Commit 52089e88 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

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
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment