@@ -883,7 +883,7 @@ For any $n \geq 0$, consider the following cocartesian square

Similarly, if $m >0$ and $ n\geq0$, then $A_{(m,n)}$ has the homotopy type of a point and is \good{}.

\end{paragr}

Combined with the result of Paragraph \ref{paragr:bubble}, we have proved the following proposition.

\begin{proposition}

\begin{proposition}\label{prop:classificationAmn}

Let $m,n \geq0$ and consider the $2$-category $A_{(m,n)}$. If $m\neq0$ or $n\neq0$, then $A_{(m,n)}$ is \good{} and has the homotopy type of a point. If $n=m=0$, then $A_{(0,0)}$ is not \good{} and has the homotopy type of a $K(\mathbb{Z},2)$\nbd{}space.

\end{proposition}

\section{Zoology of $2$-categories: more examples}

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@@ -897,9 +897,15 @@ For any $n \geq 0$, consider the following cocartesian square

In picture, this gives

\[

\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.

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

A. \ar[loop,in=130,out=50,distance=1.5cm,"f"',""{name=A,below}]\ar[loop,in=-130,out=-50,distance=1.5cm,"g",""{name=B,above}]

\ar[from=A,to=1-1,Rightarrow,"\alpha"]

\ar[from=B,to=1-1,Rightarrow,"\beta"]

\end{tikzcd}

\]

Notice that this category has many non-trivial $2$\nbd{}cells and it is not

\emph{a priori} clear what its homotopy type is and whether or not it is \good{}.

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@@ -1022,7 +1028,50 @@ For similar reasons as for $F$, the $2$\nbd{}functor $G$ is a Thomason equivalen

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}

Let $P$ be the free $2$\nbd{}category defined as follows:

\begin{itemize}[label=-]

\item generating $0$\nbd{}cell: $A$,

\item generating $1$\nbd{}cell: $f : A \to A$,

\item generating $2$\nbd{}cells: $\alpha : f \Rightarrow1_A$ and $\beta: 1_A \Rightarrow f$.

\end{itemize}

In picture, this gives

\[

\begin{tikzcd}[column sep=huge]

A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend right=75,"f"',""{name=B,above}]\ar[r,"1_A" pos=1/3,""{name=C,above},""{name=D,below}]& A

Let $F : P \to P'$ be the unique $2$\nbd{}functor such that:

\begin{itemize}[label=-]

\item$F(A)=A'$,

\item$F(f)=1_{A'}$

\item$F(\alpha)=\1^2_{A'}$ and $F(\beta)=\gamma$,

\end{itemize}

and $G : P' \to P$ be the unique $2$\nbd{}functor such that:

\begin{itemize}[label=-]

\item$G(A')=A$,

\item$G(\gamma)=\alpha\comp_1\beta$.

\end{itemize}

Notice that we have $F\circ G =\mathrm{id}_{P'}$, which means that $P'$ is a retract of $P$. In particular, $\sH^{\sing}(P)$ is a retract of $\sH^{\sing}(P')$ and since $P'$ has the homotopy type of $K(\mathbb{Z},2)$-space (\ref{paragr:bubble}), this proves that $P$ have non-trivial singular homology groups in all even dimension. But since it is a free $2$\nbd{}category, all its polygraphic homology groups are trivial above dimension $2$, which means that $P$ is \emph{not}\good{}.