@@ -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\inA}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

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?}