@@ -205,7 +205,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo

\]

is $C$ and the universal arrow $\sk_{n}(C)\to C$ is given by the co-unit of the adjunction $\tau^{s}_{\leq n}\dashv\iota$.

\end{lemma}

\begin{paragr}

\begin{paragr}\label{paragr:defglobe}

For $n \in\mathbb{N}$, the \emph{$n$-globe}$\sD_n$ is the $n$-category that has:

\begin{itemize}

\item exactly one non-trivial $n$-cell, which we refer to as the \emph{principal $n$-cell} of $\sD_n$, and which we denote by $e_n$,

...

...

@@ -305,8 +305,11 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo

\]

is cocartesian.

\end{definition}

\begin{remark}

Note that since for all $n<m$, we have $\sk_n \circ\sk_m =\sk_n$, an $\oo$-category $C$ has an $n$-basis if and only if $\sk_n(C)$ has an $n$-basis.

\end{remark}

\begin{paragr}\label{paragr:defnbasisdetailed}

Unfolding the previous definition gives that $E$ is an $n$-basis of $C$ if for every $(n-1)$-category $D$, every $(n-1)$-functor

Unfolding Definition \ref{def:nbasis} gives that $E$ is an $n$-basis of $C$ if for every $(n-1)$-category $D$, every $(n-1)$-functor

\[

F : \tau_{\leq n-1}^{s}(C)\to\tau_{\leq n-1}^{s}(D),

\]

...

...

@@ -326,6 +329,10 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo

Intuitively speaking, this means that $\sk_{n}(C)$ has been obtained from $\sk_{n-1}(C)$ by freeling adjoining the cells in $E$.

\end{paragr}

\begin{example}\label{zerobase}

Every $\oo$-category $C$ has a $0$-base which is $C_0$ itself. This is the unique $0$-base possible.

\end{example}

\begin{example}\label{dummyexample}

An $n$-category (seen as an $\oo$-category) always has a $k$-basis for every $k>n$, namely the empty set. Less trivial examples will come along soon.

\end{example}

...

...

@@ -375,7 +382,7 @@ We can now state the promised result, whose proof can be found in \cite[Section

We often say refer to the elements of the $n$-basis of a free $\oo$-category as the \emph{generating $n$-cells}. This sometimes leads to use the alternative terminology \emph{set of generating $n$-cells} instead of \emph{$n$-basis}.

\end{paragr}

So far, we have not yet seen examples of free $\oo$-categories. In order to do that, we will explain in a further section a recursive way of constructing free $\oo$-categories but first we take a little detour.

So far, we have not yet seen examples of free $\oo$-categories. In order to do that, we will explain in a further section a recursive way of constructing free $\oo$-categories; but first we take a little detour.

\section{Suspension of monoids and counting generators}

\begin{paragr}\label{paragr:suspmonoid}

...

...

@@ -436,10 +443,10 @@ This construction will turn out to be of great use many times in this dissertati

is bijective.

\end{lemma}

\begin{proof}

This is an immediate consequence of Lemma \ref{lemma:nfunctortomonoid} and the universal property of an $n$-basis (as explained in Paragraph \ref{paragr:defnbasisdetailed})

This is an immediate consequence of the universal property of an $n$-basis (as explained in Paragraph \ref{paragr:defnbasisdetailed})

\end{proof}

We can now prove the important proposition below.

\begin{proposition}

\begin{proposition}\label{prop:countingfunction}

Let $C$ be an $\oo$-category, and suppose that $C$ has a $n$-basis $E$ with $n\geq0$. For every $\alpha\in E$, there exists a unique function

\[

w_{\alpha} : C_n \to\mathbb{N}

...

...

@@ -474,13 +481,142 @@ Furthermore, this function satisfies the condition

\[

w_{\alpha}(1_x)=0.

\]

Hence, from Lemma \ref{lemma:nfunctortomonoid} we know that $w_{\alpha}$ necessarily comes from an $n$\nbd-functor $C \to B^n\mathbb{N}$ and the uniqueness follows then from conditions (a) and (b) and Lemma \ref{lemma:freencattomonoid}.

Hence, we can apply Lemma \ref{lemma:nfunctortomonoid} which shows that $w_{\alpha}$ necessarily comes from an $n$\nbd-functor $C \to B^n\mathbb{N}$. Then, the uniqueness follows from conditions (a) and (b) and Lemma \ref{lemma:freencattomonoid}.

\end{proof}

\begin{paragr}

Let $C$ be as in the previous corollary and let $x$ be an $n$-cell of $C$. We sometimes refer to $w_{\alpha}(x)$ as the \emph{weight of $\alpha$ in $x$}. Intuitively speaking, the function $w_{\alpha}$ counts how many times $\alpha$ ``appears'' in $x$. We shall give in a later section an more solid explanation for this intuition.

Let $C$ be an $n$-category with an $n$-basis $E$. For an $n$-cell $x$ of $C$, we refer to the integer $w_{\alpha}(x)$ as the \emph{weight of $\alpha$ in $x$}. The reason for such a name will become clearer later once we give an explicit construction of $w_{\alpha}$ as a function that ``counts the number of occurences of $\alpha$ in an $n$-cell''.

For later reference, let us also highlight the fact that in the proof of the previous proposition, we have shown the important property that if $n>0$, then for $y \in C_{n-1}$, we have

\[

w_{\alpha}(1_y)=0.

\]

This implies that for $n>1$, there might be $n$-cells $x$ such that

\[

x \neq\alpha\text{ and }w_{\alpha}(x)=1.

\]

Indeed, suppose that there exists a $k$-cell $z$ with $0<k<n-1$ which is not a unit on a lower dimensional cell and such that $\trgt_{k-1}(z)=\src_{k-1}(\alpha)$, then we have

\section{Recursive construction of free $\oo$-categories}

\begin{definition}\label{def:cellularextension}

Let $n \in\mathbb{N}$. A \emph{$n$-cellular extension} consists of a quadruplet $(C,\Sigma,\sigma,\tau)$ where:

\begin{itemize}[label=-]

\item$C$ is an $n$-category,

\item$\Sigma$ is a set,

\item$\sigma$ and $\tau$ are maps $\Sigma\to C_n$ such that for every element $x \in\Sigma$, the $n$-cells $\sigma(x)$ and $\tau(x)$ are parallel.

\end{itemize}

\end{definition}

\begin{paragr}

If we are given an $n$-category $C$, then we also say \emph{a cellular extension of $C$} for a cellular extension of the form $(C,\Sigma,\sigma,\tau)$.

Note also that, while not explicitely

Intuitively speaking, the set $\Sigma$ is a set of formal extra $(n+1)$-cells of $C$. The source and target of every $x \in\Sigma$ are given respectively by $\sigma(x)$ and $\tau(x)$ (which makes sense since these two $n$-cells are parallel). This is why we often adopt the notation

\[

x : a \to b

\]

for an element of $\Sigma$ such that $\sigma(x)=a$ and $\tau(x)=b$.

\end{paragr}

\begin{definition}

Let $\E=(C,\Sigma,\sigma,\tau)$ and $\E'=(C',\Sigma',\sigma',\tau')$ be two $n$-cellular extensions. A morphism of $n$-cellular extensions $\E\to\E'$ consists of a pair $(F,\varphi)$ where:

\begin{itemize}[label=-]

\item F is an $n$-functor $C \to C'$,

\item$\varphi$ is a map $\Sigma\to\Sigma'$,

\end{itemize}

such that for every $x \in\Sigma$, we have

\[

\sigma'(\varphi(x))=F(\sigma(x))\text{ and }\tau'(\varphi(x))=F(\tau(x)).

\]

\end{definition}

\begin{paragr}

We denote by $n\Cat^{+}$ the category of $n$-cellular extensions and morphisms of $n$-cellular extensions. Note that every $(n+1)$-category $C$ canonically defines an $n$-cellular extension $(\tau^s_{\leq n }(C),C_{n+1},\src,\tau)$ where $\src,\tau : C_{n+1}\to C_n$ are the source and target maps of $C$. This defines a functor

\begin{align*}

U_n : (n+1)\Cat&\to n\Cat^+\\

C &\mapsto (\tau^s_{\leq n }(C),C_{n+1},s,t).

\end{align*}

\end{paragr}

The following proposition is the key result of this section. It is slightly less trivial than it appears.

\begin{proposition}

Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension and let $\E^*$ be the $(n+1)$-category defined as the following amalgamated sum:

\[

\begin{tikzcd}[column sep=huge, row sep=huge]

\displaystyle\coprod_{x \in\Sigma}\sS^n \ar[d,"\displaystyle\coprod_{x \in\Sigma}i_{n+1}"']\ar[r,"{\langle\sigma(x),\tau(x)\rangle_{x \in\Sigma}}"]& C \ar[d]\\

\ar[from=1-1,to=2-2,very near end,phantom,"\ulcorner"]

\end{tikzcd}.

\]

Then, $\E^*$ has an $(n+1)$-basis which is isomorphic to $\Sigma$. Moreover, the identification of $\Sigma$ as a subset of $(\E^*)_{n+1}$ is natural in that, for any morphism of $n$-cellular extensions $G=(F,\varphi) : \E\to\E'$, the induced $(n+1)$-functor $G^* : \E^*\to\E^*$ is such that the square

Notice first that since the map $i_{n+1} : \sS^n \to\sD_{n+1}$ is nothing but the canonical inclusion $\sk_{n}(\sD_{n+1})\to\sk_{n+1}(\sD_{n+1})=\sD_{n+1}$, it follows easily from the fact the the skeleton functors preserve colimits that $C$ is canonically isomorphic to $\sk_n(\E^*)$ and that the map $C \to\E^*$ can be identified with the canonical inclusion $\sk_n(\E^*)\to\sk_{n+1}(\E^*)=\E^*$.

Now let $\phi : \coprod_{x \in\Sigma}\sD_{n}\to\E^*$ the bottom map of the cocartesian square of the proposition and consider the map

\begin{align*}

j: \Sigma&\to (\E^*)_{n+1}\\

x &\mapsto\phi_x(e_{n+1}),

\end{align*}

where $e_{n+1}$ is the principal $(n+1)$-cell of $\sD_{n+1}$ (\ref{paragr:defglobe}).

The only subtlety of the proposition is to show that $j$ is injective and hence, that $\Sigma$ can be identified to a subset of $(n+1)$-cells of $E^*$. Now a thorough reading of the techniques used in the proofs of Lemma \ref{lemma:nfunctortomonoid}, Lemma \ref{lemma:freencattomonoid} and Proposition \ref{prop:countingfunction} shows that the universal property defining $\E^*$ is sufficient enough to prove the existence, for each $x \in\Sigma$, of a function

\[

w_{x} : (\E^*)_{n+1}\to\mathbb{N}

\]

such that $w_{x}(j(x))=1$ and $w_{x}(j(y))=0$ for any $y \in\Sigma$ with $y\neq x$. In particular, this implies that $j$ is injective.

Altogether, we have shown that there exists a subset of $(\E^*)_{n+1}$ (which we identify with $\Sigma$ via the map $j$) such that the square

\ar[from=1-1,to=2-2,very near end,phantom,"\ulcorner"]

\end{tikzcd}

\]

is cocartesian, which proves the first part of the proposition.

The second part, concerning the naturality of the identification of $\Sigma$ as a subset of $(\E^*)_{n+1}$ is clear from the definition of $j$.

\end{proof}

\begin{corollary}

Let $n \geq0$. The functor

\begin{align*}

n\Cat^+ &\to (n+1)\Cat\\

\E&\mapsto\E^*

\end{align*}

is left adjoint to the functor $U_n : (n+1)\Cat\to n\Cat^+$.

\end{corollary}

\begin{proof}

This is simply a reformulation of the universal property defining $\E^*$.

\end{proof}

\begin{paragr}

Let $C$ be an $(n+1)$-category and $E$ be a subset $E \subseteq C_{n+1}$. This defines a cellular extension

\[

\E_E =(\tau_{\leq n}^s(C),E,\src,\trgt),

\]

where $\src$ and $\trgt$ are simply the restriction to $E$ of the source and target maps $C_{n+1}\to C_n$. The canonical inclusion $E \hookrightarrow C_{n+1}$ induces a morphism of $n$-cellular extensions

\[

\E_E \to U_n(C),

\]

and then, by adjunction, an $(n+1)$-functor

\[

\E_E^*\to C.

\]

\end{paragr}

\begin{proposition}

Let $C$ be an $(n+1)$-category. A subset $E \subseteq C_{n+1}$ is an $(n+1)$-basis of $C$ if and only if the canonical $(n+1)$-functor

\[

\E_E^*\to C

\]

is an isomorphism.

\end{proposition}

\begin{example}

Recall that for a graph $G$ (or $1$-graph in the terminology of \ref{paragr:defncat}), the free category on $G$ is the category whose objects are those of $G$ and whose arrows are strings of composable arrows of $G$; the composition being given by concatenation of strings.

...

...

@@ -498,4 +634,3 @@ Furthermore, this function satisfies the condition

\]

is not free on a $2$-graph. The reason is that the source (resp. the target) of $\alpha$ is $g \comp_0 f$ (resp. $i \comp_0 h$) which are not generating $1$-cells.