Commit 7fe0121c by Leonard Guetta

### gotta go

parent 28ef12b9
 ... ... @@ -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} ... ...
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