Commit 32d2ff06 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

edited a few typos and corrected the proof of 5.2.3

parent beac890d
...@@ -1205,7 +1205,7 @@ isomorphisms, which means by definition that $P$, $P'$ and $P''$ are \good{}. ...@@ -1205,7 +1205,7 @@ isomorphisms, which means by definition that $P$, $P'$ and $P''$ are \good{}.
N_{\oo}(P'') \ar[r] & N_{\oo}(P) N_{\oo}(P'') \ar[r] & N_{\oo}(P)
\end{tikzcd} \end{tikzcd}
\] \]
is also cocartesian. This proves that square \ref{square:bouquet} is is also cocartesian. This proves that square \eqref{square:bouquet} is
Thomason homotopy cocartesian \todo{détailler?} and in particular that $P$ has the homotopy Thomason homotopy cocartesian \todo{détailler?} and in particular that $P$ has the homotopy
type of a bouqet of two $2$\nbd{}spheres. Since $\sD_1$, $P'$ and $P''$ are type of a bouqet of two $2$\nbd{}spheres. Since $\sD_1$, $P'$ and $P''$ are
free and \good{} and since $\langle \beta \rangle : \sD_1 \to P'$ and free and \good{} and since $\langle \beta \rangle : \sD_1 \to P'$ and
......
...@@ -166,24 +166,27 @@ From these two lemmas, follows the important proposition below. ...@@ -166,24 +166,27 @@ From these two lemmas, follows the important proposition below.
For every $n \geq -1$, the $\oo$\nbd{}category $\sS_n$ is \good{}. For every $n \geq -1$, the $\oo$\nbd{}category $\sS_n$ is \good{}.
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
We proceed by induction on $n$. When $n=-1$, it is trivial to check that the %% We proceed by induction on $n$. When $n=-1$, it is trivial to check that the
empty $\oo$\nbd{}category is \good{}. Now, since $i_n : \sS_n \to \sD_{n+1}$ is %% empty $\oo$\nbd{}category is \good{}.
a folk cofibration and $\sS_{n}$ and $\sD_{n}$ are folk cofibrant, it follows Recall that the cofibrations of simplicial sets are exactly the monomorphisms, and in particular all simplicial sets are cofibrant. Since $i_n : \sS_{n-1} \to \sD_n$ is a monomorphism for every $n \geq 0$ and since $N_{\oo}$ preserves monomorphisms (as a right adjoint), it follows from Lemma \ref{lemma:squarenerve} and Lemma \ref{lemma:hmtpycocartesianreedy} that the square
from Lemma \ref{lemma:hmtpycocartesianreedy} that the cocartesian square \[
\begin{equation}\label{square}
\begin{tikzcd} \begin{tikzcd}
\sS_{n} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_{n+1} \ar[d,"j_n^+"] \\ N_{\oo}(\sS_{n-1}) \ar[r,"N_{\oo}(i_n)"] \ar[d,"N_{\oo}(i_n)"] & N_{\oo}(\sD_{n}) \ar[d,"N_{\oo}(j_n^+)"] \\
\sD_{n+1} \ar[r,"j_n^-"] & \sS_{n+1} N_{\oo}(\sD_{n}) \ar[r,"N_{\oo}(j_n^-)"] & N_{\oo}(\sS_{n})
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd} \end{tikzcd}
\end{equation} \]
is homotopy cocartesian with respect to the canonical weak equivalences on is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an
$\oo\Cat$. Besides, since $N_{\oo} : \oo\Cat \to \Psh{\Delta}$ induces an
equivalence of op-prederivators $\Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta})$ equivalence of op-prederivators $\Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta})$
(Theorem \ref{thm:gagna}), it follows from Lemma \ref{lemma:squarenerve} that (Theorem \ref{thm:gagna}), it follows that the square of $\oo\Cat$
the square \eqref{square} is also homotopy cocartesian with respect to \[
Thomason equivalences. Then, the desired result follows from the induction \begin{tikzcd}
hypothesis and Corollary \ref{cor:usefulcriterion}. \sS_{n-1} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_{n} \ar[d,"j_n^+"] \\
\sD_{n} \ar[r,"j_n^-"] & \sS_{n}
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
is Thomason homotopy cocartesian for every $n\geq 0$. Finally, since $i_n : \sS_{n-1} \to \sD_{n}$ is
a folk cofibration and $\sS_{n-1}$ and $\sD_{n}$ are folk cofibrant for every $n\geq0$, we deduce the result from Corollary \ref{cor:usefulcriterion} and an immediate induction. The base case being simply that $\sS_{-1}=\emptyset$ is obviously \good{}.
\end{proof} \end{proof}
\begin{paragr} \begin{paragr}
The previous proposition implies what we claimed in Paragraph \ref{paragr:prelimcriteriongoodcat}, which is that the morphism of op-prederivators The previous proposition implies what we claimed in Paragraph \ref{paragr:prelimcriteriongoodcat}, which is that the morphism of op-prederivators
...@@ -207,7 +210,7 @@ From these two lemmas, follows the important proposition below. ...@@ -207,7 +210,7 @@ From these two lemmas, follows the important proposition below.
$i_2$ is a cofibration for the canonical model structure, the square is also $i_2$ is a cofibration for the canonical model structure, the square is also
homotopy cocartesian with respect to folk weak equivalences. If $\J$ was homotopy cocartesian with respect to folk weak equivalences. If $\J$ was
homotopy cocontinuous, then this square would also be homotopy cocartesian homotopy cocontinuous, then this square would also be homotopy cocartesian
with respect to Thomason equivalences. Since we know that $\sS_A$, $\sD_0$ and $\sD_2$ are \good{}, this would imply that $B^2\mathbb{N}$ is \good{}. with respect to Thomason equivalences. Since we know that $\sS_1$, $\sD_0$ and $\sD_2$ are \good{}, this would imply that $B^2\mathbb{N}$ is \good{}.
From Proposition \ref{prop:spheresaregood}, we also deduce the proposition From Proposition \ref{prop:spheresaregood}, we also deduce the proposition
below which gives a criterion to detect \good{} $\oo$\nbd{}category when we below which gives a criterion to detect \good{} $\oo$\nbd{}category when we
...@@ -230,7 +233,7 @@ From these two lemmas, follows the important proposition below. ...@@ -230,7 +233,7 @@ From these two lemmas, follows the important proposition below.
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
Since the morphisms $i_k$ are folk cofibration and the $\oo$\nbd{}categories Since the morphisms $i_k$ are folk cofibration and the $\oo$\nbd{}categories
$\sS^{k-1}$ and $\sS^{k}$ are folk cofibrant and \good{}, it follows $\sS_{k-1}$ and $\sD_{k}$ are folk cofibrant and \good{}, it follows
from Corollary \ref{cor:usefulcriterion} that all $\sk_k(C)$ are \good{}. The from Corollary \ref{cor:usefulcriterion} that all $\sk_k(C)$ are \good{}. The
result follows then from Lemma \ref{lemma:filtration} and Proposition \ref{prop:sequentialhmtpycolimit}. result follows then from Lemma \ref{lemma:filtration} and Proposition \ref{prop:sequentialhmtpycolimit}.
\end{proof} \end{proof}
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