Commit 0857c269 authored by Leonard Guetta's avatar Leonard Guetta
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I am letting the last chapter aside for a bit and focusing on chapter 1 of the...

I am letting the last chapter aside for a bit and focusing on chapter 1 of the dissertation. I added a new file that contains the old version of chapter 1
parent ee8c8c77
......@@ -718,7 +718,7 @@ Before embarking on computations of homology and homotopy types of $2$-categorie
i(0)=0 \text{ and } i(1)=n.
\]
\end{paragr}
\begin{lemma}
\begin{lemma}\label{lemma:istrngdefrtract}
For $n\neq0$, the functor $i : \Delta_1 \to \Delta_n$ is a strong deformation retract (\ref{paragr:defrtract}).
\end{lemma}
\begin{proof}
......@@ -733,8 +733,10 @@ Before embarking on computations of homology and homotopy types of $2$-categorie
and it is straightforward to check that $\alpha \ast i = \mathrm{id}_i$.
\todo{Est-ce qu'il ne faudra pas dire quelque part qu'une transfo naturelle donne une transfo oplax}.
\end{proof}
\begin{paragr}
In particular, it follows from Lemma \ref{lemma:pushoutstrngdefrtract} that if $n\neq0$, then $i : \Delta_1 \to \Delta_n$ is a co-universal Thomason weak equivalence. Now consider the following cocartesian square
For any $n \geq 0$, consider the following cocartesian square
\[
\begin{tikzcd}
\Delta_1 \ar[r,"i"] \ar[d,"\tau"] & \Delta_n \ar[d] \\
......@@ -742,9 +744,9 @@ Before embarking on computations of homology and homotopy types of $2$-categorie
\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd},
\]
where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non-trival $1$\nbd-cell of $\Delta_1$ to the target of the generating $2$-cell of $A_{(1,1)}$. If $n\neq 0$, then $i$ is a co-universal Thomason weak equivalence and thus, so is the morphism $A_{(1,1)} \to A_{(1,n)}$. In particular, the square is Thomason homotopy cocartesian. Besides, the morphism $\tau : \Delta_1 \to A_{(1,1)}$ is a folk cofibration and since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ are \good{}, it follows from Corollary \ref{cor:usefulcriterion} that if $n\neq 0$, then $A_{(1,n)}$ is \good{}. Note also that, since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ have the homotopy type of a point, the fact that the previous square is Thomason homotopy cocartesian (when $n\neq 0$) implies that $A_{(1,n)}$ has the homotopy type of a point.
where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non-trival $1$\nbd-cell of $\Delta_1$ to the target of the generating $2$-cell of $A_{(1,1)}$. It is not hard to check that $\tau$ is strong deformation retract, hence a co-universal Thomason weak equivalence (Lemma \ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $A_{(1,1)} \to A_{(1,n)}$ is also a (co-universal) Thomason weak equivalence and the square is Thomason homotopy cocartesian. Now, the morphism $\tau : \Delta_1 \to A_{(1,1)}$ is also a folk cofibration and since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ are \good{}, it follows from Corollary \ref{cor:usefulcriterion} that $A_{(1,n)}$ is \good{}. Finally, since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ have the homotopy type of a point, the fact that the previous square is Thomason homotopy cocartesian implies that $A_{(1,n)}$ has the homotopy type of a point.
Similarly, by considering the cocartesian square
Similarly, for any $m \geq 0$, by considering the cocartesian square
\[
\begin{tikzcd}
\Delta_1 \ar[r,"i"] \ar[d,"\sigma"] & \Delta_m \ar[d] \\
......@@ -752,14 +754,20 @@ Before embarking on computations of homology and homotopy types of $2$-categorie
\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd},
\]
where $\sigma : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non trivial $1$\nbd-cell of $\Delta_1$ the source of the generating $2$\nbd-cell of $A_{(1,1)}$, we can prove that if $m\neq 0$, then $A_{(m,1)}$ is \good{} and has the homotopy type of a point.
where $\sigma : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non trivial $1$\nbd-cell of $\Delta_1$ the source of the generating $2$\nbd-cell of $A_{(1,1)}$, we can prove that $A_{(m,1)}$ is \good{} and has the homotopy type of a point.
Finally, for $m,n \geq 0$, consider the cocartesian square
Now, let $m\geq 0$ and $n > 0$ and consider the cocartesian square
\[
\begin{tikzcd}
A_{(1,1)} \ar[r] \ar[d] & A_{(1,n)} \\
A_{(m,1)} \ar[r] & A_{(m,n)}.
\Delta_1 \ar[r,"i"] \ar[d,"\tau"] & \Delta_n \ar[d] \\
A_{(m,1)} \ar[r] & A_{(m,n)},
\end{tikzcd}
\]
If
where $\tau$ is the $2$-functor that sends the unique non-trivial $1$-cell of $\Delta_1$ to the target of the generating $2$-cell of $A_{(m,1)}$. This $2$-functor is once again a folk cofibration, but it is \emph{not} in general a co-universal Thomason weak equivalence (it is if we make the hypothesis that $m\neq 0$, but we did not). However, since we made the hypothesis that $n\neq 0$, it follows from Lemma \ref{lemma:istrngdefrtract} that $i : \Delta \to \Delta_n$ is a co-universal Thomason weak equivalence. Hence, the previous square is Thomason homotopy cocartesian and $A_{(m,n)}$ has the homotopy type of a point. Since $A_{(m,1)}$, $\Delta_1$ and $\Delta_n$ are \good{}, this shows that for $m \geq 0$ and $n >0$, $A_{(m,n)}$ is \good{}.
Similarly, if $m >0$ and $ n\geq 0$, 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 proven the following proposition.
\begin{proposition}
Let $m,n \geq 0$ and consider the $2$-category $A_{(m,n)}$. If $m\neq 0$ or $n\neq 0$, 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 $K(\mathbb{Z},2)$.
\end{proposition}
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