@@ -526,8 +526,10 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\begin{remark}
All the results we have seen in this section are still true if we replace ``oplax'' by ``lax'' everywhere.
\end{remark}
\section{Equivalences of \texorpdfstring{$\oo$}{ω}-categories and the folk model
structure}
\section[Equivalences of \texorpdfstring{$\oo$}{ω}-categories and the folk model
structure]{Equivalences of \texorpdfstring{$\oo$}{ω}-categories and the folk model
structure%
\sectionmark{The folk model structure}}
\sectionmark{The folk model structure}
\begin{paragr}\label{paragr:ooequivalence}
Let $C$ be an $\omega$-category. We define the equivalence relation $\sim_{\omega}$ on the set $C_n$ by co-induction on $n \in\mathbb{N}$. For $x, y \in C_n$, we have $x \sim_{\omega} y $ when:
...
...
@@ -639,7 +641,11 @@ For later reference, we put here the following trivial but important lemma, whos
See \cite[Proposition 5.1.2.7]{lucas2017cubical} or \cite{ara2019folk}.
\end{proof}
\fi
\section{Equivalences of \texorpdfstring{$\oo$}{ω}-categories vs Thomason equivalences}
\section[Equivalences of \texorpdfstring{$\oo$}{ω}-categories vs Thomason
equivalences ]{Equivalences of \texorpdfstring{$\oo$}{ω}-categories vs Thomason
equivalences%
\sectionmark{Folk vs Thomason}}
\sectionmark{Folk vs Thomason}
\begin{lemma}\label{lemma:nervehomotopical}
The nerve functor $N_{\omega} : \omega\Cat\to\Psh{\Delta}$ sends the equivalences of $\oo$\nbd{}categories to weak equivalences of simplicial sets.
\end{lemma}
...
...
@@ -686,7 +692,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen
%% is the identity on objects.
This functor cannot be an equivalence since this would imply that every Thomason equivalence is an equivalence of $\oo$\nbd{}categories.
\end{paragr}
\section{Slice \texorpdfstring{$\oo$}{ω}-categories and a folk TheoremA}
\section{{Slice \texorpdfstring{$\oo$}{ω}-categories and folk Theorem~A}}
\begin{paragr}\label{paragr:slices}
Let $A$ be an $\oo$\nbd{}category and $a_0$ an object of $A$. We define the slice $\oo$\nbd{}category $A/a_0$ as the following fibred product:
