@@ -374,9 +374,9 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp

\]

We say that a morphism a bisimplicial sets, $f : X \to Y$, is a \emph{diagonal weak equivalence} when $\delta^*(f)$ is a weak equivalence of simplicial sets. By definition, $\delta^*$ induces a morphism of op-prederivators

Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category of bisimplicial sets can be equipped with a model structure whose weak equivalences are diagonal weak equivalences and whose cofibrations are monomorphisms. We shall refer to this model structure as the \emph{diagonal model structure}. Since $\delta^*$ preserves monomorphisms (and diagonal weak equivalences trivially), it is left Quillen. In fact, the following proposition tells us more. We denote by$\delta_*$ the right adjoint of $\delta^*$.

Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category of bisimplicial sets can be equipped with a model structure whose weak equivalences are the diagonal weak equivalences and whose fibrations are the diagonal fibrations in an obvious sense. We shall refer to this model structure as the \emph{diagonal model structure}. Let us write$\delta_*$for the right adjoint of $\delta^*$.

\end{paragr}

\begin{proposition}\label{prop:diageqderivator}

Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model structure. Then, the adjunction

...

...

@@ -388,7 +388,11 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp

is a Quillen equivalence.

\end{proposition}

\begin{proof}

\todo{Ça doit être contenu dans Moerdijk mais à voir.}

We know from the second part of \cite[Proposition 1.2]{moerdijk1989bisimplicial} that $\delta^*$ induces an equivalence at the level of homotopy categories

Hence, all we need to show is that the adjunction $\delta^*\dashv\delta_*$ is a Quillen adjunction. The fact that $\delta_*$ preserves weak equivalences follows easily from the above equivalence of homotopy categories and the fact that it preserves fibrations is \cite[Lemma 3.14]{goerss2009simplicial}.

\end{proof}

In particular, the morphism of op-prederivators

\[

...

...

@@ -467,7 +471,7 @@ In particular, morphisms of op-prederivators \eqref{fromverttodiag} and \eqref{f

\todo{Écrire une preuve ?}

\end{proof}

\section{Bisimplicial nerve for 2-categories}

We shall now describe a ``nerve'' for $2$-categories with value in bisimplicial sets and recall a few results that shows that this nerve is, in some sense, equivalent to the Street nerve from section \todo{ref}.

We shall now describe a ``nerve'' for $2$-categories with values in bisimplicial sets and recall a few results that shows that this nerve is, in some sense, equivalent to the Street nerve from section \todo{ref}.

\begin{notation}

\begin{itemize}

\item[-] Once again, we will use the notation $N : \Cat\to\Psh{\Delta}$ instead of $N_1$ for the usual nerve of categories. Moreover, using the usual convention for the set of $k$-simplices of a simplicial set, if $C$ is a (small) category, then

@@ -90,7 +90,7 @@ We end this section with an important result on slices $\oo$\nbd{}category (Para

\[

\alpha_{\src_k(e_n)}=\begin{cases}\trgt_{k+1}(e_n), \text{ if } k<n-1\\ e_n, \text{ if } k=n-1\end{cases}\text{ and }\alpha_{\trgt_k(e_n)}=\begin{cases}1_{\trgt_k(e_n)}, \text{ if } k<n-1\\ e_n, \text{ if } k=n-1\end{cases}.

\]

It is straightforward to check that this data define an oplax transformation $\alpha : \mathrm{id}_{\sD_n}\Rightarrow r\circ p$ (see \todo{ref}), which proves the result.

It is straightforward to check that this data define an oplax transformation $\alpha : \mathrm{id}_{\sD_n}\Rightarrow r\circ p$ (see \ref{paragr:formulasoplax} and Example \ref{example:natisoplax}), which proves the result.

%% \[

%% \alpha_{\src_k(e_n)}=\trgt_{k+1}(e_n) \text{ and } \alpha_{\trgt_k(e_n)}=1_{\trgt_k(e_n)},

%% \]

...

...

@@ -137,7 +137,7 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \re

\]

is cartesian and all of the four morphisms are monomorphisms. Since the

functor $\Hom_{\oo\Cat}(\Or_k,-)$ preserves limits, the square

\eqref{squarenervesphere} is also cartesian and all of the four morphisms are

\eqref{squarenervesphere} is a cartesian square of $\Set$and all of whose four morphisms are

monomorphisms.

Hence, what we need to show is that for every $k \geq0$ and

$\oo$\nbd{}functor $\varphi : \Or_k \to\sS_{n}$, there exists an

...

...

@@ -156,9 +156,9 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \re

Now, let $\varphi : \Or_k \to\sS_n$ be an $\oo$\nbd{}functor. There are several cases to distinguish.

\begin{description}

\item[Case $k<n$:] Since every generating cell of $\gamma$ of $\Or_k$ is of dimension lower or equal to$k$, the cell $\varphi(\gamma)$ is of dimension strictly lower than $n$. Since all cells of dimension strictly lower than $n$ are both in the image of $j^+_n$ and in the image of $j^-_n$, $\varphi$ obviously factors through $j^+_n$ (and $j^+_n$).

\item[Case $k<n$:] Since every generating cell of $\gamma$ of $\Or_k$ is of dimension non-greater than$k$, the cell $\varphi(\gamma)$ is of dimension strictly lower than $n$. Since all cells of dimension strictly lower than $n$ are both in the image of $j^+_n$ and in the image of $j^-_n$, $\varphi$ obviously factors through $j^+_n$ (and $j^+_n$).

\item[Case $k=n$:] The image of $\alpha_n$ is either a non-trivial $n$\nbd{}cell of $\sS_n$ or a unit on a strictly lower dimensional cell. In the second situation, everything works like the case $k<n$. Now suppose for example that $\varphi(\alpha_n)$ is $h^+_n$, which is in the image of $j^+_n$. Since all of the other generating cells of $\Or_n$ are of dimension strictly lower than $n$, their images by $\varphi$ are also of dimension strictly lower than $n$ and hence, are all contained in the image of $j^+_n$. Altogether this proves that $\varphi$ factors through $j^+_n$. The case where $\varphi(\alpha_n)=h^-_n$ is symmetric.

\item[Case $k>n$:] Since $\sS_n$ is an $n$\nbd{}category, the image of $\alpha_k$ is necessarily of the form $\varphi(\alpha_k)=\1^k_{x}$ with $x$ a cell of $\sS_n$ of dimension non-greater than $n$. If the dimension of $x$ is strictly lower than $n$, then everything works like in the case $k<n$. If not, this means that $x$ is a non-trivial $n$\nbd{}cell of $\sS_n$. Suppose for example that $x=h^+_n$. Now let $\gamma$ be a generator of $\Or_k$ of dimension $k-1$. Necessarily, we have $\varphi(\gamma)=1^n_y$ for $y$ an $n$\nbd{}cell of dimension non-greater than $n$ (with the convention that if the dimension of $y$ is $n$, then $1^ny:=y$). If the dimension

\item[Case $k>n$:] Since $\sS_n$ is an $n$\nbd{}category, the image of $\alpha_k$ is necessarily of the form $\varphi(\alpha_k)=\1^k_{x}$ with $x$ a cell of $\sS_n$ of dimension non-greater than $n$. If the dimension of $x$ is strictly lower than $n$, then everything works like in the case $k<n$. If not, this means that $x$ is a non-trivial $n$\nbd{}cell of $\sS_n$. Suppose for example that $x=h^+_n$. Now let $\gamma$ be a generator of $\Or_k$ of dimension $k-1$. Either we have $\varphi(\gamma)=1^n_y$ for $y$ a cell of dimension strictly lower than $n$, or we have that $\varphi(\gamma)$ is a non-degenerate $n$\nbd{}cell of $\sS_n$. In the first situation, $y$ is in the image of $j^+_n$ as in the case $k<n$, and thus, so is $1^n_y$. In the second situation, this means \emph{a priori} that either $y=h_n^+$ or $y=h_n^-$. But we know that $\gamma$ is part of a composition that is equal to either the source or the target of $\alpha_k$ (see \ref{paragr:orientals}) and thus, $f(\gamma)$ is part of a composition that is equal to either the source or the target of $x=h^+_n$. Since no composition involving $h^-_n$ can be equal to $h^+_n$ (one could invoke the function introduced in \ref{prop:countingfunction}), this implies that $y=h_n^+$ and hence, $f(\gamma)$ is in the image of $j^n_+$. This goes for all generating cells of dimension $k-1$ of $\Or_k$ and we can recursively apply the same reasoning for generating cells of dimension $k-2$, then $k-3$ and so forth. Altogether, this proves that $\varphi$ factorizes through $j^+_n$. The case where $x=h^-_n$ and $\varphi$ factorizes through $j^-_n$ is symmetric.

\end{description}

\end{proof}

From these two lemmas, follows the important proposition below.

...

...

@@ -172,8 +172,8 @@ From these two lemmas, follows the important proposition below.

from Lemma \ref{lemma:hmtpycocartesianreedy} that the cocartesian square

\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]

\end{tikzcd}

\end{equation}

...

...

@@ -395,7 +395,7 @@ higher than $1$.

In particular, when we apply the previous lemma to $\mathrm{id}_A : A \to A$,

we obtain that every small category $A$ is (canonically isomorphic to) the colimit

\[

\colim_{a_0\in A} A/a_0.

\colim_{a_0\in A}(A/a_0).

\]

We now proceed to prove that this colimit is homotopic with respect to

folk weak equivalences.

...

...

@@ -578,7 +578,7 @@ We now recall an important Theorem due to Thomason.

\]

is a Thomason 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.

\hocolim_{a \in A}(A/a)\simeq A.

\]

A thorough analysis of all the isomorphisms involved (or see ) shows that this last isomorphism is indeed induced by the co-cone $(A/a \to A)_{a \in\Ob(A)}$.

\end{proof}

...

...

@@ -603,12 +603,12 @@ Putting all the pieces together, we are now able to prove the awaited Theorem.

\begin{itemize}[label=-]

\item The canonical map of $\ho(\oo\Cat^{\folk})$

\[

\hocolim_{a \in A}^{\folk} A/a \to A

\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

\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}.

@@ -74,7 +74,7 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv

%% \end{paragr}

\begin{remark}

The adjective ``singular'' is there to avoid future confusion with another homological invariant for $\oo$\nbd{}categories that will be introduced later. As a matter of fact, the underlying point of view adopted in this thesis is that \emph{singular homology of $\oo$-categories} ought to be simply called \emph{homology of $\oo$\nbd{}categories} as it is the only ``correct'' definition of homology. This assertation will be justified later. \todo{Le faire !}

The adjective ``singular'' is there to avoid future confusion with another homological invariant for $\oo$\nbd{}categories that will be introduced later. As a matter of fact, the underlying point of view adopted in this thesis is that \emph{singular homology of $\oo$-categories} ought to be simply called \emph{homology of $\oo$\nbd{}categories} as it is the only ``correct'' definition of homology. This assertion will be justified later. \todo{Le faire !}

\end{remark}

\begin{remark}

We could also have defined the homology of $\oo$\nobreakdash-category with $K : \Psh{\Delta}\to\Ch$ instead of $\kappa : \Psh{\Delta}\to\Ch$ since these two functors are quasi-isomorphic (see \cite[Theorem 2.4]{goerss2009simplicial} for example). An advantage of the later one is that it is left Quillen.

...

...

@@ -245,7 +245,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c

\end{align*}

\end{proof}

\begin{paragr}

Let $C$ be a \emph{free}$\oo$-category and write $\Sigma=(\Sigma_n)_{n \in\mathbb{N}}$ for its basis. For every $n \geq0$ and every $\alpha\in\Sigma_n$, recall that we have proved in Proposition \ref{prop:countingfunction} the existence of a unique functionr$w_{\alpha} : C_n \to\mathbb{N}$ such that:

Let $C$ be a \emph{free}$\oo$-category and write $\Sigma=(\Sigma_n)_{n \in\mathbb{N}}$ for its basis. For every $n \geq0$ and every $\alpha\in\Sigma_n$, recall that we have proved in Proposition \ref{prop:countingfunction} the existence of a unique function $w_{\alpha} : C_n \to\mathbb{N}$ such that:

@@ -319,7 +319,11 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\end{itemize}

\end{paragr}

\begin{example}\label{example:natisoplax}

Let $u,v : C \to D$ be two functors between (small) categories. Using the formulas of the previous paragraph, it is straightforward to check that the data of an oplax transformation from $u$ to $v$ consists exactly of the data of a natural transformation from $u$ to $v$.

When $C$ and $D$ are $n$\nbd{}categories with $n$ finite and $u,v :C \to D$ are two $n$\nbd{}functors, an oplax transformation $\alpha : u \Rightarrow v$ amounts to the data of a $(k+1)$\nbd{}cell $\alpha_x$ of $D$ for each $k$\nbd{}cell $x$ of $C$ with $0\leq k \leq n$, with source and target as in the previous paragraph. These data being subject to the axioms of the previous paragraph. Note that when $x$ is an $n$\nbd{}cell of $C$, $\alpha_x$ is necessarily a unit, which can be expressed as the equality

In particular, when $n=1$ and $C$ and $D$ are thus (small) categories, an oplax transformation $u \Rightarrow v$ is nothing but a natural transformation from $u$ to $v$.

\end{example}

\begin{paragr}

Let $u : C \to D$ be an $\oo$-functor. There is an oplax transformation from $u$ to $u$, denoted by $1_u$, which is defined as