Commit 02338bc4 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

corrections de mise en page

parent d8b4cc5a
......@@ -657,7 +657,7 @@ bisimplicial sets.
\overline{\delta^*} : \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to
\Ho(\Psh{\Delta})
\]
are homotopy cocontinuous. On the other hand, the obvious identity
are homotopy cocontinuous. Now, the obvious identity
$\delta^*=\delta^* \circ \mathrm{id}_{\Psh{\Delta\times\Delta}}$ implies that
we have commutative triangles
\[
......@@ -846,8 +846,8 @@ an equivalent definition of the bisimplicial nerve which uses the other directio
\[
\alpha=(\alpha_1,\alpha_2,\cdots,\alpha_k)
\]
of $2$-cells of $C$ that are vertically composable, i.e.\ such that for
every $1 \leq i \leq k-1$,
of vertically composable $2$-cells of $C$, i.e.\ such that for
every $1 \leq i \leq k-1$, we have
\[
\src(\alpha_i)=\trgt(\alpha_{i+1}).
\]
......
......@@ -11,7 +11,7 @@ We denote by $\Ch$ the category of non-negatively graded chain complexes of abel
From now on, we will implicitly consider that the category $\Ch$ is equipped with this model structure.
\end{paragr}
\begin{paragr}
Let $X$ be a simplicial set. We denote by $K_n(X)$ the abelian group of $n$-chains of $X$, i.e.\ the free abelian group on the set $X_n$. For $n>0$, let $\partial : K_n(X) \to K_{n-1}(X)$ be the linear map defined for $x \in X_n$ by
Let $X$ be a simplicial set. We denote by $K_n(X)$ the abelian group of $n$\nbd{}chains of $X$, i.e.\ the free abelian group on the set $X_n$. For $n>0$, let $\partial : K_n(X) \to K_{n-1}(X)$ be the linear map defined for $x \in X_n$ by
\[
\partial(x):=\sum_{i=0}^n(-1)^i\partial_i(x).
\]
......@@ -455,7 +455,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c
\]
The commutativity of the two induced triangles shows what we needed to prove.
\end{proof}
From now on, when given an $\oo$\nbd{}functor $u$, we write $\sH^{\pol}(u)$ instead of $\sH^{\pol}(\gamma^{\folk}(u))$ (where $\gamma^{\folk}$ is the localization functor $\oo\Cat \to \ho(\oo\Cat^{\folk})$) for the morphism induced by $u$ at the level of polygraphic homology.
From now on, for an $\oo$\nbd{}functor $u$, we write $\sH^{\pol}(u)$ instead of $\sH^{\pol}(\gamma^{\folk}(u))$ (where $\gamma^{\folk}$ is the localization functor $\oo\Cat \to \ho(\oo\Cat^{\folk})$) for the morphism induced by $u$ at the level of polygraphic homology.
\begin{lemma}\label{lemma:oplaxpolhmlgy}
Let $u,v : C \to D$ be two $\oo$\nbd{}functors. If there exists an oplax transformation $u\Rightarrow v$, then
\[
......@@ -633,7 +633,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins
\ar[from=B,to=A,Rightarrow]\ar[from=1-1,to=2-2,phantom,"\simeq" description]
\end{tikzcd}
\]
where the bottom square is commutative (up to an isomorphism) because $\sH^{\sing}\simeq\overline{\lambda c_{\oo}} \overline{N_{\oo}}$.
where the square is commutative (up to an isomorphism) because $\sH^{\sing}\simeq\overline{\lambda c_{\oo}} \overline{N_{\oo}}$.
\end{paragr}
\section{Comparing homologies}
\begin{paragr}\label{paragr:cmparisonmap}
......@@ -687,9 +687,7 @@ Another consequence of the above counter-example is the following result, which
Suppose the converse, which is that the functor
\[
\sH^{\pol} \circ \gamma^{\folk} : \oo\Cat \to \ho(\Ch)
\]sends Thomason equivalences to isomorphisms of $\ho(\Ch)$.
Because of the inclusion $\W^{\folk} \subseteq \W^{\Th}_{\oo}$, the category $\ho(\oo\Cat^{\Th})$ may be identified with the localization of $\ho(\oo\Cat^{\folk})$ with respect to $\gamma^{\folk}(\W^{\Th}_{\oo})$ and then the localization functor is nothing but
\]sends Thomason equivalences to isomorphisms of $\ho(\Ch)$. Because of the inclusion $\W^{\folk} \subseteq \W^{\Th}_{\oo}$, the category $\ho(\oo\Cat^{\Th})$ may be identified with the localization of $\ho(\oo\Cat^{\folk})$ with respect to $\gamma^{\folk}(\W^{\Th}_{\oo})$ and then the localization functor is nothing but
\[
\J : \ho(\oo\Cat^{\folk}) \to \ho(\oo\Cat^{\Th}).
\]
......
......@@ -115,7 +115,7 @@
\]
For $m > 0$ and $0 \leq i \leq m$, the $(m-1)$-simplex $\partial_i(X)$ is obtained by composing arrows at $X_i$ (or simply deleting it for $i=0$ or $m$). For $m \geq 0$ and $0 \leq i \leq m$, the $(m+1)$-simplex $s_i(X)$ is obtained by inserting a unit map at $X_i$.
For $n=2$, the functor $N_2$ is what is sometimes known as the \emph{Duskin nerve} \cite{duskin2002simplicial} (restricted from bicategories to $2$-categories). For a $2$-category $C$, an $m$-simplex $X$ of $N_2(C)$ consists of:
For $n=2$, the functor $N_2$ is what is sometimes known as the \emph{Duskin nerve} \cite{duskin2002simplicial} (restricted from bicategories to $2$-categories). For a $2$-category $C$, an $m$\nbd{}simplex $X$ of $N_2(C)$ consists of:
\begin{itemize}[label=-]
\item for every $0\leq i \leq m$, an object $X_i$ of $C$,
\item for all $0\leq i \leq j \leq m$, an arrow $X_{i,j} : X_i \to X_j$ of $C$,
......@@ -388,7 +388,8 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\]
\end{paragr}
\begin{remark}
All the above descriptions of oplax transformations can be easily dualized for \emph{lax transformations} (that is to say $1$\nbd{}cells of the $\oo$\nbd{}category $\underline{\hom}_{\mathrm{lax}}(X,Y)$ for some $\oo$\nbd{}categories $X$ and $Y$). Habit is the only reason why we put emphasis on oplax transformations rather than lax transformations.
All the above descriptions of oplax transformations can be easily
dualized for \emph{lax transformations} (i.e.\ the $1$\nbd{}cells of the $\oo$\nbd{}category $\underline{\hom}_{\mathrm{lax}}(X,Y)$ for some $\oo$\nbd{}categories $X$ and $Y$). Habit is the only reason why we put emphasis on oplax transformations rather than lax transformations.
\end{remark}
\section{Homotopy equivalences and deformation retracts}
\begin{paragr}\label{paragr:hmtpyequiv}
......@@ -431,7 +432,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
B \ar[r,"v"] & B'\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd}
\end{equation}
be a cocartesian square. We have to show that $i'$ is also a strong oplax deformation retract. By hypothesis there exists $r : B \to A$ such that $r \circ i = \mathrm{id}_A$ and $\alpha : \sD_1 \otimes B \to B$ such that the diagrams
be a cocartesian square. We have to show that $i'$ is also a strong oplax deformation retract. By hypothesis there exist $r : B \to A$ such that $r \circ i = \mathrm{id}_A$ and $\alpha : \sD_1 \otimes B \to B$ such that the diagrams
\begin{equation}\label{diagramtransf}\tag{ii}
\begin{tikzcd}
B\ar[rd,"\mathrm{id}_B"] \ar[d,"i_0^B"']& \\
......@@ -922,7 +923,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen
& (\alpha,\Lambda)
\end{pmatrix}
\]
whose image by $u/c_0$ is equivalent for the relation $\sim_{\oo}$ to the above $(n+1)$\nbd{}cell of $B/c_0$. In particular, the source and target of $\alpha$ are respectively $f$ and $f'$. Finally, we obtain that $\alpha \sim_{\oo} \beta$ by applying the canonical $\oo$\nbd{}functor $A/b_0 \to A$.
whose image by $u/c_0$ is equivalent for the relation $\sim_{\oo}$ to the above ${(n+1)}$\nbd{}cell of $B/c_0$. In particular, the source and target of $\alpha$ are respectively $f$ and $f'$. Finally, we obtain that $\alpha \sim_{\oo} \beta$ by applying the canonical $\oo$\nbd{}functor $A/b_0 \to A$.
\end{enumerate}
\end{proof}
......
......@@ -110,8 +110,8 @@ For later reference, we put here the following definition.
\begin{remark}\label{remark:localizedfunctorobjects}
Since we always consider that for every localizer $(\C,\W)$ the categories
$\C$ and $\ho(\C)$ have the same class of objects and the localization functor
is the identity on objects, it follows that for a morphism of localizers ${F :
(\C,\W) \to (\C',\W')}$, we tautologically have
is the identity on objects, it follows that for a morphism of localizers $F \colon
(\C,\W) \to (\C',\W')$, we tautologically have
\[
\overline{F}(X)=F(X)
\]
......@@ -951,10 +951,10 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\end{theorem}
\begin{theorem}[Cisinski]\label{thm:cisinskiII}
Let $\M$ and $\M'$ be two model categories and let ${F : \M \to \M'}$ be a left
Let $\M$ and $\M'$ be model categories. Let $F \colon \M \to \M'$ be a left
Quillen functor (i.e.\ the left adjoint in a Quillen adjunction). The
functor $F$ is strongly left derivable and the morphism of
op\nbd{}prederivators $\LL F : \Ho(\M) \to \Ho(\M')$ is homotopy
op\nbd{}prederivators \[\LL F \colon \Ho(\M) \to \Ho(\M')\] is homotopy
cocontinuous.
\end{theorem}
\begin{remark}
......@@ -963,10 +963,10 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
no use whatsoever of homotopy limits in this dissertation.
\end{remark}
\begin{remark}
Since the homotopy op\nbd{}prederivator of a model category
$(\M,\W,\Cof,\Fib)$ only depends on its underlying localizer, the existence
of the classes $\Cof$ and $\Fib$ with the usual properties defining model
structure ought to be thought as a \emph{property} of the localizer
Note that for a model category
$(\M,\W,\Cof,\Fib)$, its homotopy op\nbd{}prederivator only depends on its underlying localizer. Hence, the existence
of the classes $\Cof$ and $\Fib$ with the usual properties defining a model
structure ought to be thought of as a \emph{property} of the localizer
$(\M,\W)$, which is sufficient to define a ``homotopy theory''. For
example, Theorem \ref{thm:cisinskiI} should have been stated by saying that
if a localizer $(\M,\W)$ can be extended to a model category
......
......@@ -485,7 +485,7 @@ $\mathbf{Str}\oo\Cat$ la catégorie des $\oo$\nbd{}catégories (strictes).
libre est homologiquement cohérente si et seulement si elle est sans bulles.
\end{named}
\selectlanguage{english}
\frenchspacing
%%% Local Variables:
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%%% TeX-master: "main"
......
\documentclass[12pt,a4paper]{report}
\documentclass[12pt,a4paper,twoside]{report}
\usepackage[dvipsnames]{xcolor}
\usepackage[unicode,psdextra,final]{hyperref}
......@@ -25,6 +25,8 @@
%% \usepackage[pagewise,displaymath, mathlines]{lineno}
%% \linenumbers
\frenchspacing
\begin{document}
\begin{titlepage}
......@@ -98,6 +100,7 @@
\end{titlepage}
\selectlanguage{french}
\begin{abstract}
Dans cette thèse, on compare l'homologie \og classique \fg{} d'une
......@@ -128,6 +131,7 @@ des dérivateurs de Grothendieck.
théorie de l'homotopie, polygraphes.
\end{abstract}
\selectlanguage{english}
\frenchspacing
\begin{abstract}
In this dissertation, we compare the ``classical''
homology of an $\oo$\nbd{}category (defined as the homology of its Street
......@@ -156,10 +160,12 @@ théorie de l'homotopie, polygraphes.
homotopy theory, polygraphs.
\end{abstract}
\pagenumbering{roman}
\include{remerciements}
\tableofcontents
%\include{remerciements}
\newpage
\pagenumbering{arabic}
\include{introduction}
\include{introduction_fr}
\include{omegacat}
......
......@@ -590,7 +590,7 @@ Furthermore, this function satisfies the condition
\section{Recursive construction of free \texorpdfstring{$\oo$}{ω}-categories}\label{section:freeoocataspolygraph}
\begin{definition}\label{def:cellularextension}
Let $n \in \mathbb{N}$. A \emph{$n$\nbd{}cellular extension} consists of a quadruplet $\E=(C,\Sigma,\sigma,\tau)$ where:
Let $n \in \mathbb{N}$. An \emph{$n$\nbd{}cellular extension} consists of a quadruplet $\E=(C,\Sigma,\sigma,\tau)$ where:
\begin{itemize}[label=-]
\item $C$ is an $n$\nbd{}category,
\item $\Sigma$ is a set, whose elements are referred to as the \emph{indeterminates} of $\E$,
......@@ -712,7 +712,7 @@ We can now prove the following proposition, which is the key result of this sect
\]
\end{paragr}
\begin{proposition}\label{prop:criterionnbasis}
Let $C$ be an $(n+1)$\nbd{}category. A subset $E \subseteq C_{n+1}$ is an $(n+1)$\nbd{}basis of $C$ if and only if the canonical $(n+1)$\nbd{}functor
Let $C$ be an $(n+1)$\nbd{}category. A subset $E \subseteq C_{n+1}$ is an ${(n+1)}$\nbd{}basis of $C$ if and only if the canonical $(n+1)$\nbd{}functor
\[
\E_E^* \to C
\]
......@@ -724,7 +724,7 @@ We can now prove the following proposition, which is the key result of this sect
Conversely, if $E$ is an $(n+1)$\nbd{}base of $C$, then we can define an $(n+1)$\nbd{}functor $C \to \E_E^*$ that sends $E$, seen as a subset of $C_{n+1}$, to $E$, seen as a subset of $(\E^*_E)_{n+1}$ (and which is obviously the identity on cells of dimension strictly lower than $n+1$). The fact that $C$ and $\E^*$ have $E$ as an $(n+1)$\nbd{}base implies that this $(n+1)$\nbd{}functor $C \to \E^*$ is the inverse of the canonical one $\E^* \to C$.
\end{proof}
\begin{paragr}
We extend the definitions and results from \ref{def:cellularextension} to
We extend the definitions and the results from \ref{def:cellularextension} to
\ref{prop:criterionnbasis} to the case $n=-1$ by saying that a $(-1)$-cellular
extension is simply a set $\Sigma$ (which is the set of indeterminates) and $(-1)\Cat^+$ is the category of sets. Since a $0\Cat$ is also the category of sets, it makes sense to define the functors
\[
......
......@@ -51,7 +51,7 @@ formidablement singulier, je crois
que ce que j'admire le plus c'est sa liberté d'esprit.
Merci à mes s\oe{}urs Margot, Naomi et Oriane d'êtres présentes dans ma vie. Vous
occupez une place dans mon c\oeu{}r bien plus grande que vous ne le soupçonnez
occupez une place dans mon c\oe{}ur bien plus grande que vous ne le soupçonnez
certainement. Merci Maman de m'avoir toujours poussé et encouragé dans
toutes mes entreprises. Il ne fait aucun doute que je n'en serais pas là sans
toi. Merci Papa d'avoir éveillé en moi le goût pour les
......@@ -72,6 +72,7 @@ rend toute tentative de remerciement d'une fadeur ingrate. Indirectement, cette
thèse doit autant à toi qu'à moi et c'est pourquoi elle t'est dédiée...
\selectlanguage{english}
\frenchspacing
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
......
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