omit the notation of the fully
faithful inclusions $\Theta_{m}\to\Theta_{n}$, for $m<n$.

######  Lemma 6.14.

Let $k>0$. For every $0\leq\ell<n$ and $p\geq 0$ such that $\ell+p=k$, the
object $\partial\Theta_{n}[\sigma^{\ell}[p]]$ of
$\operatorname{\mathbf{PSh}(\Theta_{n})}$ is a $(k-1)$-sphere in
$\operatorname{n\mathchar 45\relax QGpd}$.

###### Proof.

The proof follows by induction on $n$ and $k$.

For $n=1$, we have $\Theta_{1}=\Delta$, $\ell=0$, and the result is explained
in Example 6.13. Now suppose the result is true for $\Theta_{n-1}$, for $n>1$,
and we will show that it is true for $\Theta_{n}$.

We proceed by induction on $k$. For $k=1$, the statement is true, since we
have

$\partial\Theta_{n}[\sigma[0]]=\partial\Theta_{n}[1]=\Theta_{n}[0]\sqcup\Theta_{n}[0]$

Now suppose it is true for $k^{\prime}\leq k$, $k>0$. We want to show that
$\partial\Theta_{n}[\sigma^{\ell}[p]]$ is a $k$-sphere, for $\ell+p=k+1$.

For $\ell=0$ (so $p=k+1$), we have

$\partial\Theta_{n}[k+1]=i_{!}(\partial\Delta[k+1])$

which is a $k$-sphere, given the Example 6.13 and the fact that
$i_{!}:\operatorname{Sp}\to\operatorname{n\mathchar 45\relax QGpd}$ is a left
Quillen functor, preserving both cofibrations and weak equivalences (since all
objects are cofibrant) and sending the terminal object of
$\operatorname{\mathbf{PSh}(\Delta)}$ to the terminal object of
$\operatorname{\mathbf{PSh}(\Theta_{n})}$.

For $\ell>0$, we have

$\partial\Theta_{n}[\sigma^{\ell}[p]]=\Sigma(\partial\Theta_{n-1}[\sigma^{\ell-1}[p]])$

and $\partial\Theta_{n-1}[\sigma^{\ell-1}[p]]$ is a model for
$\mathbb{S}^{k-1}$ by the induction hypothesis. If $p>0$, it can be presented
as a homotopy pushout

${{\partial\Theta_{n-1}[\sigma^{\ell-1}[p-1]]}}$${{*}}$${{*}}$${{\partial\Theta_{n-1}[\sigma^{\ell-1}[p]]}}$$\scriptstyle{h}$$\scriptstyle{\lrcorner}$

For every $n$, the Quillen adjunction $\Sigma:\operatorname{(n-1)\mathchar
45\relax QCat}\rightleftarrows*\sqcup*/\operatorname{n\mathchar 45\relax
QCat}:\operatorname{Hom}$ of Proposition 4.6 descends to a Quillen adjunction
$\operatorname{(n-1)\mathchar 45\relax
QGpd}\rightleftarrows*\sqcup*/\operatorname{n\mathchar 45\relax QGpd}$ by
Proposition 1.8, since if $X$ is a $n$-quasi-groupoid, then
$\operatorname{Hom}_{X}(x,y)$ is a $(n-1)$-quasi-groupoid by definition.

Therefore, applying $\Sigma$ to the diagram above, we obtain a homotopy
pushout square

${{\partial\Theta_{n}[\sigma^{\ell}[p-1]]}}$${{D_{1}}}$${{D_{1}}}$${{\partial\Theta_{n}[\sigma^{\ell}[p]]}}$$\scriptstyle{h}$$\scriptstyle{\lrcorner}$

in $\operatorname{n\mathchar 45\relax QGpd}$, and we conclude by noting that
$\Theta_{n}[D_{1}]\to\Theta_{n}[D_{0}]$ is a weak equivalence in this model
category by definition.

It remains only to treat the case where $\ell>0$ and $p=0$, and so
$\ell=k+1<n$. In this case, we have
$\Theta_{n}[\sigma^{\ell}[0]]=\Theta_{n}[D_{k+1}]$. Consider the following
pushout square

${{\partial\Theta_{n}[D_{k}]}}$${{\Theta_{n}[D_{k}]}}$${{\Theta_{n}[D_{k}]}}$${{\partial\Theta_{n}[D_{k+1}]}}$$\scriptstyle{\lrcorner}$

Note that it is a homotopy pushout, that $\partial\Theta_{n}[D_{k}]$ is a
$(k-1)$-sphere by induction and that $\Theta_{n}[D_{k}]$ is contractible by
the definition of $\operatorname{n\mathchar 45\relax QGpd}$. Therefore,
$\partial\Theta_{n}[D_{k+1}]$ is a model for $\mathbb{S}^{k}$. ∎

######  6.15.

We denote by $\operatorname{n\mathchar 45\relax QGpd_{k}}$ the localisation of
$\operatorname{n\mathchar 45\relax QGpd}$ whose fibrant objects are
$(n+k)$-truncated $n$-quasi-groupoids. By Theorem 4.12, it is the localisation
of $\operatorname{n\mathchar 45\relax QGpd}$ with respect to the boundary
inclusion
$\partial\Theta_{n}[\sigma^{n-1}[k+3]]\to\Theta_{n}[\sigma^{n-1}[k+3]]$.

######  6.16.

Let $m\geq 0$. Homotopy $m$-types can be characterised as local objects in the
Kan-Quillen model structure on simplicial sets. Indeed, a Kan complex $X$ is
an $m$-type if and only if it is local with respect to the boundary inclusion
$\partial\Delta[m+2]\to\Delta[m+2]$ in this model structure (see for example
[CL20, Corollary 3.25]). Therefore, by Theorem 1.4, the Bousfield localisation
of the Kan-Quillen model structure with respect to
$\delta_{m+2}:\partial\Delta[m+2]\to\Delta[m+2]$ produces a model structure
whose fibrant objects are homotopy $m$-types, which we denote by
$\operatorname{Sp_{m}}$.

######  Theorem 6.17.

The adjunction (5.2) is a Quillen equivalence

$i_{!}:\operatorname{Sp_{n+k}}\rightleftarrows\operatorname{n\mathchar
45\relax QGpd_{k}}:i^{*}$

###### Proof.

Starting from the Quillen equivalence

$i_{!}:\operatorname{Sp}\rightleftarrows\operatorname{n\mathchar 45\relax
QGpd}:i^{*}$

of Theorem 6.10, we can localize $\operatorname{Sp}$ with respect to the
morphism $\delta_{[n+k+2]}:\partial\Delta[n+k+2]\to\Delta[n+k+2]$ and
$\operatorname{n\mathchar 45\relax QGpd}$ with respect to the image of this
morphism by the (left-derived) functor $i_{!}$, i.e., by

$i_{!}(\delta_{[n+k+2]}:\partial\Delta[n+k+2]\to\Delta[n+k+2])=(\delta_{[n+k+2]}:\partial\Theta_{n}[i[n+k+2]]\to\Theta_{n}[i[n+k+2]])$

After localisation, we obtain a new Quillen equivalence

$i_{!}\operatorname{Sp_{n+k}}\rightleftarrows\operatorname{n\mathchar 45\relax
QGpd}_{\operatorname{loc}}:i^{*}$

by Theorem 1.7.

We want to show that the localised model structure $\operatorname{n\mathchar
45\relax QGpd}_{\operatorname{loc}}$ is exactly $\operatorname{n\mathchar
45\relax QGpd_{k}}$. Recall from §6.15 that $\operatorname{n\mathchar 45\relax
QGpd_{k}}$ is the localisation of $\operatorname{n\mathchar 45\relax QGpd}$
with respect to the boundary inclusion
$\delta_{\sigma^{n-1}[k+3]}:\partial\Theta_{n}[\sigma^{n-1}[k+3]]\to\Theta_{n}[\sigma^{n-1}[k+3]]$.

Thanks to Lemma 6.14, we know that both $\partial\Theta_{n}[i[n+k+2]]$ and
$\partial\Theta_{n}[\sigma^{n-1}[k+3]]$ are models for the $(n+k+1)$-sphere in
$\operatorname{n\mathchar 45\relax QCat}$, so they are weakly equivalent in
this model structure. Moreover, by Proposition 5.7, all representable
$\Theta_{n}$-sets are contractible in $\operatorname{n\mathchar 45\relax
QGpd}$. We can then draw the following commutative diagram

${{\partial\Theta_{n}[i[n+k+2]]}}$${\bullet}$${\ldots}$${\bullet}$${{\partial\Theta_{n}[\sigma^{n-1}[k+3]]}}$${{\Theta_{n}[i[n+k+2]]}}$${{\Theta_{n}[0]}}$${{\Theta_{n}[\sigma^{n-1}[k+3]]}}$$\scriptstyle{\delta_{[n+k+2]}}$$\scriptstyle{\delta_{\sigma^{n-1}[k+3]}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

where the arrows marked with $\sim$ are weak equivalences in
$\operatorname{n\mathchar 45\relax QGpd}$ and the bullets represent some
objects appearing in the zigzag of weak equivalences.

In $\operatorname{n\mathchar 45\relax QGpd}_{\operatorname{loc}}$ (resp.
$\operatorname{n\mathchar 45\relax QGpd_{k}}$), the left (resp. right)
vertical arrow is a weak equivalence, so the right (resp. left) vertical arrow
is also a weak equivalence, by 2-out-of-3. Since localising a model structure
with respect to a weak equivalence does not change anything, and since
localisations commute (cf. Proposition 1.11), we have:

$\operatorname{n\mathchar 45\relax
QGpd}_{\operatorname{loc}}=L_{\delta_{\sigma^{n-1}[k+3]}}\operatorname{n\mathchar
45\relax
QGpd}_{\operatorname{loc}}=L_{\delta_{[n+k+2]}}\operatorname{n\mathchar
45\relax QGpd_{k}}=\operatorname{n\mathchar 45\relax QGpd_{k}}$

∎

We have seen that, working with different model structures on the category of
$\Theta_{n}$-sets, we are able to obtain models for the theory of
$(\infty,n)$-categories (with Ara’s model structure), of $(n+k,n)$-categories
(with the truncated model structures) and of $(n+k,0)$-categories (with the
groupoidal model structures) for $k\geq 0$. We actually expect that it is
possible to model any theory of $(m+k,m)$-categories inside the realm of
$\Theta_{n}$-sets, for $m\leq n$ and $0\leq k\leq\infty$. For example, the
passage from $(\infty,n)$ to $(\infty,m)$ should be made by localizing by all
morphisms $D_{j}\to D_{j-1}$, for $m<j\leq n$. The case $n=2$, $m=1$ is proven
by Campbell in [Cam20, Theorem 11.14]. However, for the general case one
should develop other techniques, that are not within the scope of this work.

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