Commit 8fe63b67 authored by Leonard Guetta's avatar Leonard Guetta
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security commit

parent c7b6a844
......@@ -350,7 +350,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\[
\pi : A/a_0 \to \homlax(\sD_1,A) \overset{\pi^A_0}{\longrightarrow} A.
\]
This $\oo$-category can also be described in the following way (the equivalence of definitions follows from the dual of \cite[Proposition B.5.2]{ara2016joint}):
The $\oo$-category $A/a_0$ can also be described in the following explicit way (the equivalence of definitions follows from the dual of \cite[Proposition B.5.2]{ara2016joint}):
\begin{itemize}[label=-]
\item An $n$-cell of $A/a_0$ is a matrix
\[
......@@ -421,14 +421,12 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
$c'_i=a'_i\comp_k b'_i \comp_{k-1} a'_{k-1} \comp_{k-2} a'_{k-2} \comp_{k-3} \cdots \comp_{1} a'_1\comp_0 x'_k$&for every $k+1 \leq i \leq n$.\\
\end{tabular}
\end{itemize}
We leave it to the reader to check that the formulas are well defined and the axioms of $\oo$-category are satisfied. Notice that there is a canonical forgetful $\oo$-functor
We leave it to the reader to check that the formulas are well defined and the axioms of $\oo$-category are satisfied. The canonical forgetful $\oo$-functor $\pi : A/a_0 \to A$ is simply expressed as:
\begin{align*}
A/a_0 &\to A \\
(x,a) &\mapsto x_n.
\end{align*}
\end{paragr}
\begin{paragr}
When $A$ is an $n$-category then so is $A/a_0$ for any object $a_0$ of $A$. In this case, for an $n$-cell $(x,a)$, $a_{n+1}$ is an identity, hence
Notice that if $A$ is an $n$-category then so is $A/a_0$. In this case, for an $n$-cell $(x,a)$, $a_{n+1}$ is an identity, hence
\[
a'_n \comp_{n-1} a'_{n-1} \comp_{n-2} \cdots \comp_1 a'_1 \comp_0 x_n = a_n.
\]
......@@ -438,10 +436,10 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
Let $u : A \to B$ be a morphism of $\oo\Cat$ and $b_0$ an object of $B$. We define the $\oo$-category $A/b_0$ (also denoted $u\downarrow b_0$) as the following fibred product
\[
\begin{tikzcd}
A/b_0 \ar[d] \ar[r] & A \ar[d,"u"] \\
B/b_0 \ar[r,"u/b_0"] & B
A/b_0 \ar[d,"u/b_0"'] \ar[r] & A \ar[d,"u"] \\
B/b_0 \ar[r,"\pi"'] & B.
\ar[from=1-1,to=2-2,phantom,description,very near start,"\lrcorner"]
\end{tikzcd}.
\end{tikzcd}
\]
More explicitly, an $n$-cell $(x,b)$ of $A/b_0$ is a matrix
\[
......@@ -536,7 +534,16 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\todo{Uniformiser les notations de troncations.}
For the converse, see \cite{metayer2008cofibrant}.
\end{proof}
\todo{Ai-je vraiment besoin de mettre le théorème ci-dessous ?}
\begin{proposition}
Let $f : A \to B$ and $g : C \to D$ be morphisms of $\oo\Cat$. If $f$ and $g$ are cofibrations for the canonical model structure, then so is
\[
f\otimes g : A \otimes B \to C \otimes D.
\]
\end{proposition}
\begin{proof}
See \cite[Proposition 5.1.2.7]{lucas2017cubical} or \cite{ara2019folk}.
\end{proof}
\section{Equivalences of $\omega$-categories vs Thomason weak equivalences}
\begin{lemma}\label{lemma:nervehomotopical}
The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences of $\omega$-categories to weak equivalences of simplicial sets.
......@@ -572,4 +579,20 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
\fi
As we shall see later, this morphism is \emph{not} left exact. This point is probably the most important of the work presented here \todo{Finir le blabla. Il faudrait peut-être rappeler concrètement ce que signifie le fait que le morphisme n'est pas exact à gauche.}
\end{paragr}
\section{A Folk theorem A}
\begin{proposition}(Folk Theorem $A$) Let
\[
\begin{tikzcd}[column sep=small]
A \ar[rr,"u"] \ar[dr,"w"'] & &B \ar[dl,"v"] \\
&C&
\end{tikzcd}
\]
be a commutative triangle in $\oo\Cat$. If for every object $c_0$ of $C$, the induced morphism
\[
u/c_0 : A/c_0 \to B/c_0
\]
is an equivalence of $\oo$-categories, then so is $u$.
\end{proposition}
\begin{proof}
\end{proof}
......@@ -11,6 +11,7 @@
\include{omegacat}
\include{homtheo}
\include{hmtpy}
\include{hmlgy}
\bibliographystyle{alpha}
\bibliography{memoire}
\end{document}
......@@ -75,6 +75,7 @@
\newcommand{\sD}{\ensuremath{\mathbb{D}}}
\newcommand{\sS}{\ensuremath{\mathbb{S}}}
\newcommand{\sH}{\ensuremath{\mathbb{H}}}
% oo-categories
\newcommand{\oo}{\omega}
......@@ -97,6 +98,8 @@
% Some standard categories
\newcommand{\Set}{\mathbf{Set}}
\newcommand{\Ch}{\mathbf{Ch}_{\geq 0}} %category of chain complexes
\newcommand{\Ab}{\mathbf{Ab}} %category of abelian groups
% compositions and units
\def\1^#1_#2{1^{(#1)}_{#2}} % for iterated units
......
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