@@ -166,24 +166,27 @@ From these two lemmas, follows the important proposition below.

For every $n \geq-1$, the $\oo$\nbd{}category $\sS_n$ is \good{}.

\end{proposition}

\begin{proof}

We proceed by induction on $n$. When $n=-1$, it is trivial to check that the

empty $\oo$\nbd{}category is \good{}. Now, since $i_n : \sS_n \to\sD_{n+1}$ is

a folk cofibration and $\sS_{n}$ and $\sD_{n}$ are folk cofibrant, it follows

from Lemma \ref{lemma:hmtpycocartesianreedy} that the cocartesian square

\begin{equation}\label{square}

%% We proceed by induction on $n$. When $n=-1$, it is trivial to check that the

%% empty $\oo$\nbd{}category is \good{}.

Recall that the cofibrations of simplicial sets are exactly the monomorphisms, and in particular all simplicial sets are cofibrant. Since $i_n : \sS_{n-1}\to\sD_n$ is a monomorphism for every $n \geq0$ and since $N_{\oo}$ preserves monomorphisms (as a right adjoint), it follows from Lemma \ref{lemma:squarenerve} and Lemma \ref{lemma:hmtpycocartesianreedy} that the square

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

\end{tikzcd}

\]

is Thomason homotopy cocartesian for every $n\geq0$. Finally, since $i_n : \sS_{n-1}\to\sD_{n}$ is

a folk cofibration and $\sS_{n-1}$ and $\sD_{n}$ are folk cofibrant for every $n\geq0$, we deduce the result from Corollary \ref{cor:usefulcriterion} and an immediate induction. The base case being simply that $\sS_{-1}=\emptyset$ is obviously \good{}.

\end{proof}

\begin{paragr}

The previous proposition implies what we claimed in Paragraph \ref{paragr:prelimcriteriongoodcat}, which is that the morphism of op-prederivators

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@@ -207,7 +210,7 @@ From these two lemmas, follows the important proposition below.

$i_2$ is a cofibration for the canonical model structure, the square is also

homotopy cocartesian with respect to folk weak equivalences. If $\J$ was

homotopy cocontinuous, then this square would also be homotopy cocartesian

with respect to Thomason equivalences. Since we know that $\sS_A$, $\sD_0$ and $\sD_2$ are \good{}, this would imply that $B^2\mathbb{N}$ is \good{}.

with respect to Thomason equivalences. Since we know that $\sS_1$, $\sD_0$ and $\sD_2$ are \good{}, this would imply that $B^2\mathbb{N}$ is \good{}.

From Proposition \ref{prop:spheresaregood}, we also deduce the proposition

below which gives a criterion to detect \good{}$\oo$\nbd{}category when we

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@@ -230,7 +233,7 @@ From these two lemmas, follows the important proposition below.

\end{proposition}

\begin{proof}

Since the morphisms $i_k$ are folk cofibration and the $\oo$\nbd{}categories

$\sS^{k-1}$ and $\sS^{k}$ are folk cofibrant and \good{}, it follows

$\sS_{k-1}$ and $\sD_{k}$ are folk cofibrant and \good{}, it follows

from Corollary \ref{cor:usefulcriterion} that all $\sk_k(C)$ are \good{}. The

result follows then from Lemma \ref{lemma:filtration} and Proposition \ref{prop:sequentialhmtpycolimit}.