gotta go

2cat.tex

@@ -1307,7 +1307,13 @@ with $n$ occurences of $f$ and $m$ occurences of $g$.
     \item $F(\alpha)=1_{(1,1)}$.
   \end{itemize}
   This last equation makes sense since $(1,1)=(0,1)+(1,0)=(1,0)+(0,1)$. For any
-  $1$\nbd{}cell $w$ of $P$ (encoded as a finite words on the alphabet $\{f,g\}$) such that $f$ appears $n$ times and $g$ appears $m$ times, we have $F(w)=(n,m)$. Let us now prove that $F$ is a Thomason equivalence using a dual of \cite[Corollaire 5.26]{ara2020theoreme} (see Remark 5.20 of op.\ cit.). If we write $\star$ for the only object of $B^1(\mathbb{N}\times\mathbb{N})$, what we need to show is that the canonical $2$\nbd{}functor from $P/{\star}$ (\ref{paragr:comma}) to the terminal $2$\nbd{}category 
+  $1$\nbd{}cell $w$ of $P$ (encoded as a finite words on the alphabet $\{f,g\}$)
+  such that $f$ appears $n$ times and $g$ appears $m$ times, we have
+  $F(w)=(n,m)$. Let us now prove that $F$ is a Thomason equivalence using a dual
+  of \cite[Corollaire 5.26]{ara2020theoreme} (see Remark 5.20 of op.\ cit.). If
+  we write $\star$ for the only object of $B^1(\mathbb{N}\times\mathbb{N})$,
+  what we need to show is that the canonical $2$\nbd{}functor from $P/{\star}$
+  (see \ref{paragr:comma}) to the terminal $2$\nbd{}category 
   \[
   P/{\star} \to \sD_0
   \]
@@ -1315,9 +1321,48 @@ with $n$ occurences of $f$ and $m$ occurences of $g$.
   \begin{itemize}[label=-]
   \item A $0$\nbd{}cell is a $1$\nbd{}cell of $B^1(\mathbb{N}\times \mathbb{N})$.
   \item For $(n,m)$ and $(n',m')$ two $0$\nbd{}cells of $P/{\ast}$, a
-    $1$\nbd{}cell 
-  \item
-    \end{itemize}
+    $1$\nbd{}cell from $(n,m)$ to $(n',m')$ is a $1$\nbd{}cell $w$ of $P$ such
+    that the triangle
+    \[
+      \begin{tikzcd}[column sep=small,row sep=small]
+        \star \ar[rr,"F(w)"] \ar[rd,"{(n,m)}"']& & \star \ar[dl,"{(n',m')}"]\\
+        &\star&
+      \end{tikzcd}
+    \]
+    is commutative. More explicitely, if $F(w)=(n'',m'')$, the commutativity of
+    the previous triangle means
+    \[
+      n'+n''=n \text{ and } m'+m''=m.
+    \]
+  \item Given two parallel $1$\nbd{}cells $w$ and $w'$ of $P/\star$, a $2$\nbd{}cell of
+    $P/{\star}$ from $w$ to $w'$ is simply a $2$\nbd{}cell of $P$ from $w$ to
+    $w'$ seen as $1$\nbd{}cells of $P$.
+  \end{itemize}
+  From what we said earlier on the $1$\nbd{}cells and $2$\nbd{}cells of $P$, it
+  follows easily that for every $0$\nbd{}cell $(n,m)$ of $P/{\star}$, the
+  category
+  \[
+    P/{\star}((m,n),(0,0))
+  \]
+  has a terminal object, which is given by
+  \[
+    f\cdots fg\cdots g
+  \]
+  where $f$ is repeated $n$ times and $g$ is repeated $m$ times. Then, it
+  follows from \cite[Théorème 5.27 and Remarque 5.28]{ara2020theoreme} that $P/{\star} \to \sD_0$ is a
+  Thomason equivalence of $2$\nbd{}categories and this proves that $F$ is a
+  Thomason equivalence. Since $B^1(\mathbb{N}\times\mathbb{N})\simeq
+  B^1(\mathbb{N})\times B^1(\mathbb{N})$ and $B^1(\mathbb{N})$ has the homotopy
+  type of $\sS_1$, we conclude that $P$ has the homotopy type of $\sS_1 \times
+  \sS_1$, i.e.\ the homotopy type of the torus.
+
+  Consider now the commutative square
+  \[
+    \begin{tikzcd}
+      a
+      \end{tikzcd}
+    \]
+    \todo{À finir}
 \end{paragr}
 \section{The ``Bubble-free'' conjecture}
 \begin{definition}