@@ -262,9 +262,9 @@ From the previous proposition, we deduce the following very useful corollary.

\displaystyle\coprod_{x \in E}F_x \ar[r]\ar[d]& E \ar[d]&\\

A \ar[d,"\alpha"]\ar[r]& G \ar[d]\ar[r]& B \ar[d,"\delta"]\\

C \ar[r]& H \ar[r]& D

\ar[from=1-1,to=2-2,phantom,"\ulcorner" very near end,"\textcircled{\tiny\textbf{1}}" near start, description]

\ar[from=2-1,to=3-2,phantom,"\ulcorner" very near end,"\textcircled{\tiny\textbf{2}}", description]

\ar[from=2-2,to=3-3,phantom,"\ulcorner" very near end,"\textcircled{\tiny\textbf{3}}", description]

\ar[from=1-1,to=2-2,phantom,"\ulcorner" very near end,"\text{\textcircled{\tiny\textbf{1}}}" near start, description]

\ar[from=2-1,to=3-2,phantom,"\ulcorner" very near end,"\text{\textcircled{\tiny\textbf{2}}}", description]

\ar[from=2-2,to=3-3,phantom,"\ulcorner" very near end,"\text{\textcircled{\tiny\textbf{3}}}", description]

\end{tikzcd}.

\]

What we want to prove is that the image by the functor $L$ of the pasting of squares \textcircled{\tiny\textbf{2}} and \textcircled{\tiny\textbf{3}} is homotopy cocartesian. Since the morphism $G \to B$ is a monomorphism, we deduce from Corollary \ref{cor:hmtpysquaregraph} that the image by the functor $L$ of square \textcircled{\tiny\textbf{3}} is homotopy cocartesian. Hence, all we have to show is that the image by $L$ of square \textcircled{\tiny\textbf{2}} is homotopy cocartesian. \todo{Parler du pasting lemma ?} On the other hand, we know that both morphisms

...

...

@@ -689,3 +689,61 @@ It also follows from Lemma \ref{lemma:binervthom} that the bisimplicial nerve in

This is an immediate consequence of Proposition \ref{prop:streetvsbisimplicial} and Corollary \ref{cor:bisimplicialsquare}.

\end{proof}

\section{Zoology of $2$-categories : Basic examples}

\begin{paragr}

Let $n,m \geq0$. We denote by $A_{(m,n)}$ the free $2$-category with only one generating $2$-cell whose source is a chain of length $m$ and its target a chain of length $n$:

We are going to prove that if $n\neq0$ or $m\neq0$, then $A_{(m,n)}$ is \good{} and has the homotopy type of a point. When $m\neq0$\emph{and}$n\neq0$, this result is not surprising, but when $n=0$ or $m=0$ (but not both), it is \emph{a priori} less clear what the homotopy type of $A_{(m,n)}$ is and whether it is \good{} or not. For example, $A_{(1,0)}$ can be pictured as follows

\[

\begin{tikzcd}

A \ar[r, bend left=70, "f",""{name=A,below}]\ar[r,bend right=70,"1_A"',""{name=B,above}]& A \ar[from=A,to=B,Rightarrow,"\alpha"]

\end{tikzcd},

\]

and has many non trivial $2$-cells, such as $f\comp_0\alpha\comp_0 f$.

Note that when $m=0$\emph{and}$n=0$, then the $2$-category $A_{(0,0)}$ is nothing but the commutative monoid $\mathbb{N}$ seen as a $2$-category (\ref{paragr:bubble}) and we have already seen that it is \emph{not}\good{}. We shall refer to this $2$-category as the \emph{bubble}.

\end{paragr}

\begin{paragr}

Recall that for $n\geq0$, we denote by $\Delta_n$ the linear order ${0\leq\cdots\leq n}$ seen as a small category. Let $i : \Delta_1\to\Delta_n$ be the unique functor such that

\[

i(0)=0\text{ and } i(1)=n.

\]

\end{paragr}

\begin{lemma}

For $n\neq0$, the functor $i : \Delta_1\to\Delta_n$ is a strong deformation retract (\ref{paragr:defrtract}).

\end{lemma}

\begin{proof}

Let $r : \Delta_n \to\Delta_1$ the unique functor such that

\[

r(n)=1\text{ and } r(k)=1\text{ for } k\neq n.

\]

By definition we have $r \circ i =1_{\Delta_1}$. Now, the natural order on $\Delta_n$ induces a natural transformation

\[

\alpha : i\circ r \Rightarrow\mathrm{id}_{\Delta_n},

\]

and it is straightforward to check that $\alpha\ast i =\mathrm{id}_i$.

\todo{Est-ce qu'il ne faudra pas dire quelque part qu'une transfo naturelle donne une transfo oplax}.

\end{proof}

\begin{paragr}

In particular, it follows from Lemma \ref{lemma:pushoutstrngdefrtract} that if $n\neq0$, then $i : \Delta_1\to\Delta_n$ is a co-universal Thomason weak equivalences. Now consider the following cocartesian square

\[

\begin{tikzcd}

\Delta_1\ar[r,"i"]\ar[d,"\tau"]&\Delta_n \ar[d]\\

A_{(1,1)}\ar[r]& A_{(1,n)}

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

\end{tikzcd},

\]

where $\sigma : \Delta_1\to A_{(1,1)}$ is the $2$-functor that points to the the target of the generating $2$-cell of $A_{(1,1)}$

\[

\tau=\begin{tikzcd}{0\to1}\ar[d,"\langle f \rangle"]\\ A \end{tikzcd}