@@ -707,7 +707,7 @@ From this lemma, we deduce two useful criteria to detect Thomason equivalences o

\]

The result follows then from Lemma \ref{bisimpliciallemma} and the fact that weak equivalences of simplicial sets are stable by coproducts and finite products.

\end{proof}

\begin{corollary}

\begin{corollary}\label{cor:criterionThomeqII}

Let $F : C \to D$ be a $2$-functor. If for every $k \geq0$,

\[V(F)_k : V(C)_k \to V(D)_k\] is a Thomason equivalence of $1$-categories, then $F$ is a Thomason equivalence of $2$-categories.

\end{corollary}

...

...

@@ -953,8 +953,9 @@ Observe now that $P$ is obtained as the following amalgamated sum

\ar[from=A,to=B,bend right,Rightarrow,"\alpha"']

\ar[from=B,to=A,bend right,Rightarrow,"\beta"']

\end{tikzcd}

\]

We shall see that this $2$\nbd{}category is \good{}. First, let $P'$ be the free

\]

Let $P'$ be the free

$2$\nbd{}category defined as follows:

\begin{itemize}[label=-]

\item generating $0$\nbd{}cells: $A',B'$,

...

...

@@ -1003,29 +1004,120 @@ The functor $F_{A,B}$ comes from a morphism of reflexive graphs and is obtained

\end{tikzcd}

\]

is Thomason homotopy cocartesian and thus, $F_{(A,B)}$ is a Thomason equivalence.

Now let $P''$ be a copy of $\sS_2$ labelled as follows:

Now consider (a copy of)$\sS_2$ labelled as follows:

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

and let $G : \sS_2\to P'$ be the unique $2$\nbd{}functor such that

\begin{itemize}[label=-]

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

\item$G(\overline{A})=A'$ and $G(\overline{B})=B'$,

\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 $\ho(\Ch)$

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

$\sS_2$.

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

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{}.

Since $F$, $G$ and $H$ are Thomason equivalences, the three top horizontal

morphisms are isomorphisms. Besides, a simple computation using Proposition

\ref{prop:abelianizationfreeoocat}, which we left to the reader, shows that the

three bottom horizontal morphisms are also isomorphisms. Since $\sS_2$ is

\good{} (Proposition \ref{prop:spheresaregood}), the morphism $\pi_{\sS_2}$

is an isomorphism. This implies that $\pi_{P'}$, $\pi_{P'}$ and $\pi_{P''}$ are

isomorphisms, which means by definition that $P$, $P'$ and $P''$ are \good{}.

\end{paragr}

\begin{paragr}

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

...

...

@@ -1040,7 +1132,7 @@ Since $F$ and $G$ are Thomason equivalences, the two bottom horizontal morphisms

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

@@ -1072,7 +1164,29 @@ Since $F$ and $G$ are Thomason equivalences, the two bottom horizontal morphisms

\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{}.

\end{paragr}

\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 $\beta: f \Rightarrow1_A$.

\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