# Hom weak $\omega$-categories of a weak $\omega$-category

Thomas Cottrell  Department of Mathematical Sciences, University of Bath,
Bath, United Kingdom<EMAIL_ADDRESS>and  Soichiro Fujii  Research
Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
<EMAIL_ADDRESS>Dedicated to John Power on the occasion of his 60th
birthday.

###### Abstract.

Classical definitions of weak higher-dimensional categories are given
inductively; for example, a bicategory has a set of objects and hom
categories, and a tricategory has a set of objects and hom bicategories.
However, more recent definitions of weak $n$-categories for all natural
numbers $n$, or of weak $\omega$-categories, take more sophisticated
approaches, and the nature of the “hom” is often not immediate from the
definitions. In this paper, we focus on Leinster’s definition of weak
$\omega$-category based on an earlier definition by Batanin, and construct for
each weak $\omega$-category $\mathcal{A}$, an underlying (weak
$\omega$-category)-enriched graph consisting of the same objects and for each
pair of objects $x$ and $y$, a hom weak $\omega$-category $\mathcal{A}(x,y)$.
We also show that our construction is functorial with respect to weak
$\omega$-functors introduced by Garner.

###### Key words and phrases:

Weak $\omega$-category, weak $\omega$-groupoid, weak $\omega$-functor, operad,
intensional Martin-Löf type theory, identity type.

###### 2020 Mathematics Subject Classification:

18N65, 18N20

We gratefully acknowledge the support of Royal Society grant IE160402. The
second author is supported by ERATO HASUO Metamathematics for Systems Design
Project (No. JPMJER1603), JST. No data were generated in association with this
paper.

##  1\. Introduction

_Identity types_ are one of the most interesting features of (intensional)
Martin-Löf type theory [34]. Given any type $A$ and pair of terms $a,b$ of
that type, this feature yields the type $\mathrm{Id}_{A}(a,b)$, which, under
the Curry–Howard correspondence, may be regarded as the logical proposition
$a=b$. Since identity types are themselves types, we can also obtain types of
the form $\mathrm{Id}_{\mathrm{Id}_{A}(a,b)}(p,q)$. This process leads to an
infinite hierarchy of iterated identity types, which has certain structure
reminiscent of the familiar reflexivity, symmetry and transitivity properties
of equality. In fact it forms an internal _weak $\omega$-groupoid_ [9, 29],
thus suggesting connections to higher-dimensional category theory and homotopy
theory [37].

In order to prove that certain principles (such as Uniqueness of Identity
Proofs) are undecidable in Martin-Löf type theory, Hofmann and Streicher [23]
gave a groupoid model of it. Then Warren [38] gave a strict $\omega$-groupoid
model, which refutes further principles (such as the truncation rules). In
both cases, the interpretations of identity types were given by means of homs;
in the former case hom sets of a groupoid seen as discrete groupoids, and in
the latter case hom strict $\omega$-groupoids of a strict $\omega$-groupoid.
With the abovementioned observation that types have the structure of internal
weak $\omega$-groupoids in mind, it seems natural to seek a model of Martin-
Löf type theory using (external) weak $\omega$-groupoids, or at least a
suitable subclass of them. This paper is a contribution to this goal; here we
shall establish a crucial step for it, by showing that an (external) weak
$\omega$-groupoid indeed has (external) weak $\omega$-groupoids as homs.

Weak $\omega$-groupoids are weak $\omega$-categories in which each $k$-cell
($k\geq 1$) is weakly invertible, and in turn, weak $\omega$-categories are a
higher-dimensional analogue of categories in which one has $k$-cells for each
$k\in\mathbb{N}$ and various composition operations, satisfying the usual
category axioms up to coherent higher-dimensional cells. Weak
$\omega$-categories can be thought of as a limit of weak $n$-categories for
$n\in\mathbb{N}$, in which one only has $k$-cells for $k\leq n$. Classical
definitions of weak $n$-category for small $n$ are well-known: the cases
$n=0,1,2$ and $3$ correspond to set, category, bicategory [6] and tricategory
[21] respectively.

Subsequently, several definitions of weak $n$-category for arbitrary
$n\in\mathbb{N}$, as well as of weak $\omega$-category, have been proposed by
various authors; see [27]. In this paper we shall focus on the definition
given by Leinster [28] following an earlier definition by Batanin [5], this
being the one adopted in the abovementioned papers [9, 29]. Hence, more
specifically, our main aim is to show that each weak $\omega$-category in the
sense of Leinster has weak $\omega$-categories (again, in the sense of
Leinster) as homs; a completely parallel argument also shows that a weak
$(n+1)$-category has weak $n$-categories as homs for each $n\in\mathbb{N}$. We
then observe that this result suitably restricts to weak $\omega$-groupoids.

The reader familiar with the classical definitions of bicategory or
tricategory might guess that such a result would be immediate from the
definition, as, for example, a tricategory ${\mathcal{T}}$ is _defined_ as the
data consisting of a set of objects ${\mathrm{ob}({\mathcal{T}})}$, and for
each pair $x,y$ of objects, a _hom bicategory_ ${\mathcal{T}}(x,y)$, together
with various composition operations and coherence cells. However, the weakened
enrichment approach used in the classical definitions requires the coherence
cells to be specified explicitly. The number of coherence cells needed, and
the complexity of those cells, increases rapidly as the dimension increases,
and it therefore becomes impractical to maintain this approach when defining
weak $n$-categories for arbitrary $n$ or $n=\omega$.

Consequently, the various proposed definitions of weak $n$-category—including
Leinster’s definition—do not intrinsically include hom weak
$(n-1)$-categories, and they are so different in style to the classical
definitions that our result is not at all immediate.111One arguable exception
is Trimble’s definition, which does take a weakened enrichment approach.
However, Trimble’s approach differs from the classical definitions in other
important ways. In particular the resulting $n$-categories are not fully weak
as they do not have weak interchange, and indeed Trimble described them as
“flabby $n$-categories” rather than “weak $n$-categories” [27]. Indeed, to the
best of our knowledge no one seems to have written down a detailed proof of
this result, either in the case of $\omega$ or $n\in\mathbb{N}$. Our result
would be vital not only for a semantics of identity types, but also for any
serious development of a theory of weak higher-dimensional categories.

Leinster’s definition of weak $\omega$-category starts from the underlying
structure of _$\omega$ -graph_, which simply consists of cells of various
dimensions together with suitable boundary (source and target) information.
The weak $\omega$-categories are defined as the Eilenberg–Moore algebras of a
suitable monad on the category of $\omega$-graphs; using the terminology we
shall explain in Section 2, this monad is induced by the _initial
$T^{(\omega)}$-operad with contraction_. We denote the Eilenberg–Moore
category of the monad for weak $\omega$-category by ${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}}$. Our main
construction amounts to defining a suitable “forgetful” functor

${U_{\mathrm{s}}}\colon{\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}\longrightarrow{{({\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}})}\mathchar
45\relax\mathbf{Gph}},$

where the codomain is the category of (${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}}$)-enriched graphs
(cf. Definition 2.3); applying ${U_{\mathrm{s}}}$ to a weak $\omega$-category
${\mathcal{A}}$, we obtain the (${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}}$)-enriched graph
consisting of the same objects and for each pair of objects $x$ and $y$, a hom
weak $\omega$-category ${\mathcal{A}}(x,y)$. The key to the definition of
${U_{\mathrm{s}}}$ is the observation that both the domain and the codomain of
${U_{\mathrm{s}}}$ are monadic over the category of $\omega$-graphs. We shall
induce ${U_{\mathrm{s}}}$ from a canonical monad morphism, which is ultimately
induced by the _initiality_ used to determine the monad for weak
$\omega$-category.

The morphisms of the Eilenberg–Moore category ${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}}$ are called
_strict $\omega$-functors_, because they preserve the structures of weak
$\omega$-categories strictly. In the context of weak higher-dimensional
categories, however, a more natural notion of functor is one that preserves
the structures up to coherent weakly invertible cells. Garner [20] introduced
such functors between weak $\omega$-categories, which we call _weak
$\omega$-functors_. Our construction of hom weak $\omega$-categories is
compatible with weak $\omega$-functors. That is, denoting the category of weak
$\omega$-categories and weak $\omega$-functors by ${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}}$, we show that the functor
${U_{\mathrm{s}}}$ extends to a functor

$U\colon{\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}}\longrightarrow{{({\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}})}\mathchar
45\relax\mathbf{Gph}}.$

This means that from a weak $\omega$-functor
$F\colon{\mathcal{A}}\longrightarrow{\mathcal{B}}$, we can extract its action
on homs as a family of weak $\omega$-functors
$(F_{x,y}\colon{\mathcal{A}}(x,y)\longrightarrow{\mathcal{B}}(Fx,Fy))_{x,y\in{\mathrm{ob}({\mathcal{A}})}}$.

In Section 2 we review Leinster’s definition of weak $\omega$-category. Then
in Section 3 we construct the forgetful functor ${U_{\mathrm{s}}}$. Section 4
provides a definition of weak $\omega$-functor due to Garner. In Section 5 we
construct the extension $U$ of ${U_{\mathrm{s}}}$. In the final Section 6
(which can be read independently of Sections 4 and 5) we review a definition
of weak $\omega$-groupoid and observe that our construction suitably restricts
to this case.

### Related work

At the beginning of the introduction, we mentioned several papers on the
border between higher-dimensional category theory and type theory, in order to
motivate the problem we shall treat in this paper from a computer science
perspective. Although the rest of this paper does not use any type theory, we
shall mention another, related line of research connecting these two subjects.
The papers we have in mind aim to formalise the definitions of weak
$\omega$-groupoid [1, 12] and of weak $\omega$-category [17, 7, 8] via
suitable dependent type theories. (As Cartmell [13] has shown, dependent type
theories can be used as presentations of (generalised) algebraic theories.)

In a sense, the papers in this line of research are complementary to those
mentioned at the beginning of the introduction: the former capture given
higher-dimensional categorical structures by designing suitable type theories,
whereas the latter relate suitable higher-dimensional categorical structures
to a given type theory, namely Martin-Löf type theory. The definitions of weak
$\omega$-groupoid and of weak $\omega$-category adopted in the papers [12, 17,
7, 8] are those of Grothendieck–Maltsiniotis [22, 31], whose relationship to
the Batanin–Leinster definitions has been studied by Ara [2].

### Acknowledgement

We thank John Power for the countless discussions on weak higher-dimensional
categories we have had in Bath, Tokyo and Kyoto. Anonymous referees provided
us with detailed lists of comments, which helped to improve the presentation.

##  2\. Leinster’s definition of weak $\omega$-category

In this section we review Leinster’s definition of weak $\omega$-category
[28], which was introduced as a variant of an earlier definition by Batanin
[5]. According to this definition, the weak $\omega$-categories are the
Eilenberg–Moore algebras of a monad on the category ${{\omega}\mathchar
45\relax\mathbf{Gph}}$ of _$\omega$ -graphs_, which is (up to equivalence) the
presheaf category over a simple category. The monad for weak
$\omega$-categories is defined by means of the following two notions: (i)
_globular operads_ , and (ii)  _contractions_. For conceptual clarity, we
present them as instances of simpler notions, namely: (i’)  _operads over a
cartesian monad_ , following [28], and (ii’)  _choices of diagonal-fillers_ ,
following [18].

###  2.1. Operads over a cartesian monad

For any category ${\mathcal{C}}$, the category $[{\mathcal{C}},{\mathcal{C}}]$
of all endofunctors on it admits a natural (strict) monoidal structure, given
by composition of endofunctors. Monoids in $[{\mathcal{C}},{\mathcal{C}}]$ are
monads on ${\mathcal{C}}$. Now let ${\mathcal{C}}$ be a category with finite
limits. We consider the monoidal subcategory
$[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}$ of
$[{\mathcal{C}},{\mathcal{C}}]$ defined as follows.

  * •

An endofunctor on ${\mathcal{C}}$ is in
$[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}$ if and only if it preserves
all pullbacks.

  * •

A natural transformation between pullback preserving endofunctors on
${\mathcal{C}}$ is in $[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}$ if and
only if it is cartesian, meaning that all its naturality squares are pullback
squares in ${\mathcal{C}}$.

Monoids in $[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}$ are called
cartesian monads on ${\mathcal{C}}$.

Let us recall some standard facts about slice categories of a monoidal
category.

######  Proposition 2.1.

Let ${\mathcal{M}}$ be a monoidal category and $T$ be a monoid in
${\mathcal{M}}$.

  1. (1)

The slice category ${\mathcal{M}}/T$ acquires a canonical monoidal structure
in such a way that the forgetful functor
${\mathcal{M}}/T\longrightarrow{\mathcal{M}}$ is strict monoidal.

  2. (2)

The canonical functor
${\mathbf{Mon}({\mathcal{M}}/T)}\longrightarrow{\mathbf{Mon}({\mathcal{M}})}/T$
is an isomorphism of categories. That is, to give a monoid in
${\mathcal{M}}/T$ is equivalent to give a monoid $T^{\prime}$ in
${\mathcal{M}}$ together with a monoid morphism $T^{\prime}\longrightarrow T$.

Suppose $T=(T,\eta,\mu)$ is a cartesian monad on ${\mathcal{C}}$. By
Proposition 2.1 (i), we obtain a monoidal category
$[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}/T$, and by Proposition 2.1
(ii), a monoid in $[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}/T$ is
equivalent to a cartesian monad $T^{\prime}$ on ${\mathcal{C}}$ equipped with
a _cartesian_ monad morphism $T^{\prime}\longrightarrow T$ (i.e., a monad
morphism which is cartesian as a natural transformation).

Let $1$ denote the terminal object of ${\mathcal{C}}$. The fundamental fact
for the theory of operads over a cartesian monad is that the functor

(1)
$\mathrm{ev}_{1}\colon[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}/T\longrightarrow{\mathcal{C}}/T1$

mapping $(\phi\colon F\longrightarrow
T)\in[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}/T$ to $(\phi_{1}\colon
F1\longrightarrow T1)\in{\mathcal{C}}/T1$, is an equivalence of categories
[25]. The quasi-inverse of $\mathrm{ev}_{1}$ is given by mapping
$({\mathrm{ar}_{P}}\colon P\longrightarrow T1)\in{\mathcal{C}}/T1$ to
$(\overline{{\mathrm{ar}_{P}}}\colon\overline{P}\longrightarrow
T)\in[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}/T$ determined by the
pullback for each $C\in{\mathcal{C}}$:

$\overline{P}C$$TC$$P$$T1$$(\overline{{\mathrm{ar}_{P}}})_{C}$$T!$$\overline{P}!$${\mathrm{ar}_{P}}$

(the notation ${\mathrm{ar}_{P}}$ is for _arity_ , because as we shall see
later, this morphism may be interpreted as assigning arities to operations).
We shall frequently use this equivalence.

Transporting the (strict) monoidal structure on
$[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}/T$ along the equivalence (1),
we obtain the following monoidal structure on ${\mathcal{C}}/T1$.

  * •

The unit is $I=(\eta_{1}\colon 1\longrightarrow T1)$.

  * •

For any $({\mathrm{ar}_{P}}\colon P\longrightarrow T1)$ and
$({\mathrm{ar}_{Q}}\colon Q\longrightarrow T1)$, their monoidal product is
given by the top horizontal composite in the diagram below, i.e.,
$(P,{\mathrm{ar}_{P}})\otimes(Q,{\mathrm{ar}_{Q}})=(\mu_{1}\circ
T({\mathrm{ar}_{Q}})\circ\partial_{2}\colon(P,{\mathrm{ar}_{P}})\ast
Q\longrightarrow T1)$:

(2) $(P,{\mathrm{ar}_{P}})\ast
Q$$TQ$$P$$T1$.$T^{2}1$$T1$$\partial_{2}$$T!$$\partial_{1}$${\mathrm{ar}_{P}}$$T({\mathrm{ar}_{Q}})$$\mu_{1}$

A monoid in $({\mathcal{C}}/T1,I,\otimes)$ is called a $T$-operad. Notice that
$({\mathcal{C}}/T1,I,\otimes)$ is not a strict monoidal category (in general),
since the equivalence (1) is not an isomorphism (in general).

######  Example 2.2.

The free monoid monad $(-)^{\ast}$ on $\mathbf{Set}$ is cartesian. A
$(-)^{\ast}$-operad is equivalent to a _non-symmetric operad_ [32]. In this
case, $T1\cong\mathbb{N}$, so an object of ${\mathcal{C}}/T1$ consists of a
set $P$ whose elements are assigned natural number arities. Thus we can view
an element of $P$ of arity $k$ as an operation with $k$ inputs. An element of
$(P,{\mathrm{ar}_{P}})\ast Q$ as in (2) consists of an operation $p$ in $P$
together with a list of operations in $Q$ of length ${\mathrm{ar}_{P}}(p)$.
The composite along the top of the diagram in (2) adds together the arities of
these operations in $Q$. For each $(-)^{\ast}$-operad
$((O,{\mathrm{ar}_{O}}),e,m)$, the set $O$ can be understood as the set of all
(derived) operations of the algebraic theory expressed by this operad, and
${\mathrm{ar}_{O}}\colon O\longrightarrow 1^{\ast}\cong\mathbb{N}$ as mapping
each operation to its arity. An element of $(O,{\mathrm{ar}_{O}})\ast O$ then
consists of a composable arrangement of operations; that is, an operation of,
say, arity $k$, together with $k$ operations for it to be composed with, one
for each input. The multiplication map $m$ then composes this arrangement to
give a single operation in $O$.

The monoidal category $[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}$ admits a
canonical (strict left) action
$[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}\times{\mathcal{C}}\longrightarrow{\mathcal{C}}$
on ${\mathcal{C}}$, given by evaluation. In other words, this action is the
transpose of the inclusion
$[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}\longrightarrow[{\mathcal{C}},{\mathcal{C}}]$.
We may precompose the strict monoidal forgetful functor
$[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}/T\longrightarrow[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}$
with the above to obtain a (strict left) action
$[{\mathcal{C}},{\mathcal{C}}]_{\mathrm{cart}}/T\times{\mathcal{C}}\longrightarrow{\mathcal{C}}$.
Transporting this action along the monoidal equivalence (1), we obtain an
action

$\ast\colon{\mathcal{C}}/T1\times{\mathcal{C}}\longrightarrow{\mathcal{C}}$

(sometimes written as $\ast_{T}$). Concretely, the functor $\ast$ is defined
by the pullback given in (2); note that the pullback
$(P,{\mathrm{ar}_{P}})\ast Q$ is independent of the morphism
${\mathrm{ar}_{Q}}$. This is a _pseudo_ action, in the sense that it is
equipped with the canonical coherent isomorphisms

$I\ast C\cong C\qquad((P,{\mathrm{ar}_{P}})\otimes(Q,{\mathrm{ar}_{Q}}))\ast
C\cong(P,{\mathrm{ar}_{P}})\ast((Q,{\mathrm{ar}_{Q}})\ast C)$

natural in $(P,{\mathrm{ar}_{P}}),(Q,{\mathrm{ar}_{Q}})\in{\mathcal{C}}/T1$
and $C\in{\mathcal{C}}$. The pseudo action $\ast$ is not strict (in general)
because the monoidal equivalence (1) is not a monoidal isomorphism (in
general).

Let $O=((O,{\mathrm{ar}_{O}}),e,m)$ be a $T$-operad. An $O$-algebra is an
object $A$ of ${\mathcal{C}}$ together with an action of $O$, i.e., a morphism
$\alpha\colon(O,{\mathrm{ar}_{O}})\ast A\longrightarrow A$ satisfying the
usual axioms. Note that via the monoidal equivalence (1), the $T$-operad $O$
corresponds to the cartesian monad $\overline{O}=O\ast(-)$ on ${\mathcal{C}}$
equipped with the cartesian monad morphism $\overline{{\mathrm{ar}_{O}}}\colon
O\ast(-)\longrightarrow T$. An $O$-algebra is equivalent to an Eilenberg–Moore
algebra for the monad $O\ast(-)$. We say that the monad $O\ast(-)$ is induced
by $O$.

###  2.2. $\omega$-graphs and the free strict $\omega$-category monad
$T^{(\omega)}$

In order to define weak $\omega$-categories, we shall apply the above general
theory of operads to the category ${{\omega}\mathchar 45\relax\mathbf{Gph}}$
of $\omega$-graphs and the monad $T^{(\omega)}$ for _strict_
$\omega$-category. These are obtained as limits of ${{n}\mathchar
45\relax\mathbf{Gph}}$ and $T^{(n)}$, which we now define.

######  Definition 2.3 ([39]).

For any locally small category ${\mathcal{V}}$, we define the category
${{{\mathcal{V}}}\mathchar 45\relax\mathbf{Gph}}$ as follows.

  * •

An object is a (small) ${\mathcal{V}}$-graph
$G=({\mathrm{ob}(G)},(G(x,y))_{x,y\in{\mathrm{ob}(G)}})$, consisting of a
small set ${\mathrm{ob}(G)}$ of objects and for each pair $x,y$ of objects, an
object $G(x,y)$ of ${\mathcal{V}}$.

  * •

A morphism $f$ from $G=({\mathrm{ob}(G)},(G(x,y))_{x,y\in{\mathrm{ob}(G)}})$
to
$G^{\prime}=({\mathrm{ob}(G^{\prime})},(G^{\prime}(x,y))_{x,y\in{\mathrm{ob}(G^{\prime})}})$
consists of a function
${\mathrm{ob}(f)}\colon{\mathrm{ob}(G)}\longrightarrow{\mathrm{ob}(G^{\prime})}$
(whose action we denote by $f$) and, for each pair $x,y$ of objects of $G$, a
morphism $f_{x,y}\colon G(x,y)\longrightarrow G^{\prime}(fx,fy)$ in
${\mathcal{V}}$.

The ${{(-)}\mathchar 45\relax\mathbf{Gph}}$ construction routinely extends to
an endo-2-functor on the 2-category ${\mathscr{C\\!A\\!T}}$ of locally small
categories.

######  Definition 2.4.

For each $n\in\mathbb{N}$, we define the category ${{n}\mathchar
45\relax\mathbf{Gph}}$ recursively as follows:

${{0}\mathchar 45\relax\mathbf{Gph}}=\mathbf{Set},\qquad{{(n+1)}\mathchar
45\relax\mathbf{Gph}}={{({{n}\mathchar 45\relax\mathbf{Gph}})}\mathchar
45\relax\mathbf{Gph}}.$

An object of ${{n}\mathchar 45\relax\mathbf{Gph}}$ is called an $n$-graph.

######  Remark 2.5.

The ${{(-)}\mathchar 45\relax\mathbf{Gph}}$ construction preserves presheaf
categories, i.e., for any small category ${\mathcal{D}}$, the category
${{[{\mathcal{D}},\mathbf{Set}]}\mathchar 45\relax\mathbf{Gph}}$ is equivalent
to $[{\mathcal{D}}^{+},\mathbf{Set}]$, where the category ${\mathcal{D}}^{+}$
is obtained from ${\mathcal{D}}$ by newly adding an object $\ast$ and, for
each object $D\in{\mathcal{D}}$, two morphisms $s^{(D)},t^{(D)}\colon
D\longrightarrow\ast$, such that for each morphism $f\colon D\longrightarrow
D^{\prime}$ in ${\mathcal{D}}$, $s^{(D^{\prime})}\circ f=s^{(D)}$ and
$t^{(D^{\prime})}\circ f=t^{(D)}$. Hence we see by induction that
${{n}\mathchar 45\relax\mathbf{Gph}}$ is equivalent to
$[\mathbb{G}_{n}^{\mathrm{op}},\mathbf{Set}]$, where $\mathbb{G}_{n}$ is the
category freely generated by the graph

$[0]$$[1]$$\cdots$$[n]$$s_{0}$$t_{0}$$s_{1}$$t_{1}$$s_{n-1}$$t_{n-1}$

subject to the relations

$s_{k+1}\circ s_{k}=t_{k+1}\circ s_{k},\qquad s_{k+1}\circ t_{k}=t_{k+1}\circ
t_{k}\qquad(k\in\\{0,\dots,n-2\\}).$

For any locally small category ${\mathcal{V}}$ with finite products, we have
the category ${{{\mathcal{V}}}\mathchar 45\relax\mathbf{Cat}}$ of small
${\mathcal{V}}$-categories [24]; throughout this paper, we only consider
enrichment over cartesian ${\mathcal{V}}$. The ${{(-)}\mathchar
45\relax\mathbf{Cat}}$ construction extends to an endofunctor on the category
of locally small categories with finite products and finite product preserving
functors. (${{(-)}\mathchar 45\relax\mathbf{Cat}}$ is also a 2-functor, but we
shall not use this fact.)

######  Definition 2.6.

For each $n\in\mathbb{N}$, we define the category ${{n}\mathchar
45\relax\mathbf{Cat}}$ recursively as follows:

${{0}\mathchar 45\relax\mathbf{Cat}}=\mathbf{Set},\qquad{{(n+1)}\mathchar
45\relax\mathbf{Cat}}={{({{n}\mathchar 45\relax\mathbf{Cat}})}\mathchar
45\relax\mathbf{Cat}}.$

An object of ${{n}\mathchar 45\relax\mathbf{Cat}}$ is called a strict
$n$-category.

Let ${\mathcal{V}}$ be a locally small category with finite products. There is
an evident forgetful functor
$U^{({\mathcal{V}})}\colon{{{\mathcal{V}}}\mathchar
45\relax\mathbf{Cat}}\longrightarrow{{{\mathcal{V}}}\mathchar
45\relax\mathbf{Gph}}$. If ${\mathcal{V}}$ has small coproducts distributing
over finite products, then $U^{({\mathcal{V}})}$ admits a left adjoint
$F^{({\mathcal{V}})}$ [39]. For each $n\in\mathbb{N}$, ${{n}\mathchar
45\relax\mathbf{Cat}}$ satisfies this condition (in fact, a stronger condition
of _extensivity_ [16]), so we have $F^{({{n}\mathchar
45\relax\mathbf{Cat}})}\dashv U^{({{n}\mathchar 45\relax\mathbf{Cat}})}$.

######  Definition 2.7.

For each $n\in\mathbb{N}$, we define the (monadic) adjunction $F^{(n)}\dashv
U^{(n)}\colon{{n}\mathchar 45\relax\mathbf{Cat}}\longrightarrow{{n}\mathchar
45\relax\mathbf{Gph}}$ recursively as follows:

  * •

$F^{(0)}=U^{(0)}=\mathrm{id}_{\mathbf{Set}}$,

  * •

$F^{(n+1)}\dashv U^{(n+1)}$ is the composite:

${{({{n}\mathchar 45\relax\mathbf{Gph}})}\mathchar
45\relax\mathbf{Gph}}$${{({{n}\mathchar 45\relax\mathbf{Cat}})}\mathchar
45\relax\mathbf{Gph}}$${{({{n}\mathchar 45\relax\mathbf{Cat}})}\mathchar
45\relax\mathbf{Cat}}$${{F^{(n)}}\mathchar
45\relax\mathbf{Gph}}$$F^{({{n}\mathchar
45\relax\mathbf{Cat}})}$${{U^{(n)}}\mathchar
45\relax\mathbf{Gph}}$$U^{({{n}\mathchar
45\relax\mathbf{Cat}})}$$\vdash$$\vdash$

(note that ${{(-)}\mathchar 45\relax\mathbf{Gph}}$, being a 2-functor,
preserves adjunctions).

We denote by $T^{(n)}$ the monad on ${{n}\mathchar 45\relax\mathbf{Gph}}$
induced by the adjunction $F^{(n)}\dashv U^{(n)}$. This monad is called the
free strict $n$-category monad.

######  Proposition 2.8 ([28, Theorem F.2.1], [16, Theorem 4.6]).

For each $n\in\mathbb{N}$, the monad $T^{(n)}$ is cartesian.

Let us now define ${{\omega}\mathchar 45\relax\mathbf{Gph}}$ and
$T^{(\omega)}$ as limits of ${{n}\mathchar 45\relax\mathbf{Gph}}$ and
$T^{(n)}$ respectively. First, the category ${{\omega}\mathchar
45\relax\mathbf{Gph}}$ is the limit in ${\mathscr{C\\!A\\!T}}$ of the diagram

(3) ${{0}\mathchar 45\relax\mathbf{Gph}}$${{1}\mathchar
45\relax\mathbf{Gph}}$${{2}\mathchar
45\relax\mathbf{Gph}}$$\cdots$.${\mathrm{ob}(-)}$${{({\mathrm{ob}(-)})}\mathchar
45\relax\mathbf{Gph}}$${{({{({\mathrm{ob}(-)})}\mathchar
45\relax\mathbf{Gph}})}\mathchar 45\relax\mathbf{Gph}}$

Objects of ${{\omega}\mathchar 45\relax\mathbf{Gph}}$ are called
$\omega$-graphs.

######  Remark 2.9.

By Remark 2.5, the diagram (3) is equivalent (in
${\mathscr{C\\!A\\!T}}^{(\omega^{\mathrm{op}})}$) to

(4)
$[\mathbb{G}_{0}^{\mathrm{op}},\mathbf{Set}]$$[\mathbb{G}_{1}^{\mathrm{op}},\mathbf{Set}]$$[\mathbb{G}_{2}^{\mathrm{op}},\mathbf{Set}]$$\cdots$,$[J_{0}^{\mathrm{op}},\mathbf{Set}]$$[J_{1}^{\mathrm{op}},\mathbf{Set}]$$[J_{2}^{\mathrm{op}},\mathbf{Set}]$

where $J_{n}\colon\mathbb{G}_{n}\longrightarrow\mathbb{G}_{n+1}$ is the
inclusion functor (mapping $[k]\in\mathbb{G}_{n}$ to
$[k]\in\mathbb{G}_{n+1}$). Hence ${{\omega}\mathchar 45\relax\mathbf{Gph}}$ is
equivalent to the limit of (4), which is again a presheaf category since
$\lim_{n}[\mathbb{G}_{n}^{\mathrm{op}},\mathbf{Set}]\cong[{\mathrm{colim}}_{n}\mathbb{G}_{n}^{\mathrm{op}},\mathbf{Set}]$.
The category $\mathbb{G}={\mathrm{colim}}_{n}\mathbb{G}_{n}$ is freely
generated by the graph

$[0]$$[1]$$\cdots$$[n]$$\cdots$$s_{0}$$t_{0}$$s_{1}$$t_{1}$$s_{n-1}$$t_{n-1}$$s_{n}$$t_{n}$

subject to the relations

$s_{k+1}\circ s_{k}=t_{k+1}\circ s_{k},\qquad s_{k+1}\circ t_{k}=t_{k+1}\circ
t_{k}\qquad(k\in\mathbb{N}).$

Thus $\omega$-graphs are equivalent to presheaves over $\mathbb{G}$, which are
sometimes called _globular sets_ [5, 28, 15]. Let $G$ be an $\omega$-graph,
with the corresponding globular set
$G^{\prime}\colon\mathbb{G}^{\mathrm{op}}\longrightarrow\mathbf{Set}$. For
each $k\in\mathbb{N}$, elements of the set $G^{\prime}[k]$ are called
$k$-cells of $G$. The functions $G^{\prime}s_{k}$ and $G^{\prime}t_{k}$ are
written simply as $s_{k}$ and $t_{k}$, and called the ($k$-dimensional) source
and target maps of $G$. For any $(k+1)$-cell $f$ of $G$, we write $f\colon
a\longrightarrow b$ to express that $s_{k}(f)=a$ and $t_{k}(f)=b$. Henceforth
we shall use the concepts of $\omega$-graph and globular set interchangeably.

######  Remark 2.10.

One can also give a concise _coinductive_ definition of $\omega$-graph as: an
$\omega$-graph $G$ consists of a set ${\mathrm{ob}(G)}$ of objects and, for
each pair $x,y$ of objects, an $\omega$-graph $G(x,y)$ [9]. More precisely,
the category ${{\omega}\mathchar 45\relax\mathbf{Gph}}$ is the carrier of the
terminal coalgebra for ${{(-)}\mathchar 45\relax\mathbf{Gph}}$ (seen as an
endofunctor on the category of locally small categories) [15]. The structure
map of this coalgebra is the functor ${{\omega}\mathchar
45\relax\mathbf{Gph}}\longrightarrow{{({{\omega}\mathchar
45\relax\mathbf{Gph}})}\mathchar 45\relax\mathbf{Gph}}$ mapping each
$\omega$-graph $G$ to the (${{\omega}\mathchar
45\relax\mathbf{Gph}}$)-enriched graph consisting of the same objects and for
each pair of objects $x$ and $y$ of $G$, the suitably defined hom
$\omega$-graph $G(x,y)$. By Lambek’s lemma, this functor is an isomorphism of
categories.

Note that we have a monad in the 2-category
${\mathscr{C\\!A\\!T}}^{(\omega^{\mathrm{op}})}$ on the object (3) given by
the sequence of cartesian monads $(T^{(n)})_{n\in\mathbb{N}}$. Applying the
2-functor
$\lim\colon{\mathscr{C\\!A\\!T}}^{(\omega^{\mathrm{op}})}\longrightarrow{\mathscr{C\\!A\\!T}}$,
we obtain a monad $T^{(\omega)}$ on ${{\omega}\mathchar
45\relax\mathbf{Gph}}$, called the free strict $\omega$-category monad.
$T^{(\omega)}$ is also cartesian, since a commutative square in
${{\omega}\mathchar 45\relax\mathbf{Gph}}$ is a pullback if and only if for
each $n\in\mathbb{N}$, it is mapped to a pullback in ${{n}\mathchar
45\relax\mathbf{Gph}}$ by the projection ${{\omega}\mathchar
45\relax\mathbf{Gph}}\longrightarrow{{n}\mathchar 45\relax\mathbf{Gph}}$.

As a consequence, we obtain the monoidal category ${{\omega}\mathchar
45\relax\mathbf{Gph}}/T^{(\omega)}1$, and monoids therein are
$T^{(\omega)}$-operads (also called _globular operads_). Logically, we may now
proceed to the next step, but some expository comments might be helpful at
this point. See [28, Chapter 8] for a more detailed account.

Let us start with an explicit description of the $\omega$-graph
$T^{(\omega)}1$. Using the free monoid monad (or the list monad) $(-)^{\ast}$
on $\mathbf{Set}$, $T^{(\omega)}1$ (regarded as a globular set) is given by
the following diagram of sets:

(5)
$1$$1^{\ast}$$1^{\ast\ast}$$1^{\ast\ast\ast}$$\cdots$$s_{0}={!}$$t_{0}={!}$$s_{1}={!^{\ast}}$$t_{1}={!^{\ast}}$$s_{2}={!^{\ast\ast}}$$t_{2}={!^{\ast\ast}}$$s_{3}={!^{\ast\ast\ast}}$$t_{3}={!^{\ast\ast\ast}}$

Here $1=\\{\bullet\\}$ is a singleton, $1^{\ast}$ is the set of lists of
$\bullet$, $1^{\ast\ast}$ is the set of lists of lists of $\bullet$, and so
on. It is important that the cells of $T^{(\omega)}1$ can be regarded as
_globular pasting schemes_ ; the following diagram shows some cells of
$T^{(\omega)}1$ and the corresponding globular pasting schemes.

(6) $\bullet$$\bullet\in 1$ $\bullet$$[\,]\in 1^{\ast}$
$\bullet$$\bullet$$\bullet$$[\bullet,\bullet]\in 1^{\ast}$
$\bullet$$\bullet$$\bullet$$\bullet$$[[\bullet],[\,],[\bullet,\bullet]]\in
1^{\ast\ast}$

We now explain what a $T^{(\omega)}$-operad and its algebra amount to. Let
$O=((O,{\mathrm{ar}_{O}}),e,m)$ be a $T^{(\omega)}$-operad. As in the case of
non-symmetric operads (Example 2.2), the cells of the $\omega$-graph $O$ can
be regarded as operations. The morphism ${\mathrm{ar}_{O}}\colon
O\longrightarrow T^{(\omega)}1$ maps each operation to its arity, which is a
globular pasting scheme. An $O$-algebra
$(A,\alpha\colon(O,{\mathrm{ar}_{O}})\ast A\longrightarrow A)$ consists of an
$\omega$-graph $A$ equipped with the interpretation of each operation $\sigma$
in $O$ on it. For instance, if $\sigma$ is a $1$-cell of $O$ whose arity is
$[\bullet,\bullet]$, then its interpretation on $A$ is an operation mapping
each composable pair of $1$-cells in $A$ to a $1$-cell in $A$. In summary,
$T^{(\omega)}$-operads form a notion of algebraic theory for $\omega$-graphs
whose arities are the globular pasting schemes.

###  2.3. Contractions

We now turn to the notion of contraction. A contraction is a piece of
structure on a morphism in ${{\omega}\mathchar 45\relax\mathbf{Gph}}$.
Leinster [28] introduced a set-theoretic definition of contraction, to which
Garner [18] gave diagrammatic formulation. We adopt the latter.

In order to motivate the definition, we first review the classical notion of
_lifting property_. Given morphisms $l\colon A\longrightarrow B$ and $r\colon
C\longrightarrow D$ in a category ${\mathcal{C}}$, we say that $r$ has the
right lifting property with respect to $l$ (or equivalently, $l$ has the left
lifting property with respect to $r$) if, for any pair of morphisms $u\colon
A\longrightarrow C$ and $v\colon B\longrightarrow D$ such that $v\circ
l=r\circ u$, there exists a (not necessarily unique) $w\colon B\longrightarrow
C$ making the diagram

$A$$C$$B$$D$$u$$r$$l$$v$$w$

commute.

A contraction is an algebraic version of the right lifting property, given
relative to a certain set ${\mathcal{J}}$ of morphisms.

######  Definition 2.11 ([19, Proposition 3.8]).

Let ${\mathcal{C}}$ be a locally small category and ${\mathcal{J}}$ a set of
morphisms in ${\mathcal{C}}$.

A contraction (with respect to ${\mathcal{J}}$) on a morphism $r\colon
C\longrightarrow D$ in ${\mathcal{C}}$ is a function $\kappa$ assigning, for
each element $l\colon A\longrightarrow B$ in ${\mathcal{J}}$ and each $u\colon
A\longrightarrow C$ and $v\colon B\longrightarrow D$ such that $r\circ
u=v\circ l$, a morphism $\kappa(l,u,v)\colon B\longrightarrow C$ such that
$u=\kappa(l,u,v)\circ l$ and $v=r\circ\kappa(l,u,v)$.

Given morphisms $r\colon C\longrightarrow D$ and $r^{\prime}\colon
C^{\prime}\longrightarrow D^{\prime}$ equipped with contractions $\kappa$ and
$\kappa^{\prime}$ respectively, a map of morphisms $(h\colon C\longrightarrow
C^{\prime},k\colon D\longrightarrow D^{\prime})\colon r\longrightarrow
r^{\prime}$ (i.e., a commutative square) is said to preserve contractions if
for each $(l,u,v)$ in the domain of $\kappa$,
$h\circ\kappa(l,u,v)=\kappa^{\prime}(l,h\circ u,k\circ v)$.

$A$$C$$B$$D$$C^{\prime}$$D^{\prime}$$u$$h$$r$$r^{\prime}$$k$$l$$v$$\kappa(l,u,v)$$\kappa^{\prime}(l,h\circ
u,k\circ v)$

The category of morphisms in ${\mathcal{C}}$ equipped with contractions with
respect to ${\mathcal{J}}$ and contraction preserving maps is denoted by
${\mathbf{Contr}}({\mathcal{C}},{\mathcal{J}})$. In fact, we shall be mainly
interested in certain subcategories of
${\mathbf{Contr}}({\mathcal{C}},{\mathcal{J}})$, defined as follows. Denote
the evident codomain functor by
$\mathrm{cod}\colon{\mathbf{Contr}}({\mathcal{C}},{\mathcal{J}})\longrightarrow{\mathcal{C}}$.
Then for any object $D$ of ${\mathcal{C}}$, let
${\mathbf{Contr}}({\mathcal{C}},{\mathcal{J}})_{D}$ be the fibre of
$\mathrm{cod}$ over $D$; so an object of
${\mathbf{Contr}}({\mathcal{C}},{\mathcal{J}})_{D}$ is a morphism $r$ in
${\mathcal{C}}$ with codomain $D$ equipped with a contraction, and a morphism
is a contraction preserving map whose second component is $\mathrm{id}_{D}$.
Note that there is a forgetful functor
${\mathbf{Contr}}({\mathcal{C}},{\mathcal{J}})_{D}\longrightarrow{\mathcal{C}}/D$.

######  Remark 2.12.

If ${\mathcal{C}}$ has pullbacks, then for each set ${\mathcal{J}}$ of
morphisms in ${\mathcal{C}}$, the functor
$\mathrm{cod}\colon{\mathbf{Contr}}({\mathcal{C}},{\mathcal{J}})\longrightarrow{\mathcal{C}}$
is a (Grothendieck) fibration. Indeed, given any morphism $k\colon
D\longrightarrow D^{\prime}$ in ${\mathcal{C}}$ and any object
$(r^{\prime}\colon C^{\prime}\longrightarrow
D^{\prime},\kappa^{\prime})\in{\mathbf{Contr}}({\mathcal{C}},{\mathcal{J}})_{D^{\prime}}$,
one can endow the pullback $k^{\ast}r^{\prime}$ as in

$k^{\ast}C^{\prime}$$D$$C^{\prime}$$D^{\prime}$$k^{\ast}r^{\prime}$$h$$k$$r^{\prime}$

with a contraction $k^{\ast}\kappa^{\prime}$, induced from $\kappa^{\prime}$
by the universality of pullback. Then the morphism

$(h,k)\colon(k^{\ast}r^{\prime},k^{\ast}\kappa)\longrightarrow(r^{\prime},\kappa^{\prime})$

in ${\mathbf{Contr}}({\mathcal{C}},{\mathcal{J}})$ is the required cartesian
lifting of $k$.

In order to define Leinster’s notion of contraction for morphisms in
${{\omega}\mathchar 45\relax\mathbf{Gph}}$, we define a set
${\mathcal{J}}^{(\omega)}$ of morphisms in ${{\omega}\mathchar
45\relax\mathbf{Gph}}$ (called the set of _generating cofibrations_ in [18]).
Recall from Remark 2.9 the equivalence ${{\omega}\mathchar
45\relax\mathbf{Gph}}\simeq[\mathbb{G}^{\mathrm{op}},\mathbf{Set}]$. We denote
the Yoneda embedding by
$\mathbf{y}\colon\mathbb{G}\longrightarrow[\mathbb{G}^{\mathrm{op}},\mathbf{Set}]$.
The set ${\mathcal{J}}^{(\omega)}$ is defined to be
$\\{\,m_{k}\colon\partial\mathbf{y}[k]\longrightarrow\mathbf{y}[k]\mid
k\in\mathbb{N}\,\\}$, where $\partial\mathbf{y}[k]$ is the subobject of
$\mathbf{y}[k]$ obtained by removing the unique $k$-cell $\mathrm{id}_{[k]}$
of $\mathbf{y}[k]$, and $m_{k}$ is the associated inclusion. These morphisms
may be depicted as follows:

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}\hbox{{$\bigg{(}\quad\bigg{)}$}} }}\pgfsys@invoke{\lxSVG@closescope
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The geometric idea is that $\mathbf{y}[k]$ is the $\omega$-graph representing
the (directed) _$k$ -dimensional disc_ and $\partial\mathbf{y}[k]$ is its
boundary, the (directed) _$(k-1)$ -dimensional sphere_. Note that for any
$\omega$-graph $G$ and $k\in\mathbb{N}$, a morphism
$\mathbf{y}[k]\longrightarrow G$ corresponds to a $k$-cell of $G$ by the
Yoneda lemma. Similarly, for $k\geq 1$, a morphism
$\partial\mathbf{y}[k]\longrightarrow G$ corresponds to a _parallel pair of
$(k-1)$-cells_ of $G$. Here, two $(k-1)$-cells ($k\geq 2$) $a$ and $b$ are
said to be parallel if $s_{k-2}a=s_{k-2}b$ and $t_{k-2}a=t_{k-2}b$ hold, and
we count any two $0$-cells as parallel. (We may formally extend this
correspondence to the case where $k=0$ by adopting the convention that in any
$\omega$-graph there is precisely one “parallel pair of $(-1)$-cells”.)

By a contraction on a morphism $r\colon C\longrightarrow D$ in
${{\omega}\mathchar 45\relax\mathbf{Gph}}$ we always mean a contraction (in
the sense of Definition 2.11) with respect to ${\mathcal{J}}^{(\omega)}$. So
such a contraction $\kappa$ assigns for each $k\in\mathbb{N}$, each pair
$c,c^{\prime}$ of parallel $(k-1)$-cells of $C$ and each $k$-cell $d\colon
r(c)\longrightarrow r(c^{\prime})$ in $D$, a $k$-cell
$\kappa(k,(c,c^{\prime}),d)\colon c\longrightarrow c^{\prime}$ in $C$ such
that $d=r(\kappa(k,(c,c^{\prime}),d))$. For any $\omega$-graph $D$, we write
the category ${\mathbf{Contr}}({{\omega}\mathchar
45\relax\mathbf{Gph}},{\mathcal{J}}^{(\omega)})_{D}$ simply as
${\mathbf{Contr}}_{D}$.

###  2.4. The $T^{(\omega)}$-operad $L$ for weak $\omega$-categories

We define the category ${\mathbf{OC}(T^{(\omega)})}$ of $T^{(\omega)}$-operads
with contractions as the following pullback of categories (where the arrows to
${{\omega}\mathchar 45\relax\mathbf{Gph}}/T^{(\omega)}1$ denote the forgetful
functors):

(7) ${\mathbf{OC}(T^{(\omega)})}$${\mathbf{Mon}({{\omega}\mathchar
45\relax\mathbf{Gph}}/T^{(\omega)}1)}$${\mathbf{Contr}}_{T^{(\omega)}1}$${{\omega}\mathchar
45\relax\mathbf{Gph}}/T^{(\omega)}1$.

Roughly speaking, a contraction on a $T^{(\omega)}$-operad generates both
(unbiased) composition operations and operations which yield coherence cells;
see [28, Chapter 9] for a detailed discussion. We shall use the universal
$T^{(\omega)}$-operad with a contraction for our definition of weak
$\omega$-category.

######  Proposition 2.13 ([28, Proposition 9.2.2]).

The category ${\mathbf{OC}(T^{(\omega)})}$ has an initial object.

Let $L$ be the initial object in ${\mathbf{OC}(T^{(\omega)})}$; we denote the
$T^{(\omega)}$-operad underlying $L$ also by $L$.

######  Definition 2.14 ([28, Definition 9.2.3]).

A weak $\omega$-category is an $L$-algebra.

We denote the Eilenberg–Moore category of the monad $L\ast(-)$ by
${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}$; the morphisms in ${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}}$ are called the
strict $\omega$-functors, hence the subscript ‘s’.

######  Example 2.15.

A canonical source of examples of weak $\omega$-categories is provided by
_algebras of a contractible $T^{(\omega)}$-operad_ [28, Example 9.2.4]. Here
we say that a $T^{(\omega)}$-operad $O=((O,{\mathrm{ar}_{O}}),e,m)$ is
contractible if it admits _some_ contraction; or equivalently, if the morphism
${\mathrm{ar}_{O}}\colon O\longrightarrow T^{(\omega)}1$ has the right lifting
property with respect to each $m_{k}\in{\mathcal{J}}^{(\omega)}$. Given such a
$T^{(\omega)}$-operad $O$ and a choice of a contraction $\kappa$ on it, we
obtain the unique morphism $\phi\colon L\longrightarrow O$ in
${\mathbf{OC}(T^{(\omega)})}$ by the initiality of $L$. It then induces a
monad morphism $\phi\ast(-)\colon L\ast(-)\longrightarrow O\ast(-)$, hence in
turn a functor ${{(O\ast(-))}\mathchar
45\relax\mathbf{Alg}}\longrightarrow{\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}}$. So any
$O$-algebra, together with a choice of a contraction on $O$, gives rise to a
weak $\omega$-category.

For example, the terminal $T^{(\omega)}$-operad, whose arity map is just
$\mathrm{id}\colon T^{(\omega)}1\longrightarrow T^{(\omega)}1$, admits a
unique contraction, hence any algebra for it—which is just an Eilenberg–Moore
algebra for the monad $T^{(\omega)}$, i.e., a _strict_ $\omega$-category—is
canonically a weak $\omega$-category. As a less trivial example, Leinster
constructs the _fundamental weak $\omega$-category_ of a topological space $X$
by exhibiting an action of a contractible $T^{(\omega)}$-operad on the
$\omega$-graph consisting of higher homotopies in $X$; see [28, Example
9.2.7].

One can also define the notion of algebra of a $T^{(\omega)}$-operad over more
general categories than ${{\omega}\mathchar 45\relax\mathbf{Gph}}$ [5, 29],
i.e., internally in those categories. In [9, 29], such a notion is defined on
suitable categories of globular objects in the classifying (or syntactic)
category $\mathcal{C}l(\mathbb{T})$ (cf. [35, Section 6]) of a Martin-Löf type
theory $\mathbb{T}$, and it is shown (using iterated identity types) that each
type in $\mathbb{T}$ admits an action of a contractible $T^{(\omega)}$-operad,
hence an internal weak $\omega$-category structure.

##  3\. The forgetful functor ${U_{\mathrm{s}}}\colon{\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}\longrightarrow{{({\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}})}\mathchar
45\relax\mathbf{Gph}}$

In this section, we define the forgetful functor ${U_{\mathrm{s}}}$, inducing
a _hom weak $\omega$-category_ ${\mathcal{A}}(x,y)$ over each weak
$\omega$-category ${\mathcal{A}}$ and pair of objects $x,y\in{\mathcal{A}}$.
As we shall see, our construction of ${U_{\mathrm{s}}}$ heavily depends on the
fact that ${{(-)}\mathchar 45\relax\mathbf{Gph}}$ preserves a lot of
structure. First we observe that both the domain and codomain of
${U_{\mathrm{s}}}$ are monadic over ${{\omega}\mathchar
45\relax\mathbf{Gph}}$; that the codomain is so is a consequence of the
following.

######  Proposition 3.1 (Cf. [28, Proposition F.1.1 (b)]).

The 2-functor ${{(-)}\mathchar
45\relax\mathbf{Gph}}\colon{\mathscr{C\\!A\\!T}}\longrightarrow{\mathscr{C\\!A\\!T}}$
preserves Eilenberg–Moore objects. That is, for each monad $T$ on a locally
small category ${\mathcal{V}}$, the canonical comparison functor
${{({{T}\mathchar 45\relax\mathbf{Alg}})}\mathchar
45\relax\mathbf{Gph}}\longrightarrow{{({{T}\mathchar
45\relax\mathbf{Gph}})}\mathchar 45\relax\mathbf{Alg}}$ is an isomorphism of
categories.

###### Proof.

The monad ${{T}\mathchar 45\relax\mathbf{Gph}}$ is on the category
${{{\mathcal{V}}}\mathchar 45\relax\mathbf{Gph}}$. An object of
${{({{T}\mathchar 45\relax\mathbf{Gph}})}\mathchar 45\relax\mathbf{Alg}}$
consists of a ${\mathcal{V}}$-graph $G$ together with a ${\mathcal{V}}$-graph
morphism $\gamma\colon({{T}\mathchar 45\relax\mathbf{Gph}})G\longrightarrow G$
satisfying the axioms of Eilenberg–Moore algebra. The unit axiom forces
$\gamma$ to be the identity on objects, so such a $\gamma$ consists of, for
each pair $(x,y)$ of objects of $G$, a morphism $\gamma_{x,y}\colon
TG(x,y)\longrightarrow G(x,y)$ in ${\mathcal{V}}$ satisfying the
Eilenberg–Moore axioms. These data amount to give an Eilenberg–Moore algebra
structure on each $G(x,y)$, and hence correspond to a $({{T}\mathchar
45\relax\mathbf{Alg}})$-graph. ∎

So ${{({\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}})}\mathchar
45\relax\mathbf{Gph}}={{({{(L\ast_{T^{(\omega)}}(-))}\mathchar
45\relax\mathbf{Alg}})}\mathchar 45\relax\mathbf{Gph}}$ is isomorphic to
${{({{(L\ast_{T^{(\omega)}}(-))}\mathchar 45\relax\mathbf{Gph}})}\mathchar
45\relax\mathbf{Alg}}$. The monad ${{(L\ast_{T^{(\omega)}}(-))}\mathchar
45\relax\mathbf{Gph}}$ is on ${{({{\omega}\mathchar
45\relax\mathbf{Gph}})}\mathchar 45\relax\mathbf{Gph}}$, which is canonically
isomorphic to ${{\omega}\mathchar 45\relax\mathbf{Gph}}$ by Remark 2.10.
Explicitly, the functor part of the monad
${{(L\ast_{T^{(\omega)}}(-))}\mathchar 45\relax\mathbf{Gph}}$ maps an
$\omega$-graph $G$ to the $\omega$-graph
${{(L\ast_{T^{(\omega)}}(-))}\mathchar 45\relax\mathbf{Gph}}(G)$ with the same
objects and such that for each pair $(x,y)$ of objects, the hom
$({{(L\ast_{T^{(\omega)}}(-))}\mathchar 45\relax\mathbf{Gph}}(G))(x,y)$ is
equal to $L\ast_{T^{(\omega)}}(G(x,y))$, the underlying $\omega$-graph of the
free weak $\omega$-category over the hom $\omega$-graph $G(x,y)$ of $G$. We
shall induce ${U_{\mathrm{s}}}$ from a monad morphism, that is induced by
initiality of $L$.

The monad ${{(L\ast_{T^{(\omega)}}(-))}\mathchar 45\relax\mathbf{Gph}}$ is
also induced from an operad over a cartesian monad. To show this, we use the
following fact.

######  Proposition 3.2.

  1. (1)

Let ${\mathcal{V}}$ be a locally small category with pullbacks (resp. finite
limits). Then the category ${{{\mathcal{V}}}\mathchar 45\relax\mathbf{Gph}}$
has pullbacks (resp. finite limits).

  2. (2)

Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be locally small categories with
pullbacks and $F\colon{\mathcal{V}}\longrightarrow{\mathcal{W}}$ be a pullback
preserving functor. Then the functor ${{F}\mathchar
45\relax\mathbf{Gph}}\colon{{{\mathcal{V}}}\mathchar
45\relax\mathbf{Gph}}\longrightarrow{{{\mathcal{W}}}\mathchar
45\relax\mathbf{Gph}}$ preserves pullbacks.

  3. (3)

Let ${\mathcal{V}}$ be a locally small category, ${\mathcal{W}}$ be a locally
small category with pullbacks,
$F,G\colon{\mathcal{V}}\longrightarrow{\mathcal{W}}$ be functors and
$\alpha\colon F\longrightarrow G$ be a cartesian natural transformation. Then
the natural transformation ${{\alpha}\mathchar
45\relax\mathbf{Gph}}\colon{{F}\mathchar
45\relax\mathbf{Gph}}\longrightarrow{{G}\mathchar 45\relax\mathbf{Gph}}$ is
cartesian.

The $T^{(\omega)}$-operad $L$ corresponds to the cartesian monad morphism
$\overline{{\mathrm{ar}_{L}}}\colon L\ast_{T^{(\omega)}}(-)\longrightarrow
T^{(\omega)}$. By Proposition 3.2, ${{(-)}\mathchar 45\relax\mathbf{Gph}}$
preserves cartesian monads as well as cartesian monad morphisms. Hence
${{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}}$ is a cartesian monad and
${{\overline{{\mathrm{ar}_{L}}}}\mathchar
45\relax\mathbf{Gph}}\colon{{(L\ast_{T^{(\omega)}}(-))}\mathchar
45\relax\mathbf{Gph}}\longrightarrow{{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}}$ is a cartesian monad morphism. Since a cartesian monad
morphism to ${{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}}$ corresponds to a
$({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})$-operad, it follows that the
monad ${{(L\ast_{T^{(\omega)}}(-))}\mathchar 45\relax\mathbf{Gph}}$ is induced
from a $({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})$-operad.

$({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})$-operads also form a notion
of algebraic theory for $\omega$-graphs, but their arities are more restricted
than those of $T^{(\omega)}$-operads. Here is an explicit description of the
$\omega$-graph $({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})1$ (seen as a
globular set):

$1$$1$$1^{\ast}$$1^{\ast\ast}$$\cdots$$s_{0}={\mathrm{id}}$$t_{0}={\mathrm{id}}$$s_{1}={!}$$t_{1}={!}$$s_{2}={!^{\ast}}$$t_{2}={!^{\ast}}$$s_{3}={!^{\ast\ast}}$$t_{3}={!^{\ast\ast}}$

The cells of $({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})1$ represent the
globular pasting schemes _not involving compositions along $0$-cells_. For
instance, the $3$-cell $[[\bullet],[\,],[\bullet,\bullet]]\in 1^{\ast\ast}$ of
$({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})1$ corresponds to the
following globular pasting scheme (cf. (6)):

$\bullet$$\bullet$

We turn to a concrete description of the $({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})$-operad corresponding to
${{\overline{{\mathrm{ar}_{L}}}}\mathchar 45\relax\mathbf{Gph}}$, which
induces ${{(L\ast_{T^{(\omega)}}(-))}\mathchar 45\relax\mathbf{Gph}}$. For any
locally small category ${\mathcal{V}}$, define the functor
${\lceil-\rceil}_{\mathcal{V}}\colon{\mathcal{V}}\longrightarrow{{{\mathcal{V}}}\mathchar
45\relax\mathbf{Gph}}$ (also written as ${\lceil-\rceil}$) by mapping each
object $X\in{\mathcal{V}}$ to the ${\mathcal{V}}$-graph ${\lceil X\rceil}$
with a single object $\ast$ such that ${\lceil X\rceil}(\ast,\ast)=X$. Notice
that ${\lceil-\rceil}$ preserves the terminal object when ${\mathcal{V}}$ has
one. We shall take advantage of this fact and denote the terminal object of
${{{\mathcal{V}}}\mathchar 45\relax\mathbf{Gph}}$ by ${\lceil 1\rceil}$,
distinguishing it from the terminal object $1$ of ${\mathcal{V}}$. The
terminal object of ${{\omega}\mathchar
45\relax\mathbf{Gph}}\cong{{({{\omega}\mathchar
45\relax\mathbf{Gph}})}\mathchar 45\relax\mathbf{Gph}}$ is denoted by $1$.

######  Proposition 3.3.

Let ${\mathcal{V}}$ be a locally small category with finite limits and $T$ be
a cartesian monad on ${\mathcal{V}}$. The functor
${\lceil-\rceil}_{T}\colon{\mathcal{V}}/T1\longrightarrow{{{\mathcal{V}}}\mathchar
45\relax\mathbf{Gph}}/({{T}\mathchar 45\relax\mathbf{Gph}}){\lceil 1\rceil}$
mapping $({\mathrm{ar}_{P}}\colon P\longrightarrow T1)$ to

${\lceil P\rceil}$${\lceil T1\rceil}$$({{T}\mathchar
45\relax\mathbf{Gph}}){\lceil
1\rceil}$${\lceil{\mathrm{ar}_{P}}\rceil}$$\cong$

makes the following square commute up to a natural isomorphism:

$[{\mathcal{V}},{\mathcal{V}}]_{\mathrm{cart}}/T$${\mathcal{V}}/T1$$[{{{\mathcal{V}}}\mathchar
45\relax\mathbf{Gph}},{{{\mathcal{V}}}\mathchar
45\relax\mathbf{Gph}}]_{\mathrm{cart}}/{{T}\mathchar
45\relax\mathbf{Gph}}$${{{\mathcal{V}}}\mathchar
45\relax\mathbf{Gph}}/({{T}\mathchar 45\relax\mathbf{Gph}}){\lceil
1\rceil}$.$\mathrm{ev}_{1}$$\simeq$${\lceil-\rceil}_{T}$${{(-)}\mathchar
45\relax\mathbf{Gph}}$$\mathrm{ev}_{1}$$\simeq$

It follows that the functor ${\lceil-\rceil}_{T}$ acquires the structure of a
strong monoidal functor, since the functor ${{(-)}\mathchar
45\relax\mathbf{Gph}}\colon[{\mathcal{V}},{\mathcal{V}}]_{\mathrm{cart}}/T\longrightarrow[{{{\mathcal{V}}}\mathchar
45\relax\mathbf{Gph}},{{{\mathcal{V}}}\mathchar
45\relax\mathbf{Gph}}]_{\mathrm{cart}}/{{T}\mathchar 45\relax\mathbf{Gph}}$
does.

When ${\mathcal{V}}={{\omega}\mathchar 45\relax\mathbf{Gph}}$, the functor
${\lceil-\rceil}_{{{\omega}\mathchar
45\relax\mathbf{Gph}}}\colon{{\omega}\mathchar
45\relax\mathbf{Gph}}\longrightarrow{{({{\omega}\mathchar
45\relax\mathbf{Gph}})}\mathchar 45\relax\mathbf{Gph}}\cong{{\omega}\mathchar
45\relax\mathbf{Gph}}$ maps an $\omega$-graph $G$ to the $\omega$-graph
${\lceil G\rceil}$ with a single $0$-cell $\ast$ and in which a $(k+1)$-cell
is given by a $k$-cell of $G$ for all $k\in\mathbb{N}$. In view of the
equivalence ${{\omega}\mathchar
45\relax\mathbf{Gph}}\simeq[\mathbb{G}^{\mathrm{op}},\mathbf{Set}]$, we have
the evident functor $S\colon\mathbb{G}\longrightarrow\mathbb{G}$ mapping $[k]$
to $[k+1]$, and ${\lceil-\rceil}_{{{\omega}\mathchar 45\relax\mathbf{Gph}}}$
is the right Kan extension along $S^{\mathrm{op}}$. Observe that
$({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})1\cong{\lceil
T^{(\omega)}1\rceil}_{{{\omega}\mathchar 45\relax\mathbf{Gph}}}$ is obtained
from $T^{(\omega)}1$ by this construction.

As a special case of ${\lceil-\rceil}_{T}$, we obtain

(8) ${\lceil-\rceil}_{T^{(\omega)}}\colon{{\omega}\mathchar
45\relax\mathbf{Gph}}/T^{(\omega)}1\longrightarrow{{\omega}\mathchar
45\relax\mathbf{Gph}}/({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})1;$

here again we are identifying the two canonically isomorphic categories
${{\omega}\mathchar 45\relax\mathbf{Gph}}$ and ${{({{\omega}\mathchar
45\relax\mathbf{Gph}})}\mathchar 45\relax\mathbf{Gph}}$. Given an
$\omega$-graph over $T^{(\omega)}1$, ${\lceil-\rceil}_{T^{(\omega)}}$ raises
the dimensions of cells by one. Since ${\lceil-\rceil}_{T^{(\omega)}}$ is
strong monoidal, we also obtain

(9)
${\mathbf{Mon}({\lceil-\rceil}_{T^{(\omega)}})}\colon{\mathbf{Mon}({{\omega}\mathchar
45\relax\mathbf{Gph}}/T^{(\omega)}1)}\longrightarrow{\mathbf{Mon}({{\omega}\mathchar
45\relax\mathbf{Gph}}/({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})1)}.$

Here are some properties of ${\lceil-\rceil}$. For any locally small category
${\mathcal{V}}$, ${\lceil-\rceil}_{\mathcal{V}}$ is fully faithful, and a
${\mathcal{V}}$-graph is in the essential image of
${\lceil-\rceil}_{\mathcal{V}}$ if and only if it has precisely one object.
Accordingly, for any locally small category ${\mathcal{V}}$ with finite limits
and a cartesian monad $T$ thereon, ${\lceil-\rceil}_{T}$ is also fully
faithful, and an object $({\mathrm{ar}_{G}}\colon
G\longrightarrow({{T}\mathchar 45\relax\mathbf{Gph}}){\lceil
1\rceil})\in{{{\mathcal{V}}}\mathchar 45\relax\mathbf{Gph}}/({{T}\mathchar
45\relax\mathbf{Gph}}){\lceil 1\rceil}$ is in the essential image of
${\lceil-\rceil}_{T}$ if and only if $G\in{{{\mathcal{V}}}\mathchar
45\relax\mathbf{Gph}}$ has precisely one object.

For a locally small category ${\mathcal{V}}$ with small coproducts, the
functor
${\lceil-\rceil}_{\mathcal{V}}\colon{\mathcal{V}}\longrightarrow{{{\mathcal{V}}}\mathchar
45\relax\mathbf{Gph}}$ admits a left adjoint
$\coprod_{\mathcal{V}}\colon{{{\mathcal{V}}}\mathchar
45\relax\mathbf{Gph}}\longrightarrow{\mathcal{V}}$, mapping
$G\in{{{\mathcal{V}}}\mathchar 45\relax\mathbf{Gph}}$ to
$\coprod_{x,y\in{\mathrm{ob}(G)}}G(x,y)\in{\mathcal{V}}$. In particular, the
functor ${\lceil-\rceil}_{{\omega}\mathchar
45\relax\mathbf{Gph}}\colon{{\omega}\mathchar
45\relax\mathbf{Gph}}\longrightarrow{{\omega}\mathchar 45\relax\mathbf{Gph}}$
admits a left adjoint $\coprod_{{\omega}\mathchar
45\relax\mathbf{Gph}}\colon{{\omega}\mathchar
45\relax\mathbf{Gph}}\longrightarrow{{\omega}\mathchar 45\relax\mathbf{Gph}}$.
Intuitively, given an $\omega$-graph $G$, the $\omega$-graph
$\coprod_{{{\omega}\mathchar 45\relax\mathbf{Gph}}}G$ is obtained by
_lowering_ the dimensions of cells by one: for each $k\geq 1$, a $k$-cell of
$G$ is turned to a $(k-1)$-cell of $\coprod_{{{\omega}\mathchar
45\relax\mathbf{Gph}}}G$, and the 0-cells of $G$ are thrown away. In view of
the equivalence ${{\omega}\mathchar
45\relax\mathbf{Gph}}\simeq[\mathbb{G}^{\mathrm{op}},\mathbf{Set}]$,
$\coprod_{{{\omega}\mathchar 45\relax\mathbf{Gph}}}$ can be seen as the
precomposition of $S^{\mathrm{op}}$. Recall the set
${\mathcal{J}}^{(\omega)}=\\{\,m_{k}\mid k\in\mathbb{N}\,\\}$ of morphisms in
${{\omega}\mathchar 45\relax\mathbf{Gph}}$. We have $\coprod m_{k+1}\cong
m_{k}$ for each $k\in\mathbb{N}$, whereas $\coprod m_{0}$ is the identity
morphism on the empty (= initial) $\omega$-graph. In the sequel, for
simplicity we identify $\coprod m_{k+1}$ with $m_{k}$.

######  Proposition 3.4.

For any morphism $r\colon C\longrightarrow D$ in ${{\omega}\mathchar
45\relax\mathbf{Gph}}$, the adjunction $\coprod\dashv{\lceil-\rceil}$ yields a
canonical bijective correspondence between contractions on $r$ and
contractions on ${\lceil r\rceil}$. In more detail, a contraction $\kappa$ on
$r$ corresponds to a contraction ${\lceil\kappa\rceil}$ on ${\lceil r\rceil}$
if and only if whenever $\widehat{u}$ is the transpose of $u$ and
$\widehat{v}$ is the transpose of $v$ in the outer commutative squares in
(10), $\kappa(k,u,v)$ is the transpose of
${\lceil\kappa\rceil}(k+1,\widehat{u},\widehat{v})$.

(10)
$\partial\mathbf{y}[k]$$C$$\mathbf{y}[k]$$D$${u}$$r$$m_{k}$$v$$\kappa(k,{u},{v})$
$\partial\mathbf{y}[{k+1}]$${\lceil C\rceil}$$\mathbf{y}[{k+1}]$${\lceil
D\rceil}$$\widehat{u}$${\lceil
r\rceil}$$m_{k+1}$$\widehat{v}$${{\lceil\kappa\rceil}(k+1,\widehat{u},\widehat{v})}$

This correspondence respects contraction preserving morphisms in the evident
sense. In other words, we obtain a functor

(11)
${\lceil-\rceil}\colon{\mathbf{Contr}}_{D}\longrightarrow{\mathbf{Contr}}_{{\lceil
D\rceil}}$

mapping each $(r\colon C\longrightarrow D,\kappa)\in{\mathbf{Contr}}_{D}$ to
$({\lceil r\rceil}\colon{\lceil C\rceil}\longrightarrow{\lceil
D\rceil},{\lceil\kappa\rceil})\in{\mathbf{Contr}}_{{\lceil D\rceil}}$, where
the contractions $\kappa$ and ${\lceil\kappa\rceil}$ are related as in
Proposition 3.4. The functor (11) is fully faithful and its essential image
consists of those objects $(r^{\prime}\colon C^{\prime}\longrightarrow{\lceil
D\rceil},\kappa^{\prime})\in{\mathbf{Contr}}_{{\lceil D\rceil}}$ such that
$C^{\prime}\in{{\omega}\mathchar 45\relax\mathbf{Gph}}$ has precisely one
object. As a special case of (11), we have

(12)
${\lceil-\rceil}\colon{\mathbf{Contr}}_{T^{(\omega)}1}\longrightarrow{\mathbf{Contr}}_{({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})1},$

since ${\lceil T^{(\omega)}1\rceil}\cong({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})1$.

Now define the category ${\mathbf{OC}({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})}$ of $({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})$-operads with contractions as the pullback

${\mathbf{OC}({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})}$${\mathbf{Mon}({{\omega}\mathchar
45\relax\mathbf{Gph}}/({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})1)}$${\mathbf{Contr}}_{({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})1}$${{\omega}\mathchar
45\relax\mathbf{Gph}}/({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})1$

(cf. (7)). The three fully faithful functors (8), (9) and (12) induce a
functor

${\lceil-\rceil}\colon{\mathbf{OC}(T^{(\omega)})}\longrightarrow{\mathbf{OC}({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})}.$

This functor is again fully faithful, and its essential image consists of
those $({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})$-operads with
contractions whose underlying $\omega$-graph has precisely one object. Mapping
$L\in{\mathbf{OC}(T^{(\omega)})}$ by this, we obtain ${\lceil
L\rceil}\in{\mathbf{OC}({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})}$. By
Proposition 3.3, we have an isomorphism of monads on ${{\omega}\mathchar
45\relax\mathbf{Gph}}$

(13) ${{(L\ast_{T^{(\omega)}}(-))}\mathchar 45\relax\mathbf{Gph}}\cong{\lceil
L\rceil}\ast_{{{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}}}(-).$

Next observe that for each $n\in\mathbb{N}$ there exists a canonical monad
morphism $\phi^{(n)}\colon{{T^{(n)}}\mathchar
45\relax\mathbf{Gph}}\longrightarrow T^{(n+1)}$ defined as the composition

${{({{n}\mathchar 45\relax\mathbf{Gph}})}\mathchar
45\relax\mathbf{Gph}}={{(n+1)}\mathchar
45\relax\mathbf{Gph}}$${{({{n}\mathchar 45\relax\mathbf{Cat}})}\mathchar
45\relax\mathbf{Gph}}$${{({{n}\mathchar 45\relax\mathbf{Cat}})}\mathchar
45\relax\mathbf{Cat}}$${{({{n}\mathchar 45\relax\mathbf{Cat}})}\mathchar
45\relax\mathbf{Gph}}$${{({{n}\mathchar 45\relax\mathbf{Gph}})}\mathchar
45\relax\mathbf{Gph}}={{(n+1)}\mathchar
45\relax\mathbf{Gph}}$${{F^{(n)}}\mathchar
45\relax\mathbf{Gph}}$$F^{({{n}\mathchar
45\relax\mathbf{Cat}})}$$U^{({{n}\mathchar
45\relax\mathbf{Cat}})}$$\mathrm{id}$${{U^{(n)}}\mathchar
45\relax\mathbf{Gph}}$${{T^{(n)}}\mathchar
45\relax\mathbf{Gph}}$$T^{(n+1)}$$=$$\eta^{({{n}\mathchar
45\relax\mathbf{Cat}})}$$=$

(cf. Definition 2.7), where $\eta^{({{n}\mathchar 45\relax\mathbf{Cat}})}$ is
the unit of the adjunction $F^{({{n}\mathchar 45\relax\mathbf{Cat}})}\dashv
U^{({{n}\mathchar 45\relax\mathbf{Cat}})}$. Since $\eta^{({{n}\mathchar
45\relax\mathbf{Cat}})}$ is a cartesian natural transformation ([16,
Proposition 3.5]) and the right adjoint functor ${{U^{(n)}}\mathchar
45\relax\mathbf{Gph}}$ preserves pullbacks, $\phi^{(n)}$ is a cartesian
natural transformation as well. Taking the limit, we obtain a cartesian monad
morphism $\phi^{(\omega)}\colon{{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}}\longrightarrow T^{(\omega)}$.

The following is a standard fact for slice categories of a monoidal category
(cf. Proposition 2.1).

######  Proposition 3.5.

Let ${\mathcal{M}}$ be a monoidal category with pullbacks, $T$ and $S$ be
monoids in ${\mathcal{M}}$, and $h\colon T\longrightarrow S$ be a monoid
morphism. Then there exists a monoidal adjunction

${\mathcal{M}}/T$${\mathcal{M}}/S$,${\mathcal{M}}/h$$h^{\ast}$$\vdash$

where ${\mathcal{M}}/h$ maps $(p\colon P\longrightarrow T)$ to $h\circ p$ and
$h^{\ast}$ maps $(q\colon Q\longrightarrow S)$ to the pullback of $q$ along
$h$.

Since $\phi^{(\omega)}$ is a monoid morphism from ${{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}}$ to $T^{(\omega)}$ in the monoidal category
$[{{\omega}\mathchar 45\relax\mathbf{Gph}},{{\omega}\mathchar
45\relax\mathbf{Gph}}]_{\mathrm{cart}}$, by Proposition 3.5 we obtain a
monoidal adjunction

$[{{\omega}\mathchar 45\relax\mathbf{Gph}},{{\omega}\mathchar
45\relax\mathbf{Gph}}]_{\mathrm{cart}}/{{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}}$$[{{\omega}\mathchar
45\relax\mathbf{Gph}},{{\omega}\mathchar
45\relax\mathbf{Gph}}]_{\mathrm{cart}}/T^{(\omega)}$.$[{{\omega}\mathchar
45\relax\mathbf{Gph}},{{\omega}\mathchar
45\relax\mathbf{Gph}}]_{\mathrm{cart}}/\phi^{(\omega)}$$(\phi^{(\omega)})^{\ast}$$\vdash$

Modulo the monoidal equivalences (1) induced by the evaluation at $1$, this
monoidal adjunction is

(14) ${{\omega}\mathchar 45\relax\mathbf{Gph}}/({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})1$${{\omega}\mathchar
45\relax\mathbf{Gph}}/T^{(\omega)}1$,${{{\omega}\mathchar
45\relax\mathbf{Gph}}/\phi^{(\omega)}_{1}}$$(\phi^{(\omega)}_{1})^{\ast}$$\vdash$

given by postcomposition and pullback of
$\phi^{(\omega)}_{1}\colon({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})1\longrightarrow T^{(\omega)}1$.

This induces in particular the functor

(15) $(\phi^{(\omega)}_{1})^{\ast}\colon{\mathbf{Mon}({{\omega}\mathchar
45\relax\mathbf{Gph}}/T^{(\omega)}1)}\longrightarrow{\mathbf{Mon}({{\omega}\mathchar
45\relax\mathbf{Gph}}/({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})1)}.$

On the other hand, by the universality of pullback we also obtain the functor

(16)
$(\phi^{(\omega)}_{1})^{\ast}\colon{\mathbf{Contr}}_{T^{(\omega)}1}\longrightarrow{\mathbf{Contr}}_{({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})1};$

indeed, $(\phi^{(\omega)}_{1})^{\ast}$ is the reindexing (or change-of-base)
functor between the fibres of the fibration

$\mathrm{cod}\colon{\mathbf{Contr}}({{\omega}\mathchar
45\relax\mathbf{Gph}},{\mathcal{J}}^{(\omega)})\longrightarrow{{\omega}\mathchar
45\relax\mathbf{Gph}}$

(see Remark 2.12) induced by $\phi^{(\omega)}_{1}$.

The functors $(\phi^{(\omega)}_{1})^{\ast}$ in (14), (15) and (16) induce the
functor

$(\phi^{(\omega)}_{1})^{\ast}\colon{\mathbf{OC}(T^{(\omega)})}\longrightarrow{\mathbf{OC}({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})}.$

Now, the $T^{(\omega)}$-operad $L$ is mapped by $(\phi^{(\omega)}_{1})^{\ast}$
to the $({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})$-operad
$(\phi^{(\omega)}_{1})^{\ast}L$, and by construction we have the following
pullback square in the category ${\mathbf{Mon}([{{\omega}\mathchar
45\relax\mathbf{Gph}},{{\omega}\mathchar
45\relax\mathbf{Gph}}]_{\mathrm{cart}})}$ of cartesian monads on
${{\omega}\mathchar 45\relax\mathbf{Gph}}$:

(17) $((\phi^{(\omega)}_{1})^{\ast}L)\ast_{{{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}}}(-)$${{{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}}}$$L\ast_{T^{(\omega)}}(-)$${T^{(\omega)}}.$$\phi^{(\omega)}$$\overline{{\mathrm{ar}_{L}}}$

On the other hand,
$(\phi^{(\omega)}_{1})^{\ast}L\in{\mathbf{OC}({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})}$ is in the essential image of ${\lceil-\rceil}$, and
hence by the initiality of $L$ we obtain a canonical
$({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})$-operad morphism ${\lceil
L\rceil}\longrightarrow(\phi^{(\omega)}_{1})^{\ast}L$, giving rise to a monad
morphism

(18) ${\lceil L\rceil}\ast_{{{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}}}(-)\longrightarrow((\phi_{1}^{(\omega)})^{\ast}L)\ast_{{{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}}}(-).$

Precomposing (13) and postcomposing the top horizontal arrow in (17) with
this, we obtain a monad morphism

${{(L\ast_{T^{(\omega)}}(-))}\mathchar 45\relax\mathbf{Gph}}\longrightarrow
L\ast_{T^{(\omega)}}(-),$

thus inducing the forgetful functor ${U_{\mathrm{s}}}$ as desired.

On the level of operads, we have the following diagram in ${{\omega}\mathchar
45\relax\mathbf{Gph}}$, in which the top horizontal composite captures the
essence of ${U_{\mathrm{s}}}$:

(19) ${\lceil
L\rceil}$$(\phi^{(\omega)}_{1})^{\ast}L$${({{T^{(\omega)}}\mathchar
45\relax\mathbf{Gph}})1}$$L$${T^{(\omega)}1}$.${\mathrm{ar}_{{\lceil
L\rceil}}}$$\phi^{(\omega)}_{1}$${\mathrm{ar}_{L}}$

Note that by the functoriality of ${U_{\mathrm{s}}}\colon{\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}\longrightarrow{{({\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}})}\mathchar
45\relax\mathbf{Gph}}$, we see that any strict $\omega$-functor
$F\colon{\mathcal{A}}\longrightarrow{\mathcal{B}}$ induces a family of strict
$\omega$-functors
$(F_{x,y}\colon{\mathcal{A}}(x,y)\longrightarrow{\mathcal{B}}(Fx,Fy))_{x,y\in{\mathrm{ob}({\mathcal{A}})}}$
as its action on homs.

######  Remark 3.6.

By essentially the same argument, we can also construct the finite-dimensional
versions of the forgetful functor ${U_{\mathrm{s}}}$, namely for each
$n\in\mathbb{N}$ a functor

$U_{\mathrm{s}}^{(n)}\colon{\mathbf{Wk}\mathchar 45\relax{(n+1)}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}\longrightarrow{{({\mathbf{Wk}\mathchar
45\relax{n}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}})}\mathchar
45\relax\mathbf{Gph}};$

see [28, Section 9.3] or [16] for the relevant definiton of weak $n$-category.

Also, since ${U_{\mathrm{s}}}$ (resp. $U_{\mathrm{s}}^{(n)}$ for each
$n\in\mathbb{N}$) is induced from a monad morphism between finitary monads on
a locally finitely presentable category ${{\omega}\mathchar
45\relax\mathbf{Gph}}$ (resp. ${{(n+1)}\mathchar 45\relax\mathbf{Gph}}$), it
is monadic. In particular, for finite-dimensional versions this means that one
can _in principle_ define a weak $(n+1)$-category in the sense of Leinster by
means of a set of objects, for each pair of objects, a hom weak $n$-category,
and various (horizontal) composition operations, following the same (weakened
enrichment) approach as the classical definitions of bicategory [6] and
tricategory [21]. However, an explicit description of the monad induced by
$U_{\mathrm{s}}^{(n)}$ seems challenging.

######  Remark 3.7.

The (${{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}}$)-operad with contraction
${\lceil L\rceil}$ is in fact the initial object in
${\mathbf{OC}({{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}})}$. This is
because for any (${{T^{(\omega)}}\mathchar 45\relax\mathbf{Gph}}$)-operad with
contraction $O=((O,{\mathrm{ar}_{O}}),e,m,\kappa)$, there is a unique morphism
${\lceil L\rceil}\longrightarrow O$ whose action on the unique object $\ast$
of ${\lceil L\rceil}$ is determined by $e$, and whose action on the hom
${\lceil L\rceil}(\ast,\ast)=L$ is determined by the initiality of $L$ in
${\mathbf{OC}(T^{(\omega)})}$. This gives an alternative view to the morphism
(18).

######  Remark 3.8.

Just as a monoidal category can be seen as a one-object bicategory, a possible
definition of _monoidal weak $\omega$-category_ would simply be a one-object
weak $\omega$-category; cf. [4] and [7, Chapter 5]. If we adopt this
definition, then our construction of hom weak $\omega$-categories specialises
to the expected operation of _forgetting the monoidal structure_ , i.e.,
taking the underlying weak $\omega$-category of a monoidal weak
$\omega$-category.

##  4\. Garner’s definition of weak $\omega$-functor

The morphisms of the (Eilenberg–Moore) category ${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}}$ preserve the
structures of weak $\omega$-categories on the nose, hence they are called
strict $\omega$-functors. Garner [20] introduced the more general notion of
_homomorphism_ between weak $\omega$-categories, which we call _weak
$\omega$-functor_. Weak $\omega$-functors are higher dimensional analogues of
pseudofunctors between bicategories or trihomomorphisms between tricategories,
i.e., functors preserving the structures up to coherent weakly invertible
cells. Our construction of hom weak $\omega$-categories is compatible not only
with strict $\omega$-functors, but also with weak $\omega$-functors. That is,
any weak $\omega$-functor $F\colon{\mathcal{A}}\longrightarrow{\mathcal{B}}$
induces a family
$(F_{x,y}\colon{\mathcal{A}}(x,y)\longrightarrow{\mathcal{B}}(Fx,Fy))_{x,y\in{\mathrm{ob}({\mathcal{A}})}}$
of weak $\omega$-functors between the hom weak $\omega$-categories; we shall
show this in the next section. In this section, we review the definition of
weak $\omega$-functor.

Weak $\omega$-categories and weak $\omega$-functors form a category, which we
denote by ${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}}$. The category ${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}}$ is defined as the
(co-)Kleisli category of a certain comonad $Q$ on ${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}}$. So our main
task is to define this comonad. In [20], Garner derives $Q$ from his theory of
_algebraic weak factorisation systems_ [19]; although this general perspective
is intriguing, it presupposes rather heavy machinery. Here we shall present a
more direct definition.

A certain class of strict $\omega$-functors, which we call _surjective
equivalences_ , is a key for the definition of weak $\omega$-functor.
Surjective equivalences are called _acyclic fibrations_ by Garner [20], and
indeed, the surjective equivalences between strict $\omega$-categories are the
acyclic (trivial) fibrations with respect to the _folk model structure_ on the
category of strict $\omega$-categories defined in [26] and further studied in
[3]. In the following definitions, we denote the forgetful functor
${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}\longrightarrow{{\omega}\mathchar
45\relax\mathbf{Gph}}$ by $\mathopen{|}-\mathclose{|}$. Also recall the set
${\mathcal{J}}^{(\omega)}$ of morphisms in ${{\omega}\mathchar
45\relax\mathbf{Gph}}$ and related notions introduced in Section 2.3.

######  Definition 4.1.

A strict $\omega$-functor $F\colon{\mathcal{A}}\longrightarrow{\mathcal{B}}$
is a surjective equivalence if the morphism
$|F|\colon|{\mathcal{A}}|\longrightarrow|{\mathcal{B}}|$ of $\omega$-graphs
has the right lifting property with respect to all morphisms in the set
$\mathcal{J}^{(\omega)}$.

######  Definition 4.2.

Let $F\colon{\mathcal{A}}\longrightarrow{\mathcal{B}}$ be a strict
$\omega$-functor. A contraction on $F$ is a contraction (with respect to
$\mathcal{J}^{(\omega)}$) on the morphism
$|F|\colon|{\mathcal{A}}|\longrightarrow|{\mathcal{B}}|$ of $\omega$-graphs.

For each weak $\omega$-category ${\mathcal{A}}$, define the category
${\mathbf{Contr}}_{\mathcal{A}}$ of all strict $\omega$-functors to
${\mathcal{A}}$ equipped with contractions as the following pullback of
categories:

${\mathbf{Contr}}_{{\mathcal{A}}}$${{\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}/{\mathcal{A}}}$${\mathbf{Contr}}_{|{\mathcal{A}}|}$${{\omega}\mathchar
45\relax\mathbf{Gph}}/|{\mathcal{A}}|$$\mathopen{|}-\mathclose{|}$

where ${\mathbf{Contr}}_{|{\mathcal{A}}|}={\mathbf{Contr}}({{\omega}\mathchar
45\relax\mathbf{Gph}},{\mathcal{J}}^{(\omega)})_{|{\mathcal{A}}|}$.

Of course, for every object
$(F\colon{\mathcal{B}}\longrightarrow{\mathcal{A}},\kappa)\in{\mathbf{Contr}}_{{\mathcal{A}}}$,
$F$ is a surjective equivalence.

######  Remark 4.3.

Using the left adjoint $F^{(\omega)}_{\mathrm{Wk}}\colon{{\omega}\mathchar
45\relax\mathbf{Gph}}\longrightarrow{\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}}$ of
$\mathopen{|}-\mathclose{|}$, we obtain the set

$F^{(\omega)}_{\mathrm{Wk}}{\mathcal{J}}^{(\omega)}=\\{\,F^{(\omega)}_{\mathrm{Wk}}m_{k}\colon
F^{(\omega)}_{\mathrm{Wk}}\partial\mathbf{y}[k]\longrightarrow
F^{(\omega)}_{\mathrm{Wk}}\mathbf{y}[k]\mid k\in\mathbb{N}\,\\}$

of morphisms in ${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}$. A contraction on a strict
$\omega$-functor defined above corresponds to a contraction with respect to
$F^{(\omega)}_{\mathrm{Wk}}{\mathcal{J}}^{(\omega)}$, as defined in Definition
2.11. Also, for each weak $\omega$-category ${\mathcal{A}}$, the category
${\mathbf{Contr}}_{{\mathcal{A}}}$ is isomorphic to
${\mathbf{Contr}}({\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}},F^{(\omega)}_{\mathrm{Wk}}{\mathcal{J}}^{(\omega)})_{\mathcal{A}}$.

######  Proposition 4.4.

For any weak $\omega$-category ${\mathcal{A}}$, the category
${\mathbf{Contr}}_{{\mathcal{A}}}$ has an initial object
$(\varepsilon_{\mathcal{A}}\colon
Q{\mathcal{A}}\longrightarrow{\mathcal{A}},\kappa_{\mathcal{A}})$.

###### Proof.

This is a special case of [20, Proposition 2.6]. ∎

See [20, Section 5] for a more explicit description of $Q{\mathcal{A}}$ by
means of computads. Intuitively, $Q{\mathcal{A}}$ is obtained from
${\mathcal{A}}$ by inductively replacing equalities between various composites
of cells by weakly invertible higher-dimensional cells, so that a strict
$\omega$-functor $Q{\mathcal{A}}\longrightarrow{\mathcal{B}}$ amounts to a
weak $\omega$-functor ${\mathcal{A}}\longrightarrow{\mathcal{B}}$; indeed,
this is how a weak $\omega$-functor is defined. We also remark that the
construction ${\mathcal{A}}\longmapsto Q{\mathcal{A}}$ is an extension to weak
$\omega$-categories of the _standard resolutions_ of strict
$\omega$-categories introduced in [33, Section 4]. The standard resolution
$Q{\mathcal{A}}$ of a strict $\omega$-category ${\mathcal{A}}$ is a cofibrant
replacement of ${\mathcal{A}}$ with respect to the folk model structure; see
[26].

We claim that $Q$ extends to a comonad on ${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}}$.

  * •

To describe the action of $Q$ on morphisms, suppose
$F\colon{\mathcal{A}}\longrightarrow{\mathcal{B}}$ is a strict
$\omega$-functor. We have a functor
$F^{\ast}\colon{\mathbf{Contr}}_{{\mathcal{B}}}\longrightarrow{\mathbf{Contr}}_{{\mathcal{A}}}$
defined by pulling back along $F$; see Remark 2.12. In particular,
$(\varepsilon_{\mathcal{B}}\colon
Q{\mathcal{B}}\longrightarrow{\mathcal{B}},\kappa_{\mathcal{B}})$ is mapped to
the pullback $F^{\ast}\varepsilon_{\mathcal{B}}\colon
F^{\ast}Q{\mathcal{B}}\longrightarrow{\mathcal{A}}$ of
$\varepsilon_{\mathcal{B}}$ along $F$, equipped with the contraction induced
from $\kappa_{\mathcal{B}}$ by the universality of pullback. By the initiality
of $(\varepsilon_{\mathcal{A}},\kappa_{\mathcal{A}})$, we obtain a canonical
strict $\omega$-functor $Q{\mathcal{A}}\longrightarrow F^{\ast}Q{\mathcal{B}}$
making the left triangle in (20) commute. We define $QF\colon
Q{\mathcal{A}}\longrightarrow Q{\mathcal{B}}$ as the top horizontal composite
in (20).

(20)
$Q{\mathcal{A}}$$F^{\ast}Q{\mathcal{B}}$$Q{\mathcal{B}}$${\mathcal{A}}$${\mathcal{B}}$$\varepsilon_{\mathcal{A}}$$\varepsilon_{\mathcal{B}}$$F^{\ast}\varepsilon_{\mathcal{B}}$$F$

  * •

The counit of $Q$ at ${\mathcal{A}}$ is given by
$\varepsilon_{\mathcal{A}}\colon Q{\mathcal{A}}\longrightarrow{\mathcal{A}}$.

  * •

The comultiplication of $Q$ at ${\mathcal{A}}$ is given by
$\delta_{\mathcal{A}}\colon QA\longrightarrow Q^{2}A$ as below, induced by the
initiality of $(\varepsilon_{\mathcal{A}},\kappa_{\mathcal{A}})$ (we equip the
morphism $\varepsilon_{\mathcal{A}}\circ\varepsilon_{Q{\mathcal{A}}}$ with the
contraction induced from $\kappa_{\mathcal{A}}$ and $\kappa_{Q{\mathcal{A}}}$;
see [10, Section 2.8]).

$Q{\mathcal{A}}$$Q^{2}{\mathcal{A}}$${\mathcal{A}}$$\delta_{\mathcal{A}}$$\varepsilon_{\mathcal{A}}$$\varepsilon_{\mathcal{A}}\circ\varepsilon_{Q{\mathcal{A}}}$

It is routine to check that the data $(Q,\varepsilon,\delta)$ defines a
comonad on ${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}$.

######  Definition 4.5.

A weak $\omega$-functor from ${\mathcal{A}}$ to ${\mathcal{B}}$ is a strict
$\omega$-functor $Q{\mathcal{A}}\longrightarrow{\mathcal{B}}$. The category
${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar 45\relax\mathbf{Cat}}$ of all
weak $\omega$-categories and weak $\omega$-functors is defined as the Kleisli
category of the comonad $Q$.

In particular, this means that the identity weak $\omega$-functor on
${\mathcal{A}}$ is $\varepsilon_{\mathcal{A}}\colon
Q{\mathcal{A}}\longrightarrow{\mathcal{A}}$, and that the composite of
$F\colon Q{\mathcal{A}}\longrightarrow{\mathcal{B}}$ and $G\colon
Q{\mathcal{B}}\longrightarrow{\mathcal{C}}$ is given by

$Q{\mathcal{A}}$$Q^{2}{\mathcal{A}}$$Q{\mathcal{B}}$${\mathcal{C}}.$$\delta_{\mathcal{A}}$$QF$$G$

######  Remark 4.6.

By definition, a weak $\omega$-functor
$F\colon{\mathcal{A}}\longrightarrow{\mathcal{B}}$ gives rise to a span in
${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}$

${\mathcal{A}}$$Q{\mathcal{A}}$${\mathcal{B}}$$\varepsilon_{\mathcal{A}}$$F$

whose left leg $\varepsilon_{\mathcal{A}}$ is a surjective equivalence
equipped with the universal contraction $\kappa_{\mathcal{A}}$ (see also
[11]). Conversely, every span in ${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}}$

(21) ${\mathcal{A}}$${\mathcal{E}}$${\mathcal{B}}$$J$$G$

whose left leg $J$ is a surjective equivalence, together with a choice of a
contraction $\kappa$ on $J$, gives rise to a weak $\omega$-functor from
${\mathcal{A}}$ to ${\mathcal{B}}$ (because we obtain a unique morphism
$H\colon(\varepsilon_{\mathcal{A}}\colon
Q{\mathcal{A}}\longrightarrow{\mathcal{A}},\kappa_{\mathcal{A}})\longrightarrow(J\colon{\mathcal{E}}\longrightarrow{\mathcal{A}},\kappa)$
in ${\mathbf{Contr}}_{\mathcal{A}}$ by initiality and hence a weak
$\omega$-functor $G\circ H\colon{\mathcal{A}}\longrightarrow{\mathcal{B}}$).

In ordinary category theory, surjective equivalences amount to surjective-on-
objects equivalences of categories, and the spans (21) in $\mathbf{Cat}$ with
$J$ a surjective equivalence (without a choice of a contraction) correspond to
the _anafunctors_ of Makkai [30].

######  Remark 4.7.

Now that we have the hom weak $\omega$-categories ${\mathcal{A}}(x,y)$ of a
weak $\omega$-category ${\mathcal{A}}$, in light of the weakened enrichment
approach to weak higher-dimensional categories, it seems natural to seek a way
to extract the _composition_ weak $\omega$-functors

$\circ\colon{\mathcal{A}}(y,z)\times{\mathcal{A}}(x,y)\longrightarrow{\mathcal{A}}(x,z)$

from the weak $\omega$-category structure of ${\mathcal{A}}$. Currently we do
not know how to define such weak $\omega$-functors, and leave this important
construction as future work.

##  5\. The forgetful functor $U\colon{\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar
45\relax\mathbf{Cat}}\longrightarrow{{({\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}})}\mathchar
45\relax\mathbf{Gph}}$

Let us denote the canonical (bijective-on-objects) right adjoint functor
associated with the Kleisli category ${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}}$ by
$J\colon{\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}\longrightarrow{\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}}$. In this section we extend
the action of the forgetful functor
${U_{\mathrm{s}}}\colon{\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}\longrightarrow{{({\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}})}\mathchar
45\relax\mathbf{Gph}}$ defined in Section 3 to weak $\omega$-functors. That
is, we construct an extension $U$ of ${U_{\mathrm{s}}}$ making the diagram

(22) ${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}$${{({\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}})}\mathchar
45\relax\mathbf{Gph}}$${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}}$${{({\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}})}\mathchar
45\relax\mathbf{Gph}}$${U_{\mathrm{s}}}$${{J}\mathchar
45\relax\mathbf{Gph}}$$J$$U$

commute.

The key observation is that ${{J}\mathchar 45\relax\mathbf{Gph}}$ exhibits
${{({\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}})}\mathchar 45\relax\mathbf{Gph}}$ as the Kleisli
category of the comonad ${{Q}\mathchar 45\relax\mathbf{Gph}}$ on
${{({\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}})}\mathchar 45\relax\mathbf{Gph}}$. This is
a consequence of the following facts. First, an adjunction $F\dashv
U\colon{\mathcal{C}}\to{\mathcal{D}}$ in ${\mathscr{C\\!A\\!T}}$ is isomorphic
to the Kleisli adjunction for the comonad $FU$ on the category ${\mathcal{C}}$
(hence in particular ${\mathcal{D}}$ is isomorphic to the Kleisli category of
$FU$), if and only if $U$ is bijective on objects. Next, the 2-functor
${{(-)}\mathchar
45\relax\mathbf{Gph}}\colon{\mathscr{C\\!A\\!T}}\longrightarrow{\mathscr{C\\!A\\!T}}$
preserves (adjunctions and) bijective-on-objects functors. Hence
${{(-)}\mathchar 45\relax\mathbf{Gph}}$ preserves the Kleisli categories of
comonads (as well as of monads).

So (22) is a map between Kleisli categories, and in order to obtain $U$, it
suffices to equip ${U_{\mathrm{s}}}$ with the structure of a comonad opfunctor
[36], i.e., a suitable natural transformation $\alpha$ as in

${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}$${{({\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}})}\mathchar
45\relax\mathbf{Gph}}$${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}$${{({\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}_{\mathrm{s}}})}\mathchar
45\relax\mathbf{Gph}}$${U_{\mathrm{s}}}$${{Q}\mathchar
45\relax\mathbf{Gph}}$$Q$${U_{\mathrm{s}}}$$\alpha$

respecting the structures of the comonads $Q$ and ${{Q}\mathchar
45\relax\mathbf{Gph}}$. The natural transformation $\alpha$ consists of, for
each weak $\omega$-category ${\mathcal{A}}$, a morphism

$\alpha_{\mathcal{A}}\colon({{Q}\mathchar
45\relax\mathbf{Gph}}){U_{\mathrm{s}}}{\mathcal{A}}\longrightarrow{U_{\mathrm{s}}}Q{\mathcal{A}}$

in ${{({\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}})}\mathchar 45\relax\mathbf{Gph}}$. We
define $\alpha_{\mathcal{A}}$ to be the identity on objects. So it remains to
define, for each pair $(x,y)$ of objects of ${\mathcal{A}}$, a strict
$\omega$-functor

$(\alpha_{\mathcal{A}})_{x,y}\colon
Q({\mathcal{A}}(x,y))\longrightarrow(Q{\mathcal{A}})(x,y).$

We induce $(\alpha_{\mathcal{A}})_{x,y}$ from the initiality used to determine
$Q({\mathcal{A}}(x,y))$. Recall that, by definition, the weak
$\omega$-category $Q({\mathcal{A}}(x,y))$ is equipped with a strict
$\omega$-functor $\varepsilon_{{\mathcal{A}}(x,y)}\colon
Q({\mathcal{A}}(x,y))\longrightarrow{\mathcal{A}}(x,y)$ and a contraction
$\kappa_{{\mathcal{A}}(x,y)}$ thereon, such that
$(\varepsilon_{{\mathcal{A}}(x,y)},\kappa_{{\mathcal{A}}(x,y)})$ is the
initial object of ${\mathbf{Contr}}_{{\mathcal{A}}(x,y)}$. On the other hand,
we also have a strict $\omega$-functor $\varepsilon_{\mathcal{A}}\colon
Q{\mathcal{A}}\longrightarrow{\mathcal{A}}$ equipped with a contraction
$\kappa_{\mathcal{A}}$. We obtain a strict $\omega$-functor
$(\varepsilon_{\mathcal{A}})_{x,y}\colon(Q{\mathcal{A}})(x,y)\longrightarrow{\mathcal{A}}(x,y)$
as a part of its action on homs. Moreover, it is easy to see that the
contraction $\kappa_{\mathcal{A}}$ restricts to give a contraction
$(\kappa_{\mathcal{A}})_{x,y}$ on $(\varepsilon_{\mathcal{A}})_{x,y}$. So we
obtain an object
$((\varepsilon_{\mathcal{A}})_{x,y},(\kappa_{\mathcal{A}})_{x,y})$ of
${\mathbf{Contr}}_{{\mathcal{A}}(x,y)}$. Now define the strict
$\omega$-functor $(\alpha_{\mathcal{A}})_{x,y}$ as the unique morphism
$(\varepsilon_{{\mathcal{A}}(x,y)},\kappa_{{\mathcal{A}}(x,y)})\longrightarrow((\varepsilon_{\mathcal{A}})_{x,y},(\kappa_{\mathcal{A}})_{x,y})$
from the initial object in ${\mathbf{Contr}}_{{\mathcal{A}}(x,y)}$. One can
check that $({U_{\mathrm{s}}},\alpha)$ is a comonad opfunctor by a
straightforward calculation.

##  6\. Restriction to weak $\omega$-groupoids

We have shown that a weak $\omega$-category ${\mathcal{A}}$ has weak
$\omega$-categories ${\mathcal{A}}(x,y)$ as homs. In this section we briefly
sketch that if ${\mathcal{A}}$ is a weak $\omega$-_groupoid_ , then so are the
homs.

Weak $\omega$-groupoids are weak $\omega$-categories in which each $k$-cell
($k\geq 1$) is weakly invertible, so we start with a definition of weakly
invertible cell in a weak $\omega$-category. In fact, one can define weakly
invertible cells in any $\omega$-graph equipped with suitable identity and
binary composition operations.

######  Definition 6.1 ([14, Definition 1]).

An $\omega$-precategory is an $\omega$-graph $P$ equipped with the following
structure:

  1. (1)

for any $k\geq 1$ and a $(k-1)$-cell $a$ of $P$, a specified $k$-cell
$\mathrm{id}_{a}\colon a\longrightarrow a$;

  2. (2)

for any $k\geq 1$ and a pair of $k$-cells $f\colon a\longrightarrow b$ and
$g\colon b\longrightarrow c$ of $P$, a specified $k$-cell $g\circ f\colon
a\longrightarrow c$.

Note that an $\omega$-precategory has compositions of $k$-cells only along
$(k-1)$-dimensional boundaries.

######  Definition 6.2 ([14, 9]).

Let $P$ be an $\omega$-precategory. The set of weakly invertible cells in $P$
is the set of cells of dimension $\geq 1$ defined _coinductively_ as follows:
for $k\geq 1$, a $k$-cell $f\colon a\longrightarrow b$ of $P$ is weakly
invertible if and only if there exist a $k$-cell $g\colon b\longrightarrow a$
and weakly invertible $(k+1)$-cells $\eta\colon\mathrm{id}_{a}\longrightarrow
g\circ f$ and $\varepsilon\colon f\circ g\longrightarrow\mathrm{id}_{b}$.

Since this is a coinductive definition, in order to show that a $k$-cell $f$
is weakly invertible, it suffices to exhibit a set $W$ of cells in $P$ (called
a set of _witnesses_ in [14, Definition 6]) such that $f\in W$ and, for any
$f^{\prime}\colon a^{\prime}\longrightarrow b^{\prime}$ in $W$, there exist
cells $g^{\prime}\colon b^{\prime}\longrightarrow
a^{\prime},\eta^{\prime}\colon\mathrm{id}_{a^{\prime}}\longrightarrow
g^{\prime}\circ f^{\prime}$ and $\varepsilon^{\prime}\colon f^{\prime}\circ
g^{\prime}\longrightarrow\mathrm{id}_{b^{\prime}}$ in $W$.

Each weak $\omega$-category ${\mathcal{A}}$ has a canonical
$\omega$-precategory structure underlying it. To see this, it suffices to find
$k$-cells $i_{k}$ and $m_{k}$ in $L$ ($k\geq 1$) of suitable arities, for then
we can use their interpretations on the underlying $\omega$-graph
$|{\mathcal{A}}|$ to define an $\omega$-precategory structure on
$|{\mathcal{A}}|$. So we first introduce $k$-cells $0_{k}$ and $2_{k}$ in
$T^{(\omega)}1$ ($k\geq 1$) which are the arities of $i_{k}$ and $m_{k}$
respectively (cf. [9, Section 2.1]). For convenience, we also introduce a
$k$-cell $1_{k}$ in $T^{(\omega)}1$ ($k\geq 0$). In low dimensions, these
cells represent the following globular pasting schemes (cf. (6)).

$\bullet$$1_{0}$
$\bullet$$0_{1}$$\bullet$$\bullet$$1_{1}$$\bullet$$\bullet$$\bullet$$2_{1}$
$\bullet$$\bullet$$0_{2}$$\bullet$$\bullet$$1_{2}$$\bullet$$\bullet$$2_{2}$
$\bullet$$\bullet$$0_{3}$$\bullet$$\bullet$$1_{3}$$\bullet$$\bullet$$2_{3}$

Formally, in terms of lists (cf. (5)), we define recursively $1_{0}=\bullet$,
$1_{k+1}=[1_{k}]$, $0_{1}=[\,]$, $0_{k+1}=[0_{k}]$, $2_{1}=[\bullet,\bullet]$
and $2_{k+1}=[2_{k}]$. Note that we have $0_{k}\colon 1_{k-1}\longrightarrow
1_{k-1}$ and $2_{k}\colon 1_{k-1}\longrightarrow 1_{k-1}$ in $T^{(\omega)}1$.

In order to define the cells $i_{k}$ and $m_{k}$ of $L$, we use the
contraction $\kappa$ associated with $L$. Note that the unit morphism $e\colon
I\longrightarrow(L,{\mathrm{ar}_{L}})$ of the $T^{(\omega)}$-operad $L$ yields
a $k$-cell $e_{k}$ in $L$ of arity $1_{k}$ for each $k\in\mathbb{N}$. We
define $i_{k}=\kappa(k,(e_{k-1},e_{k-1}),0_{k})$ and
$m_{k}=\kappa(k,(e_{k-1},e_{k-1}),2_{k})$.

We define the weakly invertible cells in a weak $\omega$-category to be the
weakly invertible cells in its underlying $\omega$-precategory, and define a
weak $\omega$-groupoid to be a weak $\omega$-category in which all cells of
dimension $\geq 1$ are weakly invertible [14, 9].

We claim that for any weak $\omega$-groupoid ${\mathcal{A}}$ and pair of
objects $x$ and $y$, the hom weak $\omega$-category ${\mathcal{A}}(x,y)$ given
by the forgetful functor ${U_{\mathrm{s}}}$ of Section 3 (or equivalently, by
$U$ of Section 5) is again a weak $\omega$-groupoid. In order to show this, it
suffices to show that the canonical identity and binary composition operations
in ${\mathcal{A}}(x,y)$ agrees with those in ${\mathcal{A}}$. This follows
from the construction in Section 3; since the morphism ${\lceil
L\rceil}\longrightarrow L$ in (19) preserves contractions and units, for each
$k\geq 2$ it maps the $k$-cell $i_{k-1}$ (resp. $m_{k-1}$) in ${\lceil
L\rceil}$ to the $k$-cell $i_{k}$ (resp. $m_{k}$) in $L$.

If we denote by ${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Gpd}_{\mathrm{s}}}$ (resp. ${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Gpd}}$) the full subcategory of
${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Cat}_{\mathrm{s}}}$ (resp. ${\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Cat}}$) consisting of all weak
$\omega$-groupoids, then it follows that the forgetful functor
${U_{\mathrm{s}}}$ (resp. $U$) restricts to

${\mathbf{Wk}\mathchar 45\relax{\omega}\mathchar
45\relax\mathbf{Gpd}_{\mathrm{s}}}\longrightarrow{{({\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Gpd}_{\mathrm{s}}})}\mathchar
45\relax\mathbf{Gph}}\qquad(\text{resp.~{}}{\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar
45\relax\mathbf{Gpd}}\longrightarrow{{({\mathbf{Wk}\mathchar
45\relax{\omega}\mathchar 45\relax\mathbf{Gpd}})}\mathchar
45\relax\mathbf{Gph}}).$

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