@@ -219,7 +219,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

...

@@ -219,7 +219,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\begin{remark}

\begin{remark}

Corollaries \ref{cor:thomhmtpycocomplete} and \ref{cor:thomsaturated} would also follow from the existence of a model structure on $n\Cat$ with $\W^{\Th}_n$ as the weak equivalences. For $n=1$, this was established by Thomason \cite{thomason1980cat}, and for $n=2$ by Ara and Maltsiniotis \cite{ara2014vers}. For $n>3$, the existence of such a model structure is conjectured but not yet established.

Corollaries \ref{cor:thomhmtpycocomplete} and \ref{cor:thomsaturated} would also follow from the existence of a model structure on $n\Cat$ with $\W^{\Th}_n$ as the weak equivalences. For $n=1$, this was established by Thomason \cite{thomason1980cat}, and for $n=2$ by Ara and Maltsiniotis \cite{ara2014vers}. For $n>3$, the existence of such a model structure is conjectured but not yet established.

\end{remark}

\end{remark}

By definition, for all $1\leq n \leq m \leq\omega$, the canonical inclusion \[n\Cat\hookrightarrow m\Cat\] sends Thomason equivalences of $n\Cat$ to Thomason equivalences of $m\Cat$. Hence, it induces a morphism of localizers and then a morphism of op\nbd{}prederivator $\Ho(n\Cat^\Th)\to\Ho(m\Cat^{\Th})$.

By definition, for all $1\leq n \leq m \leq\omega$, the canonical inclusion \[n\Cat\hookrightarrow m\Cat\] sends the Thomason equivalences of $n\Cat$ to Thomason equivalences of $m\Cat$. Hence, it induces a morphism of localizers and then a morphism of op\nbd{}prederivator $\Ho(n\Cat^\Th)\to\Ho(m\Cat^{\Th})$.

\begin{proposition}

\begin{proposition}

For all $1\leq n \leq m \leq\omega$, the canonical morphism

For all $1\leq n \leq m \leq\omega$, the canonical morphism

\[

\[

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@@ -605,12 +605,12 @@ For later reference, we put here the following trivial but important lemma, whos

...

@@ -605,12 +605,12 @@ For later reference, we put here the following trivial but important lemma, whos

\fi

\fi

\section{Equivalences of \texorpdfstring{$\oo$}{ω}-categories vs Thomason equivalences}

\section{Equivalences of \texorpdfstring{$\oo$}{ω}-categories vs Thomason equivalences}

\begin{lemma}\label{lemma:nervehomotopical}

\begin{lemma}\label{lemma:nervehomotopical}

The nerve functor $N_{\omega} : \omega\Cat\to\Psh{\Delta}$ sends equivalences of $\omega$-categories to weak equivalences of simplicial sets.

The nerve functor $N_{\omega} : \omega\Cat\to\Psh{\Delta}$ sends the equivalences of $\omega$-categories to weak equivalences of simplicial sets.

\end{lemma}

\end{lemma}

\begin{proof}

\begin{proof}

Since every $\omega$-category is fibrant for the folk model structure \cite[Proposition 9]{lafont2010folk}, it follows from Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} that it suffices to show that the nerve sends folk trivial fibrations to weak equivalences of simplicial sets. In particular, it suffices to show the stronger condition that the nerve sends folk trivial fibrations to trivial fibrations of simplicial sets.

Since every $\omega$-category is fibrant for the folk model structure \cite[Proposition 9]{lafont2010folk}, it follows from Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} that it suffices to show that the nerve sends folk the trivial fibrations to weak equivalences of simplicial sets. In particular, it suffices to show the stronger condition that the nerve sends the folk trivial fibrations to trivial fibrations of simplicial sets.

By adjunction, this is equivalent to showing that the functor $c_{\omega} : \Psh{\Delta}\to\omega\Cat$ sends cofibrations of simplicial sets to folk cofibrations. Since $c_{\omega}$ is cocontinuous and the cofibrations of simplicial sets are generated by the inclusions

By adjunction, this is equivalent to showing that the functor $c_{\omega} : \Psh{\Delta}\to\omega\Cat$ sends the cofibrations of simplicial sets to folk cofibrations. Since $c_{\omega}$ is cocontinuous and the cofibrations of simplicial sets are generated by the inclusions