@@ -297,7 +297,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
\]
where the morphism $\sS_1\to\sD_1$ is the one that sends the two generating arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square is homotopy cocartesian in $\Cat$ (when equipped with Thomason equivalences). Indeed, it is the image by the functor $L$ of a cocartesian square in $\Rgrph$, the morphism $\sS_1\to\sD_1$ is injective on objects and the morphism $\sS_1\to C$ is quasi-injective on arrows. Hence, we can apply Proposition \ref{prop:hmtpysquaregraphbetter}. Note that since we did \emph{not} suppose that $A\neq B$, the top morphism of the previous square is not necessarily a monomorphism and we cannot always apply Corollary \ref{cor:hmtpysquaregraph}.
\end{example}
\begin{example}[Killing a generator]
\begin{example}[Killing a generator]\label{example:killinggenerator}
Let $C$ be a free category and let $f : A \to B$ one of its generating arrow such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by ``killing'' $f$, i.e. defined with the following cocartesian square:
\[
\begin{tikzcd}
...
...
@@ -688,7 +688,7 @@ From this lemma, we deduce two useful criteria to detect Thomason equivalences o
\begin{corollary}\label{cor:criterionThomeqI}
Let $F : C \to D$ be a $2$-functor. If
\begin{enumerate}[label=\alph*)]
\item$F_0 : C_0\to D_0$ is an bijection,
\item$F_0 : C_0\to D_0$ is a bijection,
\end{enumerate}
and
\begin{enumerate}[resume*]
...
...
@@ -898,7 +898,7 @@ For any $n \geq 0$, consider the following cocartesian square
\[
\begin{tikzcd}[column sep=huge]
A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend right=75,"g"',""{name=B,above}]\ar[r,"1_A" pos=1/3,""{name=C,above},""{name=D,below}]& A.
and let $F : P \to P'$ be the unique $2$\nbd{}functor such that
\begin{itemize}[item=-]
\begin{itemize}[label=-]
\item$F(A)=A'$ and $F(B')=B$,
\item$F(f)=F(g)=h$,
\item$F(\alpha)=\gamma$ and $F(\beta)=1_h$.
\end{itemize}
We wish to prove that this $2$\nbd{}functor is a Thomason equivalence. Since it is an isomorphism on objects, it suffices to prove that the functor induced by $F$
\[
F_{A,B} : P(A,B)\to P'(F(A),F(B))
\]
is a Thomason equivalence of categories (Corollary \ref{cor:criterionThomeqI}). The category $P(A,B)$ is the free category on the graph
\[
\begin{tikzcd}
f \ar[r,shift left,"\alpha"]& g \ar[l,shift left,"\beta"]
\end{tikzcd}
\]
($2$\nbd{}cells of $P$ become $1$\nbd{}cells of $P(A,B)$ and $1$\nbd{}cells of $P$ become $0$\nbd{}cells of $P(A,B)$) and the category $P'(F(A),F(B))=P'(A',B')$ is the free category on the graph
\[
\begin{tikzcd}
h. \ar[loop above,"\gamma"]
\end{tikzcd}
\]
The functor $F_{A,B}$ comes from a morphism of reflexive graphs and is obtained by ``killing the generator $\beta$'' (see Example \ref{example:killinggenerator}). In particular, the square
and let $G : P'' \to P'$ be the unique $2$\nbd{}functor such that
\begin{itemize}[label=-]
\item$G(A'')=A'$ and $G(B'')=B'$,
\item$G(i)=G(j)=h$,
\item$G(\delta)=\gamma$ and $G(\epsilon)=1_h$.
\end{itemize}
For similar reasons as for $F$, the $2$\nbd{}functor $G$ is a Thomason equivalence. This proves that both $P$ and $P''$ have the homotopy type of $\sS_2$. Finally, consider the commutative diagram of $\ho(\Ch)$
Since $F$ and $G$ are Thomason equivalences, the two bottom horizontal morphisms are isomorphisms. Besides, a simple computation using Proposition \ref{prop:abelianizationfreeoocat}, which we left to the reader, shows that the two top horizontal morphisms are isomorphisms. Since $P''$ is \good{} (Proposition \ref{prop:spheresaregood}), the morphism is $\pi_{P''}$ is an isomorphism. This implies that $\pi_{P'}$ and $\pi_{P'}$ are isomorphisms, which means by definition that $P$ and $P'$ are \good{}.
\end{paragr}
\begin{paragr}
\end{paragr}
\section{The ``Bubble-free'' conjecture}
\begin{definition}
Let $C$ be a $2$\nbd{}category. A \emph{bubble} (in $C$) is a $2$\nbd{}cell
