@@ -576,71 +576,84 @@ Furthermore, this function satisfies the condition

\[

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

\]

\end{definition}

\end{definition}

\begin{paragr}

We denote by $n\Cat^{+}$ the category of $n$-cellular extensions and morphisms of $n$-cellular extensions. Every $(n+1)$-category $C$ canonically defines an $n$-cellular extension $(\tau^s_{\leq n }(C),C_{n+1},\src,\trgt)$ where $\src,\trgt : C_{n+1}\to C_n$ 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},\src,\trgt).

\end{align*}

\end{paragr}

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

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

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:

\[

On the other hand, any $\E=(D,\Sigma,\sigma,\tau)$, cellular extension of an $n$-category $D$, yields an $(n+1)$-category $\E^*$ defined as the following amalgamated sum:

\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

\end{equation}

This actually defines a functor

\begin{align*}

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

\E&\mapsto\E^*,

\end{align*}

which is easily checked to be left adjoint to $U_n$.

Now let $\phi : \coprod_{x \in\Sigma}\sD_{n}\to\E^*$ the bottom map of square \eqref{squarefreecext}. It induces a canonical 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}). Notice that this map is natural in that, for any morphism of $n$-cellular extensions $G=(F,\varphi) : \E\to\E'$, the square

is commutative. Notice also that $j$ is compatible with source and targets in the sense that for every $x \in\Sigma$, we have

\[

\src(j(x))=\sigma(x)\text{ and }\trgt(j(x))=\tau(x).

\]

\end{paragr}

\begin{lemma}\label{lemma:basisfreecext}

Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension. The canonical map

\[

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

\]

is injective.

\end{lemma}

\begin{proof}

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 $\sk_n(\E^*)\to\sk_{n+1}(\E^*)=\E^*$.

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 $\E^*$ as the amalgamated sum \eqref{squarefreecext} 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.

\end{proof}

\begin{paragr}

In particular, the previous lemma tells us that for any $n$-cellular extension $\E=(C,\Sigma,\sigma,\tau)$, the set of indeterminates $\Sigma$ canonically defines a subset of the $(n+1)$-cells of $\E^*$. As of now, we will consider $\Sigma$ as a subset of $(\E^{*})_{n+1}$.

\end{paragr}

We can now prove the following proposition, which is the key result of this section. It is slightly less trivial than it appears.

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

For any $n$-cellular extension $\E=(C,\Sigma,\sigma,\tau)$, the subset $\Sigma\subseteq(\E^{*})_{n+1}$ is an $(n+1)$-basis of $\E^*$.

\end{proposition}

\begin{proof}

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 square \eqref{squarefreecext} and 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

where $e_{n+1}$ is the principal $(n+1)$-cell of $\sD_{n+1}$ (\ref{paragr:defglobe}). Hence, cocartesian square \eqref{squarefreecext} can be identified with

\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 $\E^*$.

Since from $j : \Sigma\to(\E^{*})_{n+1}$ is injective, we can consider $\Sigma$ as a subset of $(\E^{*})_{n+1})$, and then, by definition, $\E^*$ has $\Sigma$ as an $(n+1)$-basis.

\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

...

...

@@ -733,7 +746,7 @@ The following proposition is the key result of this section. It is slightly less

The previous proposition admits an obvious truncated version for free $n$-categories with $n$ finite. In that case, we only need a finite sequence $(\E^{(k)}))_{-1\leq k \leq n-1}$ of cellular extensions.

\end{remark}

\begin{remark}

The data of a sequence $(\E^{(n)})_{n \geq-1}$ as in Proposition \ref{prop:freeonpolygraph} is commonly referred to in the litterature of the filed as a \emph{computad}\cite{street1976limits} or \emph{polygraph}\cite{burroni1993higher}; and consequently a $\oo$-category which is free in the sense of definition \ref{def:freeoocat} is sometimes referred to as \emph{free on a computad}-or-\emph{polygraph} in the litterature. However, the underlying polygraph of a free $\oo$-category is uniquely determined by the free $\oo$-category itself (a straightforward consequence of Proposition \ref{prop:uniquebasis}), and this is why we chose the shorter terminology \emph{free $\oo$-category}.

The data of a sequence $(\E^{(n)})_{n \geq-1}$ as in Proposition \ref{prop:freeonpolygraph} is commonly referred to in the litterature of the field as a \emph{computad}\cite{street1976limits} or \emph{polygraph}\cite{burroni1993higher}; and consequently a $\oo$-category which is free in the sense of definition \ref{def:freeoocat} is sometimes referred to as \emph{free on a computad}-or-\emph{polygraph} in the litterature. However, the underlying polygraph of a free $\oo$-category is uniquely determined by the free $\oo$-category itself (a straightforward consequence of Proposition \ref{prop:uniquebasis}), and this is why we chose the shorter terminology \emph{free $\oo$-category}.

\end{remark}

\begin{paragr}

Concretely, Proposition \ref{prop:freeonpolygraph} gives us a recipe to construct free $\oo$-categories. It suffices to give a formal list of generating cells of the form:

...

...

@@ -859,7 +872,7 @@ In the following definition, we consider that a binary relation $\R$ on a set $E

\]

for $x$ and $y$$k$-composable $n$-cells of $X$. Altoghether, this defines an $n$-magma, which we denote by $X/{\R}$, whose set $k$-cells is $X_k$ for $0\leq k < n$, and $X_n/{\R}$ for $k=n$. The composition, source, target and units of cells of dimension strictly lower than $n$ being those of $X$ and the composition, source and target of $n$-cells being given by the above formulas.

Let $\R$ be a congruence on an $n$-magma $X$ with $n \geq1$. We say that $\R$ is \emph{categorical} if it satisfies all four following axioms:

\begin{enumerate}

\item for every $k<n$ and every $n$-cell $x$ of $X$, we have

...

...

@@ -885,16 +898,55 @@ In the following definition, we consider that a binary relation $\R$ on a set $E

\]

\end{enumerate}

\end{definition}

\begin{example} Let $C$ be an $n$-category with $n>1$, which we consider as an $n$-magma. The equality on the set of $n$-cells of $C$, is, by definition, categorical.

\begin{example} Let $C$ be an $n$-category with $n>1$, which we consider as an $n$-magma. The equality on the set of $n$-cells of $C$ is, by definition, categorical.

\end{example}

\begin{example}

Another important example of categorical congruence is the following. Let $F : X \to Y$ be a morphism of $n$-magmas with $n>1$ and suppose that $Y$ is an $n$-category. Then the binary relation $\R$ on $X_n$ defined as

Another important example is the following. Let $F : X \to Y$ be a morphism of $n$-magmas with $n>1$ and suppose that $Y$ is an $n$-category. Then the binary relation $\R$ on $X_n$ defined as

\[

x \R y \text{ if } F(x)=F(y)

x\;\R\;y \text{ if } F(x)=F(y)

\]

is obviously a \emph{categorical} congruence.

\end{example}

In the following lemma, we use the notation $\tau_{\leq n}^s(X)$ for an $n$-mag

Similarly to the ``stupid'' truncation of $\oo$-categories (\ref{paragr:defncat}), for an $(n+1)$-magma $X$, we write $\tau_{\leq n}^s(X)$ for the $n$-magma obtained by simply forgetting the cells of dimension $(n+1)$.

The following lemma is trivial but nonetheless important. Its immediate proof is omitted.

\begin{lemma}

Let $X$ be an $n$-magma with $n>1$ and c

Let $X$ be an $n$-magma with $n>1$ and $\R$ a congruence on $X$. If $\tau_{\leq n}^s(X)$ is an $(n-1)$-category and $\R$ is categorical, then $X/{\R}$ is an $n$-category.

\end{lemma}

We wish now to see on which conditions there exists a ``smallest'' congruence on an $n$-magma that contains a given binary relation $\R$ on the set of $n$-cells.

\begin{lemma}\label{lemma:intersectioncongruence}

Let $X$ be an $n$-magma with $n \geq1$ and $(\R_i)_{i \in I}$ a \emph{non-empty} family of congruences on $X$ (i.e.\ $I$ is not empty). Then, the binary relation

\[

\R:=\bigcap_{i \in I}\R_i

\]

is a congruence.

\end{lemma}

\begin{proof}

The fact that $\R$ satisfies that first and third axiom of Definition \ref{def:congruence} is immediate and do not even require that $I$ be non empty. The only left thing to prove is that if $x \;\R\; x'$, then $x$ and $x'$ are parallel. To see that, notice that since $I$ is not empty, we can choose $i \in I$. Then, by definition, we have $x \;\R_i \; x'$ and thus, $x$ and $x'$ are parallel.

\end{proof}

\begin{remark}

The hypothesis that $I$ be non empty in the previous lemma cannot be ommited because, in this case, we would have

\[

\R=\bigcap_{\emptyset}\R_i = X_n\times X_n,

\]

which means that \emph{all}$n$-cells would be equivalent under $\R$. This certainly does not guarantee that the second axiom of Definition \ref{def:congruence}, i.e.\ that equivalent cells are parallel, is satisfied.

\end{remark}

\begin{lemma}\label{lemma:congruencegenerated}

Let $X$ be an $n$-magma with $n\geq1$ and $E$ a set of pairs of parallel $n$-cells of $X$. There exists a smallest congruence $\R$ on $X$ such that for every $(x,y)\in E$, we have $x \;\R\;y$.

\end{lemma}

\begin{proof}

Let $I$ be the set of congruence $\mathcal{S}$ on $X$ such that for every $(x,y)\in E$, we have $x\;\mathcal{S}\; y$. All we have to prove is that $I$ is not empty, since in that case, we can apply Lemma \ref{lemma:intersectioncongruence} to the binary relation

\[

\R:=\bigcup_{\mathcal{S}\in I}\S,

\]

which will obviously be the smallest congruence satisfying the desired condition. To see that $I$ is not empty, it suffices to notice that the binary relation ``being parallel $n$-cells'' is a congruence, which obviously is in $I$.

\end{proof}

\begin{proposition}

Let $X$ be an $n$-magma with $n\geq1$. There exists a smallest categorical congruence on $X$.

\end{proposition}

\begin{proof}

Each four axioms of Definition \ref{def:categoricalcongruence} says that some pairs of parallel $n$-cells must be equivalent under a congruence $\R$ for it to be categorical. The result follows then from Lemma \ref{lemma:congruencegenerated}.

\end{proof}

\begin{paragr}

Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension