OMG! 6.5.2 is complete

2cat.tex

@@ -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 $k \geq 0$,
     \[V(F)_k : V(C)_k \to V(D)_k\] 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:
 \[
 \begin{tikzcd}
-  A'' \ar[r,bend left=75,"i",""{name=A,below}] \ar[r,bend
-    right=75,"j"',""{name=B,above}] & B''
+  \overline{A} \ar[r,bend left=75,"i",""{name=A,below}] \ar[r,bend
+    right=75,"j"',""{name=B,above}] & \overline{B}
   \ar[from=A,to=B,bend right,Rightarrow,"\delta"']
   \ar[from=A,to=B,bend left ,Rightarrow,"\epsilon"]
 \end{tikzcd}
 \]
-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:
+\begin{itemize}[label=-]
+  \item generating $0$\nbd{}cell: $A''$,
+  \item generating $1$\nbd{}cell: $l : A'' \to A''$,
+    \item generating $2$\nbd{}cells: $\lambda : l \Rightarrow 1_{A''}$ and $\mu: l \Rightarrow 1_{A''}$.
+    \end{itemize}
+    In picture, this gives
+      \[
+    \begin{tikzcd}[column sep=huge]
+    A'' \ar[r,bend left=75,"l",""{name=A'',below}]\ar[r,bend right=75,"l"',""{name=B,above}] \ar[r,"1_{A''}" pos=1/3,""{name=C,above},""{name=D,below}]& A''
+    \ar[from=A'',to=C,Rightarrow,"\lambda"] \ar[from=B,to=D,"\mu" pos=9/20,Rightarrow]
+    \end{tikzcd}
+    \qquad \quad \text{ or }
+    \begin{tikzcd}
+      A''. \ar[loop,in=30,out=150,distance=3cm,"l",""{name=A,below}]
+      \ar[from=A,to=1-1,bend right,Rightarrow,"\lambda"']
+      \ar[from=A,to=1-1,bend left,Rightarrow,"\mu"]
+      \end{tikzcd}
+    \]
+    Let $H : \sS_2 \to P''$ be the unique $2$\nbd{}functor such that:
+    \begin{itemize}[label=-]
+    \item $H(\overline{A})=(\overline{B})=A''$,
+      \item $H(i)=l$ and $H(j)=1_{A''}$,
+      \item $H(\delta)=\lambda$ and $H(\epsilon)=\lambda$.
+      \end{itemize}
+Let us prove that $H$ is a Thomason equivalence using Corollary
+\ref{cor:criterionThomeqII}. To do that, we have to compute $V(H)_k : V(\sS_2)_k
+\to V(P'')_k$ for every $k\leq 0$. For $k=0$, the category $V(\sS_2)_0$ is the
+free category on the graph
+\[
+  \begin{tikzcd}
+    \overline{A} \ar[r,"i",shift left] \ar[r,"j"',shift right] & \overline{B},
+  \end{tikzcd}
+\]
+the category $V(P'')$ is the free category on the graph
+\[
+  \begin{tikzcd}
+    A \ar[loop above,"l"]
+  \end{tikzcd}
+\]
+and $V(H)_0$ comes from a morphism of reflexive graphs obtained by ``killing the
+generator $j$''. Hence, it is a Thomason equivalence of categories. For $k>0$,
+the category $V(\sS_2)_k$ has two objects $\overline{A}$ and $\overline{B}$ and
+an arrow $\overline{A} \to \overline{B}$ is a $k$\nbd{}uple of either one of the
+following form
+\begin{itemize}[label=-]
+\item $(1_i,\cdots,1_i,\delta,1_j,\cdots,1_j)$
+\item $(1_i,\cdots,1_i,\epsilon,1_j,\cdots,1_j)$
+\item $(1_i,\cdots,1_i)$,
+\item $(1_j,\cdots,1_j)$,
+  \end{itemize}
+  and these are the only non-trivial arrows. In other words, $V(\sS_2)_k$ is the
+  free category on the graph with two objects and $2k+2$ parallel arrows between
+  these two objects. In order to compute $V(P'')_k$, let us first notice that
+  every 
+  $2$\nbd{}cell of $P''$ (except for $\1^2_{A''}$) is uniquely encoded as a
+  finite word
+  on the alphabet that has three symbols : $1_l$, $\lambda$ and $\mu$.
+  Concatenation corresponding to the $0$\nbd{}composition of these cells. From
+  this observation, it is easily seen that the category $V(P'')_k$ is the free
+  category on the graph that has one objects $A''$ and $2k+1$ generating arrows
+  which are of either one of the following form
+  \begin{itemize}[label=-]
+  \item $(1_l,\cdots,1_l,\lambda,1^2_{A''},\cdots,1^2_{A''})$
+  \item $(1_l,\cdots,1_l,\mu,1^2_{A''},\cdots,1^2_{A''})$
+  \item $(1_l,\cdots,1_l)$.
+  \end{itemize}
+  Once again, the functor $V(H)_k$ comes from a morphism a reflexive graphs and
+  is obtained by ``killing the generator $(1_j,\cdots,1_j)$''. Hence, it is a
+  Thomason equivalence and thus, so is $H$. This proves that $P''$ has the
+  homotopy type of $\sS_2$.
+  
+  % is of exactly one of the following form
+  % \begin{itemize}[label=-]
+  % \item $\1^2_{A''}$,
+  % \item $1_{l}\comp_0 cdots\comp_0 1_l}$ (including the case $1_{l}$),
+  % \item $1_{l}\comp_0 \cdots $
+  % \end{itemize}
+Finally, consider the commutative diagram of $\ho(\Ch)$
 \[
 \begin{tikzcd}[column sep=huge]
-  \sH^{\sing}(P) \ar[r,"\sH^{\sing}(F)"] \ar[d,"\pi_P"] & \sH^{\sing}(P') \ar[d,"\pi_{P'}"] & \sH^{\sing}(P'') \ar[l,"\sH^{\sing}(G)"'] \ar[d,"\pi_{P''}"] \\
-  \sH^{\pol}(P) \ar[r,"\sH^{\pol}(F)"'] & \sH^{\pol}(P') & \sH^{\pol}(P''). \ar[l,"\sH^{\pol}(G)"] 
+  \sH^{\sing}(P) \ar[r,"\sH^{\sing}(F)"] \ar[d,"\pi_P"] & \sH^{\sing}(P')
+  \ar[d,"\pi_{P'}"] & \sH^{\sing}(\sS_2) \ar[l,"\sH^{\sing}(G)"']
+  \ar[d,"\pi_{\sS_2}"] \ar[r,"\sH^{\sing}(H)"]&\sH^{\sing}(P'') \ar[d,"\pi_{P''}"]\\
+  \sH^{\pol}(P) \ar[r,"\sH^{\pol}(F)"'] & \sH^{\pol}(P') & \sH^{\pol}(\sS_2)
+  \ar[l,"\sH^{\pol}(G)"] \ar[r,"\sH^{\pol}(H)"']&\sH^{\pol}(P'').
 \end{tikzcd}
 \]
-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
     \ar[from=A,to=C,Rightarrow,"\alpha"] \ar[from=D,to=B,"\beta" pos=9/20,Rightarrow]
     \end{tikzcd}
-    \text{ or }
+    \qquad \quad \text{ or }
     \begin{tikzcd}
       A. \ar[loop,in=30,out=150,distance=3cm,"f",""{name=A,below}]
       \ar[from=A,to=1-1,bend right,Rightarrow,"\alpha"']
@@ -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: $\alpha : f \Rightarrow 1_A$ and $\beta: f \Rightarrow 1_A$.
+    \end{itemize}
+    In picture, this gives
+      \[
+    \begin{tikzcd}[column sep=huge]
+    A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend right=75,"f"',""{name=B,above}] \ar[r,"1_A" pos=1/3,""{name=C,above},""{name=D,below}]& A
+    \ar[from=A,to=C,Rightarrow,"\alpha"] \ar[from=B,to=D,"\beta" pos=9/20,Rightarrow]
+    \end{tikzcd}
+    \qquad \quad \text{ or }
+    \begin{tikzcd}
+      A. \ar[loop,in=30,out=150,distance=3cm,"f",""{name=A,below}]
+      \ar[from=A,to=1-1,bend right,Rightarrow,"\alpha"']
+      \ar[from=A,to=1-1,bend left,Rightarrow,"\beta"]
+      \end{tikzcd}
+    \]
+    We shall now prove that $P$ is \good{} using the techniques introduced in 
+  \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