@@ -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