Commit 52089e88 by Leonard Guetta

### version du depot 1 au SCD

parent fae18601
 ... ... @@ -326,8 +326,8 @@ From the previous proposition, we deduce the following very useful corollary. obtained from $A$ by collapsing the objects that are identified through $\beta$. It admits the following explicit description: $G_0$ is (isomorphic to) $E$ and the set of non-unit arrows of $G$ is (isomorphic to) the set of non-unit arrows of $A$; the source (resp. target) of a non-unit arrow $f$ of $G$ is the source (resp. target) of $\beta(f)$. This completely describes $G$. non-unit arrows of $A$; the source (resp.\ target) of a non-unit arrow $f$ of $G$ is the source (resp.\ target) of $\beta(f)$. This completely describes $G$. % Notice also for later reference that the morphism $\coprod_{x \in E}F_x % \to A$ is a monomorphism, i.e. injective on objects and arrows. ... ...
 ... ... @@ -323,7 +323,9 @@ higher than $1$. When $n>0$, the source and target of an $n$\nbd{}cell $(x,p)$ of $X/a$ are given by $\src((x,p))=(\src(x),p) \text{ and } \trgt((x,p))=(\trgt(x),p).$ \] %% LA DESCRIPTION DU BUT AU DESSUS N'EST PAS BONNE POUR LA DIMENSION 1 %% A CORRIGER !! Moreover, the $\oo$\nbd{}functor $f/a$ is described as $(x,p) \mapsto (f(x),p), ... ... @@ -551,7 +553,7 @@ Beware that in the previous corollary, we did \emph{not} suppose that X was fr Since trivial fibrations are stable by pullback, g/a is a trivial fibration. This proves that the most left vertical arrow of diagram \eqref{comsquare} is an isomorphism. From Proposition \ref{prop:sliceiscofibrant} and Corollary Now, from Proposition \ref{prop:sliceiscofibrant} and Corollary \ref{cor:cofprojms}, we deduce that the arrow \[\hocolim_{a \in A}^{\folk}(P/a)\to \colim_{a \in A}(P/a)$ is an isomorphism. Moreover, from Lemma \ref{lemma:colimslice}, we know that the arrows $\colim_{a \in A}(P/a)\to P$ and $\colim_{a \in A}(X/a)\to X$ are ... ... @@ -669,7 +671,9 @@ Putting all the pieces together, we are now able to prove the awaited Theorem. $\hocolim_{a \in A}^{\Th} (A/a) \to A$ is an isomorphism thanks to Corollary \ref{cor:thomhmtpycol}. is an isomorphism thanks to Corollary \ref{cor:thomhmtpycol} and the fact that the canonical morphisms of op-prederivators $\Ho(\Cat^{\Th}) \to \Ho(\oo\Cat^{\Th})$ is homotopy cocontinuous (see \ref{paragr:thomhmtpycol}). \item Every $A/a$ is \good{} thanks to Proposition \ref{prop:contractibleisgood} and Proposition \ref{prop:slicecontractible}. \end{itemize} Thus, Proposition \ref{prop:criteriongoodcat} applies and this proves that $A$ is \good{}. ... ...
 ... ... @@ -121,6 +121,11 @@ cette thèse établit un cadre général dans lequel étudier l'homologie des $\oo$\nbd{}catégories en faisant appels à des outils d'algèbre homotopique abstraite, tels que la théorie des catégorie de modèles de Quillen ou la théorie des dérivateurs de Grothendieck. \bigskip \noindent\textbf{Mots-clés : } Catégories supérieures, $\oo$\nbd{}catégories, homologie, théorie de l'homotopie, polygraphes. \end{abstract} \selectlanguage{english} \begin{abstract} ... ... @@ -144,6 +149,12 @@ des dérivateurs de Grothendieck. homology of strict $\oo$\nbd{}categories using tools of abstract homotopical algebra such as Quillen's theory of model categories or Grothendieck's theory of derivators. \bigskip \noindent\textbf{Keywords : } Higher categories, $\oo$\nbd{}categories, homology, homotopy theory, polygraphs. \end{abstract} \tableofcontents ... ...
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