2. Leinster’s Definition of Weak $\omega$ -Category

In this section, we review Leinster’s definition of weak $\omega$ -category (Leinster Reference Leinster2004) , which was introduced as a variant of an earlier definition by Batanin (Reference Batanin1998). According to this definition, the weak $\omega$ -categories are the Eilenberg–Moore algebras of a monad on the category ${{\omega}-\mathbf{Gph}}$ 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 Leinster (Reference Leinster2004) , and (ii’) choices of diagonal-fillers, following Garner (Reference Garner2009a).

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’ in $\mathcal{M}$ together with a monoid morphism $T'\longrightarrow T$ .

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’ on $\mathcal{C}$ equipped with a cartesian monad morphism $T'\longrightarrow T$ (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) \begin{equation}\mathrm{ev}_1\colon [\mathcal{C},\mathcal{C}]_{\mathrm{cart}}/T\longrightarrow \mathcal{C}/T1\end{equation}

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 (Kelly Reference Kelly1992). 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 $C\in \mathcal{C}$ :

(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, that is, $(P,{\mathrm{ar}_{P}})\otimes(Q,{\mathrm{ar}_{Q}})=(\mu_1\circ T({\mathrm{ar}_{Q}})\circ \partial_2\colon(P,{\mathrm{ar}_{P}})\ast Q\longrightarrow T1)$ :

  (2)

  

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 (May Reference May1972). 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, that is, 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 ${{\omega}-\mathbf{Gph}}$ of $\omega$ -graphs and the monad $T^{(\omega)}$ for strict $\omega$ -category. These are obtained as limits of ${{n}-\mathbf{Gph}}$ and $T^{(n)}$ , which we now define.

Definition 2.3 ( Reference Wolff Wolff 1974 ) For any locally small category $\mathcal{V}$ , we define the category ${{\mathcal{V}}-\mathbf{Gph}}$ 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 $G'=({\mathrm{ob}(G')},(G'(x,y))_{x,y\in{\mathrm{ob}(G')}})$ consists of a function ${\mathrm{ob}(f)}\colon {\mathrm{ob}(G)}\longrightarrow{\mathrm{ob}(G')}$ (whose action we denote by f) and, for each pair x,y of objects of G, a morphism $f_{x,y}\colon G(x,y)\longrightarrow G'(fx,fy)$ in $\mathcal{V}$ .

The ${{(-)}-\mathbf{Gph}}$ 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 ${{n}-\mathbf{Gph}}$ recursively as follows:

\[{{0}-\mathbf{Gph}}=\mathbf{Set},\qquad {{(n+1)}-\mathbf{Gph}}={{({{n}-\mathbf{Gph}})}-\mathbf{Gph}}.\]

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

Remark 2.5 The ${{(-)}-\mathbf{Gph}}$ construction preserves presheaf categories, that is, for any small category $\mathcal{D}$ , the category ${{[\mathcal{D},\mathbf{Set}]}-\mathbf{Gph}}$ 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 $f\colon D\longrightarrow D'$ in $\mathcal{D}$ , $s^{(D')}\circ f=s^{(D)}$ and $ t^{(D')}\circ f=t^{(D)}$ . Hence, we see by induction that ${{n}-\mathbf{Gph}}$ is equivalent to $[\mathbb{G}_n^\mathrm{op},\mathbf{Set}]$ , where $\mathbb{G}_n$ is the category freely generated by the graph

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 ${{\mathcal{V}}-\mathbf{Cat}}$ of small $\mathcal{V}$ -categories (Kelly Reference Kelly1982); throughout this paper, we only consider enrichment over cartesian $\mathcal{V}$ . The ${{(-)}-\mathbf{Cat}}$ construction extends to an endofunctor on the category of locally small categories with finite products and finite product preserving functors. ( ${{(-)}-\mathbf{Cat}}$ 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 ${{n}-\mathbf{Cat}}$ recursively as follows:

\[{{0}-\mathbf{Cat}}=\mathbf{Set},\qquad {{(n+1)}-\mathbf{Cat}}={{({{n}-\mathbf{Cat}})}-\mathbf{Cat}}.\]

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

Let $\mathcal{V}$ be a locally small category with finite products. There is an evident forgetful functor $U^{(\mathcal{V})}\colon {{\mathcal{V}}-\mathbf{Cat}}\longrightarrow{{\mathcal{V}}-\mathbf{Gph}}$ . If $\mathcal{V}$ has small coproducts distributing over finite products, then $U^{(\mathcal{V})}$ admits a left adjoint $F^{(\mathcal{V})}$ (Wolff Reference Wolff1974). For each $n\in\mathbb{N}$ , ${{n}-\mathbf{Cat}}$ satisfies this condition (in fact, a stronger condition of extensivity Cottrell et al. Reference Cottrell, Fujii and Power2018), so we have $F^{({{n}-\mathbf{Cat}})}\dashv U^{({{n}-\mathbf{Cat}})}$ .

Definition 2.7 For each $n\in\mathbb{N}$ , we define the (monadic) adjunction $F^{(n)}\dashv U^{(n)}\colon {{n}-\mathbf{Cat}}\longrightarrow{{n}-\mathbf{Gph}}$ recursively as follows:

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

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

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

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

Proposition 2.8 (Leinster Reference Leinster2004, Theorem F.2.1, Cottrell et al. 2018, Theorem 4.6) For each $n\in\mathbb{N}$ , the monad $T^{(n)}$ is cartesian.

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

(3)

Objects of ${{\omega}-\mathbf{Gph}}$ 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)

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, ${{\omega}-\mathbf{Gph}}$ 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

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 (Batanin Reference Batanin1998; Leinster Reference Leinster2004; Cheng and Leinster Reference Cheng and Leinster2019). Let G be an $\omega$ -graph, with the corresponding globular set $G'\colon \mathbb{G}^\mathrm{op}\longrightarrow \mathbf{Set}$ . For each $k\in\mathbb{N}$ , elements of the set G’[k] are called k-cells of G. The functions $G's_k$ and $G't_k$ 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)(van den Berg and Garner Reference van den Berg and Garner2011). More precisely, the category ${{\omega}-\mathbf{Gph}}$ is the carrier of the terminal coalgebra for ${{(-)}-\mathbf{Gph}}$ (seen as an endofunctor on the category of locally small categories) (Cheng and Leinster Reference Cheng and Leinster2019). The structure map of this coalgebra is the functor ${{\omega}-\mathbf{Gph}}\longrightarrow {{({{\omega}-\mathbf{Gph}})}-\mathbf{Gph}}$ mapping each $\omega$ -graph G to the ( ${{\omega}-\mathbf{Gph}}$ )-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 ${{\omega}-\mathbf{Gph}}$ , called the free strict $\omega$ -category monad. $T^{(\omega)}$ is also cartesian, since a commutative square in ${{\omega}-\mathbf{Gph}}$ is a pullback if and only if for each $n\in\mathbb{N}$ , it is mapped to a pullback in ${{n}-\mathbf{Gph}}$ by the projection ${{\omega}-\mathbf{Gph}}\longrightarrow {{n}-\mathbf{Gph}}$ .

As a consequence, we obtain the monoidal category ${{\omega}-\mathbf{Gph}}/T^{(\omega)}1$ , 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 Leinster (Reference Leinster2004) , 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)

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)

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 ${{\omega}-\mathbf{Gph}}$ . Leinster (Reference Leinster2004) ntroduced a set-theoretic definition of contraction, to which Garner (Reference Garner2009a) 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

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 (Reference GarnerGarner 2009b, 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 $r'\colon C'\longrightarrow D'$ equipped with contractions $\kappa$ and $\kappa'$ , respectively, a map of morphisms $(h\colon C\longrightarrow C', k\colon D\longrightarrow D' )\colon r\longrightarrow r'$ (i.e., a commutative square) is said to preserve contractions if for each (l,u,v) in the domain of $\kappa$ , $h\circ \kappa(l,u,v)= \kappa' (l,h\circ u,k\circ v)$ .

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 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 $k\colon D\longrightarrow D'$ in $\mathcal{C}$ and any object $(r'\colon C'\longrightarrow D',\kappa')\in \mathbf{Contr}(\mathcal{C},\mathcal{J})_{D'}$ , one can endow the pullback $k^\ast r'$ as in

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

\[(h,k)\colon (k^\ast r',k^\ast \kappa)\longrightarrow (r',\kappa')\]

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 ${{\omega}-\mathbf{Gph}}$ , we define a set $\mathcal{J}^{(\omega)}$ of morphisms in ${{\omega}-\mathbf{Gph}}$ (called the set of generating cofibrations in Garner Reference Garner2009a). Recall from Remark 2.9 the equivalence ${{\omega}-\mathbf{Gph}}\simeq [\mathbb{G}^\mathrm{op},\mathbf{Set}]$ . 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:

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 ${{\omega}-\mathbf{Gph}}$ 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’ of parallel $(k-1)$ -cells of C and each k-cell $d\colon r(c) \longrightarrow r(c')$ in D, a k-cell $\kappa(k,(c,c'), d)\colon c\longrightarrow c'$ in C such that $d=r(\kappa (k, (c,c'),d))$ . For any $\omega$ -graph D, we write the category $\mathbf{Contr}({{\omega}-\mathbf{Gph}},\mathcal{J}^{(\omega)})_D$ simply as $\mathbf{Contr}_D$ .

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

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

(7)

Roughly speaking, a contraction on a $T^{(\omega)}$ -operad generates both (unbiased) composition operations and operations which yield coherence cells, see Leinster (Reference Leinster2004), 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 (Leinster Reference Leinster2004, Proposition 9.2.2) The category ${\mathbf{OC}({T^{(\omega)}})}{\omega}$ has an initial object.

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

Definition 2.14 (Leinster Reference Leinster2004, Definition 9.2.3) A weak $\omega$ -category is an L-algebra.

We denote the Eilenberg–Moore category of the monad $L\ast(-)$ by ${\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}$ ; the morphisms in ${\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}$ 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 Leinster (Reference Leinster2004), 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 ${{(O\ast(-))}-\mathbf{Alg}}\longrightarrow {\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}$ . 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 $T^{(\omega)}$ , that is, 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 Leinster (Reference Leinster2004, Example 9.2.7.)

One can also define the notion of algebra of a $T^{(\omega)}$ -operad over more general categories than ${{\omega}-\mathbf{Gph}}$ (Batanin Reference Batanin1998; Lumsdaine Reference Lumsdaine2010), that is, internally in those categories. In van den Berg and Garner (Reference van den Berg and Garner2011), Lumsdaine (Reference Lumsdaine2010), such a notion is defined on suitable categories of globular objects in the classifying (or syntactic) category $\mathcal{C}l(\mathbb{T})$ (cf. Pitts Reference Pitts, Abramsky, Gabbay and Maibaum2000, 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 ${U_\mathrm{s}}\colon {\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}\longrightarrow {{({\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}})}-\mathbf{Gph}}$

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 ${{(-)}-\mathbf{Gph}}$ preserves a lot of structure. First we observe that both the domain and codomain of ${U_\mathrm{s}}$ are monadic over ${{\omega}-\mathbf{Gph}}$ ; that the codomain is so is a consequence of the following.

Proof. The monad ${{T}-\mathbf{Gph}}$ is on the category ${{\mathcal{V}}-\mathbf{Gph}}$ . An object of ${{({{T}-\mathbf{Gph}})}-\mathbf{Alg}}$ consists of a $\mathcal{V}$ -graph G together with a $\mathcal{V}$ -graph morphism $\gamma\colon ({{T}-\mathbf{Gph}})G\longrightarrow G$ 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 $({{T}-\mathbf{Alg}})$ -graph.

So ${{({\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}})}-\mathbf{Gph}}={{({{(L\ast_{T^{(\omega)}}(-))}-\mathbf{Alg}})}-\mathbf{Gph}}$ is isomorphic to ${{({{(L\ast_{T^{(\omega)}}(-))}-\mathbf{Gph}})}-\mathbf{Alg}}$ . The monad ${{(L\ast_{T^{(\omega)}}(-))}-\mathbf{Gph}}$ is on ${{({{\omega}-\mathbf{Gph}})}-\mathbf{Gph}}$ , which is canonically isomorphic to ${{\omega}-\mathbf{Gph}}$ by Remark 2.10. Explicitly, the functor part of the monad ${{(L\ast_{T^{(\omega)}}(-))}-\mathbf{Gph}}$ maps an $\omega$ -graph G to the $\omega$ -graph ${{(L\ast_{T^{(\omega)}}(-))}-\mathbf{Gph}}(G)$ with the same objects and such that for each pair (x,y) of objects, the hom $({{(L\ast_{T^{(\omega)}}(-))}-\mathbf{Gph}}(G))(x,y)$ 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 ${{(L\ast_{T^{(\omega)}}(-))}-\mathbf{Gph}}$ is also induced from an operad over a cartesian monad. To show this, we use the following fact.

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

$({{T^{(\omega)}}-\mathbf{Gph}})$ -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 $({{T^{(\omega)}}-\mathbf{Gph}})1$ (seen as a globular set):

The cells of $({{T^{(\omega)}}-\mathbf{Gph}})1$ represent the globular pasting schemes not involving compositions along 0-cells. For instance, the 3-cell $[[\bullet],[\,],[\bullet,\bullet]]\in 1^{\ast\ast}$ of $({{T^{(\omega)}}-\mathbf{Gph}})1$ corresponds to the following globular pasting scheme (cf. (6)):

We turn to a concrete description of the $({{T^{(\omega)}}-\mathbf{Gph}})$ -operad corresponding to ${{\overline{{\mathrm{ar}_{L}}}}-\mathbf{Gph}}$ , which induces ${{(L\ast_{T^{(\omega)}}(-))}-\mathbf{Gph}}$ . For any locally small category $\mathcal{V}$ , define the functor $\lceil -\rceil_\mathcal{V}\colon \mathcal{V}\longrightarrow {{\mathcal{V}}-\mathbf{Gph}}$ (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 ${{\mathcal{V}}-\mathbf{Gph}}$ by $\lceil 1\rceil$ , distinguishing it from the terminal object 1 of $\mathcal{V}$ . The terminal object of ${{\omega}-\mathbf{Gph}}\cong {{({{\omega}-\mathbf{Gph}})}-\mathbf{Gph}}$ 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 $\lceil -\rceil_T\colon\mathcal{V}/T1\longrightarrow {{\mathcal{V}}-\mathbf{Gph}}/({{T}-\mathbf{Gph}})\lceil 1\rceil$ mapping $({\mathrm{ar}_{P}}\colon P\longrightarrow T1)$ to

makes the following square commute up to a natural isomorphism:

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

When $\mathcal{V}={{\omega}-\mathbf{Gph}}$ , the functor $\lceil -\rceil_{{{\omega}-\mathbf{Gph}}}\colon {{\omega}-\mathbf{Gph}}\longrightarrow {{({{\omega}-\mathbf{Gph}})}-\mathbf{Gph}}\cong {{\omega}-\mathbf{Gph}}$ 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 ${{\omega}-\mathbf{Gph}}\simeq [\mathbb{G}^\mathrm{op},\mathbf{Set}]$ , we have the evident functor $S\colon \mathbb{G}\longrightarrow \mathbb{G}$ mapping [k] to $[k+1]$ , and $\lceil -\rceil_{{{\omega}-\mathbf{Gph}}}$ is the right Kan extension along $S^\mathrm{op}$ . Observe that $({{T^{(\omega)}}-\mathbf{Gph}})1\cong \lceil T^{(\omega)}1\rceil _{{{\omega}-\mathbf{Gph}}}$ is obtained from $T^{(\omega)}1$ by this construction.

As a special case of $\lceil -\rceil_T$ , we obtain

(8) \begin{equation}\lceil -\rceil_{T^{(\omega)}}\colon {{\omega}-\mathbf{Gph}}/T^{(\omega)}1\longrightarrow {{\omega}-\mathbf{Gph}}/({{T^{(\omega)}}-\mathbf{Gph}})1;\end{equation}

here again we are identifying the two canonically isomorphic categories ${{\omega}-\mathbf{Gph}}$ and ${{({{\omega}-\mathbf{Gph}})}-\mathbf{Gph}}$ . 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) \begin{equation}{\mathbf{Mon}(\lceil -\rceil_{T^{(\omega)}})}\colon {\mathbf{Mon}({{\omega}-\mathbf{Gph}}/T^{(\omega)}1)}\longrightarrow {\mathbf{Mon}({{\omega}-\mathbf{Gph}}/({{T^{(\omega)}}-\mathbf{Gph}})1)}.\end{equation}

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 $({\mathrm{ar}_{G}}\colon G\longrightarrow ({{T}-\mathbf{Gph}})\lceil 1\rceil)\in {{\mathcal{V}}-\mathbf{Gph}}/({{T}-\mathbf{Gph}})\lceil 1\rceil$ is in the essential image of $\lceil -\rceil_T$ if and only if $G\in {{\mathcal{V}}-\mathbf{Gph}}$ has precisely one object.

For a locally small category $\mathcal{V}$ with small coproducts, the functor $\lceil -\rceil_\mathcal{V}\colon \mathcal{V}\longrightarrow{{\mathcal{V}}-\mathbf{Gph}}$ admits a left adjoint $\coprod_\mathcal{V}\colon{{\mathcal{V}}-\mathbf{Gph}}\longrightarrow \mathcal{V}$ , mapping $G\in{{\mathcal{V}}-\mathbf{Gph}}$ to $\coprod_{x,y\in{\mathrm{ob}(G)}}G(x,y)\in\mathcal{V}$ . In particular, the functor $\lceil -\rceil_{{\omega}-\mathbf{Gph}}\colon {{\omega}-\mathbf{Gph}}\longrightarrow {{\omega}-\mathbf{Gph}}$ admits a left adjoint $\coprod_{{\omega}-\mathbf{Gph}}\colon{{\omega}-\mathbf{Gph}}\longrightarrow {{\omega}-\mathbf{Gph}}$ . Intuitively, given an $\omega$ -graph G, the $\omega$ -graph $\coprod_{{{\omega}-\mathbf{Gph}}}G$ 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 $\coprod_{{{\omega}-\mathbf{Gph}}}G$ , and the 0-cells of G are thrown away. In view of the equivalence ${{\omega}-\mathbf{Gph}}\simeq [\mathbb{G}^\mathrm{op},\mathbf{Set}]$ , $\coprod_{{{\omega}-\mathbf{Gph}}}$ 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 ${{\omega}-\mathbf{Gph}}$ . 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 ${{\omega}-\mathbf{Gph}}$ , 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)

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

(11) \begin{equation}\lceil -\rceil\colon \mathbf{Contr}_{D}\longrightarrow \mathbf{Contr}_{\lceil D\rceil}\end{equation}

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 $(r'\colon C'\longrightarrow\lceil D\rceil,\kappa')\in\mathbf{Contr}_{\lceil D\rceil}$ such that $C'\in {{\omega}-\mathbf{Gph}}$ has precisely one object. As a special case of (11), we have

(12) \begin{equation}\lceil -\rceil\colon \mathbf{Contr}_{T^{(\omega)}1}\longrightarrow \mathbf{Contr}_{({{T^{(\omega)}}-\mathbf{Gph}})1},\end{equation}

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

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

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

\begin{equation*}\lceil -\rceil\colon {\mathbf{OC}({T^{(\omega)}})}{\omega}\longrightarrow {\mathbf{OC}({{{T^{(\omega)}}-\mathbf{Gph}}})}{\omega}.\end{equation*}

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

(13) \begin{equation}{{(L\ast_{T^{(\omega)}}(-))}-\mathbf{Gph}}\cong \lceil L\rceil\ast_{{{T^{(\omega)}}-\mathbf{Gph}}}(-).\end{equation}

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

(cf. Definition 2.7), where $\eta^{({{n}-\mathbf{Cat}})}$ is the unit of the adjunction $F^{({{n}-\mathbf{Cat}})}\dashv U^{({{n}-\mathbf{Cat}})}$ . Since $\eta^{({{n}-\mathbf{Cat}})}$ is a cartesian natural transformation (Cottrell et al. Reference Cottrell, Fujii and Power2018, Proposition 3.5) and the right adjoint functor ${{U^{(n)}}-\mathbf{Gph}}$ preserves pullbacks, $\phi^{(n)}$ is a cartesian natural transformation as well. Taking the limit, we obtain a cartesian monad morphism $\phi^{(\omega)}\colon {{T^{(\omega)}}-\mathbf{Gph}}\longrightarrow T^{(\omega)}$ .

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

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 ${{T^{(\omega)}}-\mathbf{Gph}}$ to $T^{(\omega)}$ in the monoidal category $[{{\omega}-\mathbf{Gph}},{{\omega}-\mathbf{Gph}}]_{\mathrm{cart}}$ , by Proposition 3.5 we obtain a monoidal adjunction

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

(14)

given by postcomposition and pullback of $\phi^{(\omega)}_1\colon ({{T^{(\omega)}}-\mathbf{Gph}})1\longrightarrow T^{(\omega)}1$ .

This induces in particular the functor

(15) \begin{equation}(\phi^{(\omega)}_1)^\ast\colon{\mathbf{Mon}({{\omega}-\mathbf{Gph}}/T^{(\omega)}1)}\longrightarrow{\mathbf{Mon}({{\omega}-\mathbf{Gph}}/({{T^{(\omega)}}-\mathbf{Gph}})1)}.\end{equation}

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

(16) \begin{equation}(\phi^{(\omega)}_1)^\ast\colon \mathbf{Contr}_{T^{(\omega)}1}\longrightarrow \mathbf{Contr}_{({{T^{(\omega)}}-\mathbf{Gph}})1};\end{equation}

indeed, $(\phi^{(\omega)}_1)^\ast$ is the reindexing (or change-of-base) functor between the fibres of the fibration

\[\mathrm{cod}\colon \mathbf{Contr}({{\omega}-\mathbf{Gph}},\mathcal{J}^{(\omega)})\longrightarrow {{\omega}-\mathbf{Gph}}\]

(see Remark 2.12) induced by $\phi^{(\omega)}_1$ .The functors $(\phi^{(\omega)}_1)^\ast$ in (14), (15) and (16) induce the functor

\begin{equation*}(\phi^{(\omega)}_1)^\ast\colon{\mathbf{OC}({T^{(\omega)}})}{n+1}\longrightarrow {\mathbf{OC}({{{T^{(\omega)}}-\mathbf{Gph}}})}{n+1}.\end{equation*}

Now, the $T^{(\omega)}$ -operad L is mapped by $(\phi^{(\omega)}_1)^\ast$ to the $({{T^{(\omega)}}-\mathbf{Gph}})$ -operad $(\phi^{(\omega)}_1)^\ast L$ , and by construction we have the following pullback square in the category ${\mathbf{Mon}([{{\omega}-\mathbf{Gph}},{{\omega}-\mathbf{Gph}}]_{\mathrm{cart}})}$ of cartesian monads on ${{\omega}-\mathbf{Gph}}$ :

(17)

On the other hand, $(\phi^{(\omega)}_1)^\ast L\in{\mathbf{OC}({{{T^{(\omega)}}-\mathbf{Gph}}})}{}$ is in the essential image of $\lceil -\rceil$ , and hence by the initiality of L we obtain a canonical $({{T^{(\omega)}}-\mathbf{Gph}})$ -operad morphism $\lceil L\rceil\longrightarrow(\phi^{(\omega)}_1)^\ast L$ , giving rise to a monad morphism

(18) \begin{equation}\lceil L\rceil\ast_{{{T^{(\omega)}}-\mathbf{Gph}}}(-)\longrightarrow ((\phi_1^{(\omega)})^\ast L)\ast_{{{T^{(\omega)}}-\mathbf{Gph}}}(-).\end{equation}

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

\[{{(L\ast_{T^{(\omega)}}(-))}-\mathbf{Gph}}\longrightarrow L\ast_{T^{(\omega)}}(-),\]

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

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

(19)

Note that by the functoriality of ${U_\mathrm{s}}\colon {\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}\longrightarrow {{({\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}})}-\mathbf{Gph}}$ , 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

\[U_{\mathrm{s}}^{(n)}\colon {\mathbf{Wk}-{(n+1)}-\mathbf{Cat}_\mathrm{s}}\longrightarrow {{({\mathbf{Wk}-{n}-\mathbf{Cat}_\mathrm{s}})}-\mathbf{Gph}};\]

see Leinster (Reference Leinster2004), Section 9.3 or Cottrell et al. (Reference Cottrell, Fujii and Power2018) for the relevant definition 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 ${{\omega}-\mathbf{Gph}}$ (resp. ${{(n+1)}-\mathbf{Gph}}$ ), 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 (Bénabou Reference Bénabou1967) and tricategory (Gordon et al. Reference Gordon, Power and Street1995). However, an explicit description of the monad induced by $U_{\mathrm{s}}^{(n)}$ seems challenging.

4. Garner’s Definition of Weak $\omega$ -Functor

The morphisms of the (Eilenberg–Moore) category ${\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}$ preserve the structures of weak $\omega$ -categories on the nose, hence they are called strict $\omega$ -functors. Garner (Reference Garner2010) 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, that is, 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 ${\mathbf{Wk}-{\omega}-\mathbf{Cat}}$ . The category ${\mathbf{Wk}-{\omega}-\mathbf{Cat}}$ is defined as the (co-)Kleisli category of a certain comonad Q on ${\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}$ . So our main task is to define this comonad. In Garner (Reference Garner2010), Garner derives Q from his theory of algebraic weak factorisation systems (Garner Reference Garner2009b); 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 (Reference Garner2010), 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 Lafont et al. (Reference Lafont, Métayer and Worytkiewicz2010) and further studied in Ara and Lucas (Reference Ara and Lucas2020). In the following definitions, we denote the forgetful functor ${\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}\longrightarrow{{\omega}-\mathbf{Gph}}$ by $\mathopen{|}-\mathclose{|}$ . Also recall the set $\mathcal{J}^{(\omega)}$ of morphisms in ${{\omega}-\mathbf{Gph}}$ and related notions introduced in Section 2.3.

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:

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

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 $F^{(\omega)}_\mathrm{Wk}\colon {{\omega}-\mathbf{Gph}}\longrightarrow{\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}$ 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 ${\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}$ . 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 $\mathbf{Contr} ({\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}},F^{(\omega)}_\mathrm{Wk}\mathcal{J}^{(\omega)})_\mathcal{A}$ .

Proof. This is a special case of Garner (Reference Garner2010), Proposition 2.6.

See Garner (Reference Garner2010), 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 Métayer (Reference Métayer2003), 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 Lafont et al. (Reference Lafont, Métayer and Worytkiewicz2010).

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

• 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)

  

• 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 Bourke and Garner Reference Bourke and Garner2016a, Section 2.8).

  

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

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

Remark 4.6 By definition, a weak $\omega$ -functor $F\colon \mathcal{A}\longrightarrow\mathcal{B}$ gives rise to a span in ${\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}$

whose left leg $\varepsilon_\mathcal{A}$ is a surjective equivalence equipped with the universal contraction $\kappa_\mathcal{A}$ (see also Bourke and Garner Reference Bourke and Garner2016b). Conversely, every span in ${\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}$

(21)

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 (Reference Makkai1996).

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 $U\colon {\mathbf{Wk}-{\omega}-\mathbf{Cat}}\longrightarrow {{({\mathbf{Wk}-{\omega}-\mathbf{Cat}})}-\mathbf{Gph}}$ U

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

(22)

commute.

The key observation is that ${{J}-\mathbf{Gph}}$ exhibits ${{({\mathbf{Wk}-{\omega}-\mathbf{Cat}})}-\mathbf{Gph}}$ as the Kleisli category of the comonad ${{Q}-\mathbf{Gph}}$ on ${{({\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}})}-\mathbf{Gph}}$ . 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 ${{(-)}-\mathbf{Gph}}\colon{\mathscr{C\!A\!T}}\longrightarrow{\mathscr{C\!A\!T}}$ preserves (adjunctions and) bijective-on-objects functors. Hence, ${{(-)}-\mathbf{Gph}}$ 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 (Street Reference Street1972), that is, a suitable natural transformation $\alpha$ as in

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

\[\alpha_\mathcal{A}\colon ({{Q}-\mathbf{Gph}}){U_\mathrm{s}}\mathcal{A}\longrightarrow {U_\mathrm{s}} Q\mathcal{A}\]

in ${{({\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}})}-\mathbf{Gph}}$ . 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.

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.

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 Cheng (Reference Cheng2007), Definition 6) such that $f\in W$ and, for any $f'\colon a'\longrightarrow b'$ in W, there exist cells $g'\colon b'\longrightarrow a',\eta'\colon \mathrm{id}_{a'}\longrightarrow g'\circ f'$ and $\varepsilon'\colon f'\circ g' \longrightarrow \mathrm{id}_{b'}$ 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. van den Berg and Garner Reference van den Berg and Garner2011, 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)).

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 (Cheng Reference Cheng2007; van den Berg and Garner Reference van den Berg and Garner2011).

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 $\mathcal{A}(x,y)$ agree 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 ${\mathbf{Wk}-{\omega}-\mathbf{Gpd}_\mathrm{s}}$ (resp. ${\mathbf{Wk}-{\omega}-\mathbf{Gpd}}$ ) the full subcategory of ${\mathbf{Wk}-{\omega}-\mathbf{Cat}_\mathrm{s}}$ (resp. ${\mathbf{Wk}-{\omega}-\mathbf{Cat}}$ ) consisting of all weak $\omega$ -groupoids, then it follows that the forgetful functor ${U_\mathrm{s}}$ (resp. U) restricts to

\[{\mathbf{Wk}-{\omega}-\mathbf{Gpd}_\mathrm{s}}\longrightarrow {{({\mathbf{Wk}-{\omega}-\mathbf{Gpd}_\mathrm{s}})}-\mathbf{Gph}}\qquad(\text{resp.~}{\mathbf{Wk}-{\omega}-\mathbf{Gpd}}\longrightarrow{{({\mathbf{Wk}-{\omega}-\mathbf{Gpd}})}-\mathbf{Gph}}).\]