Commit 3de1399d authored by Leonard Guetta's avatar Leonard Guetta
Browse files

added a new chapter of the dissertation

parent b50ba4f4
\chapter{Homotopy and homology type of free $2$-categories}
......@@ -119,7 +119,7 @@ In the following lemma, $\gamma_{\folk}$ is the localization functor $\oo\Cat \t
is an isomorphism of $\ho(\Ch)$.
\end{corollary}
We can now prove the main result of this section.
\begin{proposition}
\begin{proposition}\label{prop:contractibleisgood}
Every contractible $\oo$\nbd-category is \good{}.
\end{proposition}
\begin{proof}
......@@ -288,7 +288,7 @@ induces a morphism of localizers and then a morphism of op-prederivators
This map is natural in $X$ in the following sense. Let $f' : X' \to A$ be another $\oo$\nbd-functor and let
\[
\begin{tikzcd}
X \arr[rr,"g"] \ar[dr,"f"] && X' \ar[dl,"f'"] \\
X \ar[rr,"g"] \ar[dr,"f"] && X' \ar[dl,"f'"] \\
&A&
\end{tikzcd}
\]
......@@ -305,7 +305,7 @@ induces a morphism of localizers and then a morphism of op-prederivators
\[
\begin{tikzcd}
\displaystyle \colim_{a_0 \in A}X/a_0 \ar[d] \ar[r] & X \ar[d,"g"] \\
\displaystyle\colim_{a_0 \in }X'/a_0 \ar[r] & X'
\displaystyle\colim_{a_0 \in A}X'/a_0 \ar[r] & X'
\end{tikzcd},
\]
is commutative.
......@@ -327,7 +327,7 @@ induces a morphism of localizers and then a morphism of op-prederivators
\]
We shall now proceed to prove that this colimit is ``folk homotopic'' and then ``Thomason homotopic''. Since each $A/a_0$ is contractible (Proposition \ref{prop:slicecontractible}), it will imply that every $1$-category is \good{}.
\end{paragr}
\textit{Up to \todo{ref}, we fix an $\oo$\nbd-functor $f : X \to A$ with $A$ a $1$\nbd-category.
\textit{Up to \todo{ref}, we fix an $\oo$\nbd-functor $f : X \to A$ with $A$ a $1$\nbd-category.}
\begin{lemma}
If $X$ is free, then for every object $a_0$ of $A$, the $\oo$\nbd-category $X/a_0$ is free.
......@@ -342,7 +342,7 @@ induces a morphism of localizers and then a morphism of op-prederivators
\begin{paragr}
In particular, if $X$ is free, then any arrow $\beta : a_0 \to a'_0$ of $A$ induces a map
\begin{align*}
\Sigma^{X/a'_0}_n &\to \Sigma^{X/a_0}_n \\
\Sigma^{X/a_0}_n &\to \Sigma^{X/a'_0}_n \\
(x,p) &\mapsto (x,\beta\circ p).
\end{align*}
This defines a functor
......@@ -370,5 +370,136 @@ induces a morphism of localizers and then a morphism of op-prederivators
which is natural in $a_0$. A simple verification shows that it is a bijection.
\end{proof}
\begin{proposition}
Let $A$ a $1$\nbd-category, $X$ a free $\oo$\nbd-category and $f : X \to A$ be an $\oo$\nbd-functor.
Let $A$ a $1$\nbd-category, $X$ a free $\oo$\nbd-category and $f : X \to A$ be an $\oo$\nbd-functor. The functor
\begin{align*}
A &\to \oo\Cat \\
a_0 &\mapsto A/a_0
\end{align*}
is a cofibrant object for the projective model structure on $\oo\Cat(A)$ (\todo{ref}) induced by the canonical model structure on $\oo\Cat$.
\end{proposition}
\begin{proof}
\todo{À écrire}
\end{proof}
\begin{corollary}\label{cor:folkhmptycol}
Let $A$ be a $1$\nbd-category and $f : X \to A$ an $\oo$\nbd-functor. The canonical arrow of $\ho(\oo\Cat^{\folk})$
\[
\hocolim^{\folk}_{a_0 \in A}X/a_0 \to X,
\]
induced by the co-cone $(X/a_0 \to X)_{a_0 \in \Ob(A)}$, is an isomorphism.
\end{corollary}
Beware that in the previous corollary, we did \emph{not} suppose that $X$ was free.
\begin{proof}
\todo{À écrire}
\end{proof}
We now move on to the next step needed to prove that every $1$-category is \good{}. For that purpose, let us recall a construction commonly referred to as the ``Grothendieck construction''.
\begin{paragr}
Let $A$ be a small category and $F : A \to \Cat$ a functor. We denote by $\int F$ or $\int_{a \in A}F(a)$ the category such that:
\begin{itemize}
\item An object of $\int F$ is a pair $(a,x)$ where $a$ is an object of $A$ and $x$ is an object of $F(a)$.
\item An arrow $(a,x) \to (a',x')$ of $\int F$ is a pair $(f,k)$ where
\[
f : a \to a'
\]
is an arrow of $A$, and
\[
k : F(f)(x) \to x'.
\]
\end{itemize}
The identity arrow on $(a,x)$ is the pair $(1_a,1_x)$ and the composition of $(f,k) : (a,x) \to (a',x')$ and $(f',k') : (a',x') \to (a'',x'')$ is given by:
\[
(f',k')\circ(f,k)=(f'\circ f,k'\circ F(f')(k)).
\]
Now, every morphism of functors
\[
\begin{tikzcd}
A \ar[r,bend left,"F",""{name=A,below}]\ar[r,bend right,"G"',""{name=B, above}] & \Cat \ar[from=A,to=B,Rightarrow,"\alpha"]
\end{tikzcd}
\]
induces a functor
\begin{align*}
\int \alpha : \int F &\to \int G\\
(a,x) &\mapsto (a,\alpha_a(x)).
\end{align*}
This defines a functor
\begin{align*}
\int : \Cat(A)&\to \Cat \\
F&\mapsto \int F,
\end{align*}
where $\Cat(A)$ is the category of functors from $A$ to $\Cat$.
\end{paragr}
We now recall an important Theorem due to Thomason.
\begin{theorem}[Thomason]\label{thm:Thomason}
The functor $\int : \Cat(A) \to \Cat$ sends pointwise Thomason equivalence (\todo{ref}) to Thomason equivalence and the induced functor
\[
\overline{\int} : \ho(\Cat^{\Th}(A)) \to \ho(\Cat^{\Th})
\]
is canonically isomorphic to the homotopy colimit functor
\[
\hocolim_A : \ho(\Cat^{\Th}(A)) \to \ho(\Cat^{\Th}).
\]
\end{theorem}
\begin{proof}
The original source for this Theorem is \cite{thomason1979homotopy}. However, the definition of homotopy colimit used by Thomason, albeit equivalent, is not the same as the one we used in this dissertation and is slightly outdated. A more modern proof of the theorem can be found in \cite[Proposition 2.3.1 and Théorème 1.3.7]{maltsiniotis2005theorie}.
\end{proof}
\begin{corollary}\label{cor:thomhmtpycol}
Let $A$ be a small category. The canonical map
\[
\hocolim_{a \in A}A/a \to A
\]
induced by the co-cone $(A/a \to A)_{a \in \Ob(A)}$, is an isomorphism of $\ho(\Cat^{\Th})$.
\end{corollary}
\begin{proof}
For every object $a$ of $A$, the canonical map to the terminal category
\[
A/a \to \sD_0
\]
is a Thomason weak equivalence. This comes from the fact that $A/a$ is contractible (Proposition \ref{prop:slicecontractible}), or from \cite[Section 1, Corollary 2]{quillen1973higher} and the fact that $A/a$ has a terminal object.
In particular, the morphism of functors
\[
A/(-) \Rightarrow k_{\sD_0},
\]
where $k_{\sD_0}$ is the constant functor $A \to \Cat$ with value the terminal category $\sD_0$, is a pointwise Thomason weak equivalence. It follows from the first part of Theorem \ref{thm:Thomason} that
\[
\int_{a \in A}A/a \to \int_{a \in A}k_{\sD_0}
\]
is a Thomason weak equivalence. An immediate computation shows that $\int_{a \in A}k_{\sD_0} \simeq A$. From the second part of Theorem \ref{thm:Thomason}, we have that
\[
\hocolim_{a \in A}A/a \simeq A.
\]
A thorough analysis of all the isomorphisms involved (\todo{détailler ou ref à Maltsiniotis}) shows that this last isomorphism is indeed induced by the co-cone $(A/a \to A)_{a \in \Ob(A)}$.
\end{proof}
\begin{remark}
It is possible to extend the previous corollary to prove that for any functor $f : X \to A$ ($X$ and $A$ being $1$-categories), we have $\hocolim^{\Th}_{a \in A} X/a \simeq X$. However, to prove that it is also the case when $X$ is an $\oo$\nbd-category and $f$ an $\oo$\nbd-functor, as in Corollary \ref{cor:folkhmtpycol}, would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd-categorical analoguous of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go way beyond the scope of this dissertation.
\end{remark}
Putting all the pieces together, we are now able to prove the awaited Theorem.
\begin{theorem}
Every $1$-category is \good{}.
\end{theorem}
\begin{proof}
All the arguments of the proof have already been given and we sum them up here essentially for the sake of clarity.
Let $A$ be a $1$-category. Consider the diagram
\begin{align*}
A &\to \oo\Cat\\
a &\mapsto A/a
\end{align*}
and the co-cone
\[
(A/a \to A)_{a \in \Ob(A)}.
\]
\begin{itemize}[label=-]
\item The canonical map of $\ho(\oo\Cat^{\folk})$
\[
\hocolim_{a \in A}^{\folk} A/a \to A
\]
is an isomorphism thanks to Corollary \ref{cor:folkhmtpycol} applied to $\mathrm{id}_A : A \to A$.
\item The canonical map of $\ho(\oo\Cat^{\Th})$
\[
\hocolim_{a \in A}^{\Th} A/a \to A
\]
is an isomorphism thanks to Corollary \ref{cor:thomhmtpycol}.
\item Every $A/a$ is \good{} thanks to Proposition \ref{prop:contractibleisgood}.
\end{itemize}
Thus, Proposition \ref{prop:criteriongoodcat} applies and this proves that $A$ is \good{}.
\end{proof}
\todo{Section supplémentaire qui parle de la généralisation aux ncat?}
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