Commit 7ccf5e48 authored by Leonard Guetta's avatar Leonard Guetta
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Typos edited. This should be it

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...@@ -591,7 +591,7 @@ Furthermore, this function satisfies the condition ...@@ -591,7 +591,7 @@ Furthermore, this function satisfies the condition
\section{Recursive construction of free \texorpdfstring{$\oo$}{ω}-categories}\label{section:freeoocataspolygraph} \section{Recursive construction of free \texorpdfstring{$\oo$}{ω}-categories}\label{section:freeoocataspolygraph}
\begin{definition}\label{def:cellularextension} \begin{definition}\label{def:cellularextension}
Let $n \in \mathbb{N}$. An \emph{$n$\nbd{}cellular extension} consists of a quadruplet $\E=(C,\Sigma,\sigma,\tau)$ where: Let $n \in \mathbb{N}$. An \emph{$n$\nbd{}cellular extension} is a quadruplet $\E=(C,\Sigma,\sigma,\tau)$ where:
\begin{itemize}[label=-] \begin{itemize}[label=-]
\item $C$ is an $n$\nbd{}category, \item $C$ is an $n$\nbd{}category,
\item $\Sigma$ is a set, whose elements are referred to as the \emph{indeterminates} of $\E$, \item $\Sigma$ is a set, whose elements are referred to as the \emph{indeterminates} of $\E$,
...@@ -859,7 +859,7 @@ Recall that an $n$\nbd{}category is a particular case of $n$\nbd{}magma. ...@@ -859,7 +859,7 @@ Recall that an $n$\nbd{}category is a particular case of $n$\nbd{}magma.
\[ \[
\src((w\fcomp_n w'))=s(w') \text{ and } \trgt((w\fcomp_n w'))=t(w), \src((w\fcomp_n w'))=s(w') \text{ and } \trgt((w\fcomp_n w'))=t(w),
\] \]
\item if $w$ and $w'$ are well formed words such that $\src_k(\src(w))=\trgt_k(\trgt(w))$ for some $0 \leq k < n$, then the word $(w\fcomp_k w')$ is well formed and we have \item if $w$ and $w'$ are well formed words such that $\src_k(\src(w))=\trgt_k(\trgt(w))$ with $0 \leq k < n$, then the word $(w\fcomp_k w')$ is well formed and we have
\[ \[
\src((w \fcomp_k w'))=\src(w)\comp_k s(w') \text{ and } \trgt((w \fcomp_k w'))=\trgt(w) \comp_k \trgt(w'). \src((w \fcomp_k w'))=\src(w)\comp_k s(w') \text{ and } \trgt((w \fcomp_k w'))=\trgt(w) \comp_k \trgt(w').
\] \]
...@@ -2487,7 +2487,8 @@ If $k <n-1$, we need first to use the fact that $f$ is right orthogonal to $\kap ...@@ -2487,7 +2487,8 @@ If $k <n-1$, we need first to use the fact that $f$ is right orthogonal to $\kap
we conclude that $v=u'$. we conclude that $v=u'$.
\end{proof} \end{proof}
\begin{proposition}\label{prop:conduchenbasis} \begin{proposition}\label{prop:conduchenbasis}
Let $F : C \to D$ be an $\omega$-functor, $n \in \mathbb{N}$, $\Sigma^D \subseteq D_{n}$ and $\Sigma^C := F^{-1}(\Sigma^D)$. If $\tau_{\leq n}^s(F)$ is a discrete Conduché $n$\nbd{}functor, then: Let $F : C \to D$ be an $\omega$-functor, $n \in \mathbb{N}$, $\Sigma^D
\subseteq D_{n}$ and define $\Sigma^C := F^{-1}(\Sigma^D)$. If $\tau_{\leq n}^s(F)$ is a discrete Conduché $n$\nbd{}functor, then:
\begin{enumerate} \begin{enumerate}
\item if $\Sigma^D$ is an $n$\nbd{}basis then so is $\Sigma^C$, \item if $\Sigma^D$ is an $n$\nbd{}basis then so is $\Sigma^C$,
\item if $F_{n} : C_{n}\to D_{n}$ is surjective and $\Sigma^C$ is an $n$\nbd{}basis then so is $\Sigma^D$. \item if $F_{n} : C_{n}\to D_{n}$ is surjective and $\Sigma^C$ is an $n$\nbd{}basis then so is $\Sigma^D$.
......
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