### 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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