@@ -577,7 +577,7 @@ Furthermore, this function satisfies the condition

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

\]

\end{definition}

\begin{paragr}

\begin{paragr}\label{paragr:freecext}

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^+\\

...

...

@@ -630,7 +630,7 @@ Furthermore, this function satisfies the condition

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 $(\E^{*})_{n+1}$.

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 $(\E^{*})_{n+1}$ and $j$ as the canonical inclusion.

\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}

...

...

@@ -638,14 +638,7 @@ We can now prove the following proposition, which is the key result of this sect

\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}). Hence, cocartesian square \eqref{squarefreecext} can be identified with

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 $\sk_n(\E^*)\to\sk_{n+1}(\E^*)=\E^*$. Hence, cocartesian square \eqref{squarefreecext} can be identified with

@@ -653,7 +646,7 @@ We can now prove the following proposition, which is the key result of this sect

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

\end{tikzcd}

\]

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.

Since we have identified$\Sigma$to a subset of the $n$-cells of $\E^*$ via $j$, the above cocartesian square means exactly that$\Sigma$is an $(n+1)$-basis of $\E^*$.

\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

...

...

@@ -677,7 +670,7 @@ We can now prove the following proposition, which is the key result of this sect

is an isomorphism.

\end{proposition}

\begin{proof}

It is clear that the canonical $(n+1)$-functor $\E^*_E \to C$ sends $E$, seen as a subset of $(\E^*_E)_{n+1}$, to $E$, seen as a subset of $C_{n+1}$. It follows then from Proposition \ref{prop:fromcexttocat} that if this $(n+1)$-functor is an isomorphism, then $E$ is an $(n+1)$-base of $C$.

It is clear that the canonical $(n+1)$-functor $\E^*_E \to C$ sends $E$, seen as a subset of $(\E^*_E)_{n+1}$, to $E$, seen as a subset of $C_{n+1}$. Hence, it follows from Proposition \ref{prop:fromcexttocat} that if this $(n+1)$-functor is an isomorphism, then $E$ is an $(n+1)$-base of $C$.

Conversely, if $E$ is an $(n+1)$-base of $C$, then we can define an $(n+1)$-functor $C \to\E_E^*$ that sends $E$, seen as a subset of $C_{n+1}$, to $E$, seen as a subset of $(\E^*_E)_{n+1}$ (and which is obviously the identity on cells of dimension strictly lower than $n+1$). Then, the fact that $C$ and $\E^*$ have $E$ as an $(n+1)$-base implies that this $(n+1)$-functor $C \to\E^*$ is the inverse of the canonical one $\E^*\to C$.

\end{proof}

...

...

@@ -728,8 +721,8 @@ We can now prove the following proposition, which is the key result of this sect

\]

for every $k \geq0$. Altogether, this proves that $C$ is free and its $k$-basis is the set of indeterminates of $\E^{k-1}$ for every $k \geq0$.

The fact that a sequence of morphism of cellular extension that satisfy the hypothesis given in the statement of the proposition induces a rigid $\oo$-functor is proven in a similar fashion using, this time, the second part of Proposition \ref{prop:fromcexttocat}.

The fact that a sequence of morphism of cellular extension that satisfy the hypothesis given in the statement of the proposition induces a rigid $\oo$-functor is proven in a similar fashion.

For the converse part, notice that a free $\oo$-category $C$, whose basis is denoted by $(\Sigma_k)_{k \in\mathbb{N}}$, induces a sequence of cellular extensions:

\[

\E_C^{(-1)}:=\Sigma_0

...

...

@@ -746,7 +739,7 @@ We can now prove the following proposition, which is the key result of this sect

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 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}.

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}; 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:

...

...

@@ -844,7 +837,7 @@ In the following definition, we consider that a binary relation $\R$ on a set $E

\begin{enumerate}[label=(\alph*)]

\item$\R$ is an equivalence relation,

\item if $x\;\R\; x'$ then $x$ and $x'$ are parallel,

\item if $x \;\R\;x'$, $y\;\R\; y'$ and if $x$ and $x'$ are $k$-composable for some $0\leq k <n$ then

\item if $x \;\R\;y$ and $x'\;\R\; y'$, and if $x$ and $x'$ are $k$-composable for some $0\leq k <n$ then

\[

x \comp_k x' \;\R\; y\comp_k y'

\]

...

...

@@ -857,7 +850,7 @@ In the following definition, we consider that a binary relation $\R$ on a set $E

\begin{example}

Let $F : X \to Y$ be a morphism of $n$-magmas with $n>1$. 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 a congruence.

\end{example}

...

...

@@ -913,7 +906,7 @@ The following lemma is trivial but nonetheless important. Its immediate proof is

\begin{lemma}

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.

We wish now to see how to prove the existence of a congruence defined with a condition such as ``the smallest congruence that contains a given binary relation on the $(n+1)$-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

\[

...

...

@@ -937,7 +930,7 @@ We wish now to see on which conditions there exists a ``smallest'' congruence on

\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,

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

...

...

@@ -948,5 +941,5 @@ We wish now to see on which conditions there exists a ``smallest'' congruence on

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

Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension and consider the injective map $j : \Sigma\to(\E^*)_{n+1}$ constructed in Paragraph \ref{paragr:freecext}. By induction, we define a map