@@ -238,8 +238,7 @@ From the previous proposition, we deduce the following very useful corollary.

...

@@ -238,8 +238,7 @@ From the previous proposition, we deduce the following very useful corollary.

L(C)\ar[r,"L(\gamma)"]& L(D)

L(C)\ar[r,"L(\gamma)"]& L(D)

\end{tikzcd}

\end{tikzcd}

\]

\]

is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with the Thomason

is a Thomason \emph{homotopy} cocartesian square of $\Cat$.

equivalences.

\end{corollary}

\end{corollary}

\begin{proof}

\begin{proof}

Since the nerve $N$ induces an equivalence of op-prederivators

Since the nerve $N$ induces an equivalence of op-prederivators

...

@@ -1629,7 +1628,7 @@ Now let $\sS_2$ be labelled as

...

@@ -1629,7 +1628,7 @@ Now let $\sS_2$ be labelled as

In particular, square \eqref{eq:squarebouquethybridvertical} is again a

In particular, square \eqref{eq:squarebouquethybridvertical} is again a

cocartesian square of identification of two objects of a free category, and

cocartesian square of identification of two objects of a free category, and

thus, it is Thomason homotopy cocartesian. This implies that square

thus, it is Thomason homotopy cocartesian. This implies that square

\eqref{eq:squarebouquethybrid} is homotopy cocartesian. Since $\sS_0$, $\sD_1$ and $\sS_2$ are \good{} and since the morphisms $\langle C, D \rangle : \sS_0\to\sS_2$ and $i_1 : \sS_0\to\sD_1$ are folk cofibrations, this proves that $P$ is \good{}

\eqref{eq:squarebouquethybrid} is Thomason homotopy cocartesian. Since $\sS_0$, $\sD_1$ and $\sS_2$ are \good{} and since the morphisms $\langle C, D \rangle : \sS_0\to\sS_2$ and $i_1 : \sS_0\to\sD_1$ are folk cofibrations, this proves that $P$ is \good{}

and has the homotopy type of the bouquet of a $1$\nbd{}sphere with a $2$\nbd{}sphere.

and has the homotopy type of the bouquet of a $1$\nbd{}sphere with a $2$\nbd{}sphere.