\chapter{Homotopy and homology type of free 2-categories}
\chaptermark{Homology of free $2$-categories}
\section{Preliminaries: the case of free 1-categories}\label{section:prelimfreecat}
In this section, we review some homotopical results on free
($1$-)categories that will be of great help in the sequel.
\begin{paragr}
A \emph{reflexive graph} $G$ consists of the data of two sets $G_0$ and $G_1$
together with
\begin{itemize}[label=-]
\item a ``source'' map $\src : G_1 \to G_0$,
\item a ``target'' map $\trgt : G_1 \to G_0$,
\item a ``unit'' map $1_{(-)} : G_0 \to G_1$,
\end{itemize}
such that for every $x \in G_0$,
\[
\src(1_{x}) = \trgt (1_{x}) = x.
\]
The vocabulary of categories is used: elements of $G_0$ are \emph{objects} or
\emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells},
arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A
\emph{morphism of reflexive graphs} $ f : G \to G'$ consists of maps $f_0 :
G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and
units in an obvious sense. This defines the category $\Rgrph$ of reflexive
graphs. Later we will make use of monomorphisms in the category $\Rgrph$; they
are the morphisms $f : G \to G'$ that are injective on objects and on arrows,
i.e. such that $f_0 : G_0 \to G_0'$ and $f_1 : G_1 \to G'_1$ are injective.
There is a ``underlying reflexive graph'' functor
\[
U : \Cat \to \Rgrph,
\]
which has a left adjoint
\[
L : \Rgrph \to \Cat.
\]
For a reflexive graph $G$, the objects of $L(G)$ are exactly the objects of
$G$ and an arrow $f$ of $L(G)$ is a chain
\[
\begin{tikzcd}
X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1}
\ar[r,"f_n"]& X_{n}
\end{tikzcd}
\]
of arrows of $G$, such that \emph{none} of the $f_k$ are units. The integer
$n$ is referred to as the \emph{length} of $f$ and is denoted by $\ell(f)$.
Composition is given by concatenation of chains.
\end{paragr}
\begin{lemma}
A category $C$ is free in the sense of \ref{def:freeoocat} if and only if
there exists a reflexive graph $G$ such that
\[
C \simeq L(G).
\]
\end{lemma}
\begin{proof}
If $C$ is free, consider the reflexive graph $G$ such that $G_0 = C_0$ and
$G_1$ is the subset of $C_1$ whose elements are either generating $1$-cells of
$C$ or units. It is straightforward to check that $C\simeq L(G)$.
Conversely, if $C \simeq L(G)$ for some reflexive graph $G$, then the
description of the arrows of $L(G)$ given in the previous paragraph shows that
$C$ is free and that its set of generating $1$-cells is (isomorphic to) the
set of non unital $1$-cells of $G$.
\end{proof}
\begin{remark}
In other words, a category is free on a graph if and only if it is free on a
reflexive graph. The difference between these two notions is at the level of
morphisms: there are more morphisms of reflexive graphs because (generating)
$1$\nbd{}cells may be sent to units. Hence, for a morphism of reflexive graphs
$f : G \to G'$, the induced functor $L(f)$ is not necessarily rigid in the
sense of Definition \ref{def:rigidmorphism}.
\end{remark}
\begin{paragr}
There is another important description of the category $\Rgrph$. Write
$\Delta_{\leq 1}$ for the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$.
The category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq 1}}$, the
category of pre-sheaves on $\Delta_{\leq 1}$. In particular, the canonical
inclusion $i : \Delta_{\leq 1} \rightarrow \Delta$ induces by pre-composition
a functor
\[
i^* : \Psh{\Delta} \to \Rgrph,
\]
which, by the usual technique of Kan extensions, has a left adjoint
\[
i_! : \Rgrph \to \Psh{\Delta}.
\]
For a graph $G$, the simplicial set $i_!(G)$ has $G_0$ as its set of
$0$-simplices, $G_1$ as its set of $1$-simplices and all $k$-simplices are
degenerate for $k>1$. For future reference, we put here the following lemma.
\end{paragr}
\begin{lemma}\label{lemma:monopreserved}
The functor $i_! : \Rgrph \to \Psh{\Delta}$ preserves monomorphisms.
\end{lemma}
\begin{proof}
What we need to show is that, given a morphism of simplicial sets
\[
f : X \to Y,
\]
if $f_0 : X_0 \to Y_0$ and $f_1 : X_1 \to Y_1$ are monomorphisms and if all
$n$\nbd{}simplices of $X$ are degenerate for $n\geq 2$, then $f$ is a
monomorphism. A proof of this assertion is contained in \cite[Paragraph
3.4]{gabriel1967calculus}. The key argument is the Eilenberg--Zilber Lemma
(Proposition 3.1 of op. cit.).
\end{proof}
\begin{paragr}
Let us denote by $N : \Psh{\Delta} \to \Cat$ (instead of $N_1$ as in Paragraph
\ref{paragr:nerve}) the usual nerve of categories and by $c : \Cat \to
\Psh{\Delta}$ its left adjoint. Recall that for a (small) category $C$, an
$n$-simplex of $N(C)$ is a chain
\[
\begin{tikzcd}
X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1}
\ar[r,"f_n"]& X_{n}
\end{tikzcd}
\]
of arrows of $C$. Such an $n$-simplex is degenerate if and only if at least
one of the $f_k$ is a unit. It is straightforward to check that the composite
of
\[
\Cat \overset{N}{\rightarrow} \Psh{\Delta} \overset{i^*}{\rightarrow} \Rgrph
\]
is nothing but the forgetful functor $U : \Cat \to \Rgrph$. Thus, the functor
$L : \Rgrph \to \Cat$ is (isomorphic to) the composite of
\[
\Rgrph \overset{i_!}{\rightarrow} \Psh{\Delta} \overset{c}{\rightarrow}
\Cat.
\]
We now review a construction due to Dwyer and Kan
(\cite{dwyer1980simplicial}). Let $G$ be a reflexive graph. For
every $k\geq 1$, we define the simplicial set $N^k(G)$ as the
sub-simplicial set of $N(L(G))$ whose $n$-simplices are chains
\[
\begin{tikzcd}
X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1}
\ar[r,"f_n"]& X_{n}
\end{tikzcd}
\]
of arrows of $L(G)$ such that
\[
\sum_{1 \leq i \leq n}\ell(f_i) \leq k.
\] In particular, we have
\[
N^1(G)=i_!(G)
\]
and the transfinite composition of
\[
i_!(G) = N^1(G) \hookrightarrow N^2(G) \hookrightarrow \cdots
\hookrightarrow N^{k}(G) \hookrightarrow N^{k+1}(G) \hookrightarrow \cdots
\]
is easily seen to be the map
\[
\eta_{i_!(G)} : i_!(G) \to Nci_!(G),
\]
where $\eta$ is the unit of the adjunction $c \dashv N$.
\end{paragr}
\begin{lemma}[Dwyer--Kan]\label{lemma:dwyerkan}
For every $k\geq 1$, the canonical inclusion map
\[
N^{k}(G) \to N^{k+1}(G)
\]
is a trivial cofibration of simplicial sets.
\end{lemma}
\begin{proof}
Let $A_{k+1}=\mathrm{Im}(\partial_0)\cup\mathrm{Im}(\partial_{k+1})$ be the
union of the first and last face of the standard $(k+1)$-simplex
$\Delta_{k+1}$. Notice that the canonical inclusion
\[
A_{k+1} \hookrightarrow \Delta_{k+1}
\]
is a trivial cofibration. Let $I_{k+1}$ be the set of chains
\[
\begin{tikzcd}
f = X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1}
\ar[r,"f_k"]& X_{k}\ar[r,"f_{k+1}"]& X_{k+1}
\end{tikzcd}
\]
of arrows of $L(G)$ such that for every $1 \leq i \leq k+1$
\[
\ell(f_i)=1,
\]
i.e.\ each $f_i$ is a non-unit arrow of $G$. For every $f \in I_{k+1}$, we
define a morphism $\varphi_f : A_{k+1} \to N^{k}(G)$ in the following fashion:
\begin{itemize}
\item[-]$\varphi_{f}\vert_{\mathrm{Im}(\partial_0)}$ is the $k$-simplex of
$N^{k}(G)$
\[
\begin{tikzcd}
X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k} \ar[r,"f_{k+1}"]&
X_{k+1},
\end{tikzcd}
\]
\item[-] $\varphi_{f}\vert_{\mathrm{Im}(\partial_{k+1})}$ is the $k$-simplex
of $N^{k}(G)$
\[
\begin{tikzcd}
X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1}
\ar[r,"f_k"]& X_{k}.
\end{tikzcd}
\]
\end{itemize}
All in all, we have a cocartesian square
\[
\begin{tikzcd}
\displaystyle \coprod_{f \in I_{k+1}}A_{k+1} \ar[d] \ar[r,"(\varphi_f)_f"] & N^{k}(G)\ar[d] \\
\displaystyle \coprod_{f \in I_{k+1}}\Delta_{k+1} \ar[r] & N^{k+1}(G),
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
which proves that the right vertical arrow is a trivial cofibration.
\end{proof}
From this lemma, we deduce the following proposition.
\begin{proposition}
Let $G$ be a reflexive graph. The map
\[
\eta_{i_!(G)} : i_!(G) \to Nci_!(G),
\]
where $\eta$ is the unit of the adjunction $c \dashv N$, is a trivial
cofibration of simplicial sets.
\end{proposition}
\begin{proof}
This follows from the fact that trivial cofibrations are stable by transfinite
composition.
\end{proof}
From the previous proposition, we deduce the following very useful corollary.
\begin{corollary}\label{cor:hmtpysquaregraph}
Let
\[
\begin{tikzcd}
A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\
C \ar[r,"\gamma"]& D
\end{tikzcd}
\]
be a cocartesian square in $\Rgrph$. If either $\alpha$ or $\beta$ is a
monomorphism, then the induced square of $\Cat$
\[
\begin{tikzcd}
L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"]& L(B) \ar[d,"L(\delta)"] \\
L(C) \ar[r,"L(\gamma)"]& L(D)
\end{tikzcd}
\]
is a Thomason homotopy cocartesian.
\end{corollary}
\begin{proof}
Since the nerve $N$ induces an equivalence of op-prederivators
\[
\Ho(\Cat^{\Th}) \to \Ho(\Psh{\Delta}),
\]
it suffices to prove that the induced square of simplicial sets
\[
\begin{tikzcd}
NL(A) \ar[d,"NL(\alpha)"] \ar[r,"NL(\beta)"]& NL(B) \ar[d,"NL(\delta)"] \\
NL(C) \ar[r,"NL(\gamma)"]& NL(D)
\end{tikzcd}
\]
is homotopy cocartesian. But, since $L \simeq c\circ i_!$, it follows from
Lemma \ref{cor:hmtpysquaregraph} that this last square is weakly equivalent to
the square of simplicial sets
\[
\begin{tikzcd}
i_!(A) \ar[d,"i_!(\alpha)"] \ar[r,"i_!(\beta)"] &i_!(B) \ar[d,"i_!(\delta)"] \\
i_!(C) \ar[r,"i_!(\gamma)"]& i_!(D).
\end{tikzcd}
\]
This square is cocartesian because $i_!$ is a left adjoint. Since
$i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the
result follows from the fact that the monomorphisms are the
cofibrations of the standard Quillen model structure on simplicial
sets and from Lemma \ref{lemma:hmtpycocartesianreedy}.
\end{proof}
\begin{paragr}
By working a little more, we obtain the more general result stated
in the proposition below. Let us say that a morphism of reflexive
graphs $\alpha : A \to B$ is \emph{quasi-injective on arrows} when
for all $f$ and $g$ arrows of $A$, if
\[
\alpha(f)=\alpha(g),
\]
then either $f=g$ or $f$ and $g$ are both units. In other words, $\alpha$
never sends a non-unit arrow to a unit arrow and $\alpha$ never identifies two
non-unit arrows. It follows that if $\alpha$ is quasi-injective on arrows and
injective on objects, then it is also injective on arrows and hence, a
monomorphism of $\Rgrph$.
\end{paragr}
\begin{proposition}\label{prop:hmtpysquaregraphbetter}
Let
\[
\begin{tikzcd}
A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\
C \ar[r,"\gamma"]& D
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
be a cocartesian square in $\Rgrph$. Suppose that the following two conditions
are satisfied
\begin{enumerate}[label=\alph*)]
\item Either $\alpha$ or $\beta$ is injective on objects.
\item Either $\alpha$ or $\beta$ is quasi-injective on arrows.
\end{enumerate}
Then, the induced square of $\Cat$
\[
\begin{tikzcd}
L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] &L(B) \ar[d,"L(\delta)"] \\
L(C) \ar[r,"L(\gamma)"] &L(D)
\end{tikzcd}
\]
is Thomason homotopy cocartesian square.
\end{proposition}
\begin{proof}
The case where $\alpha$ or $\beta$ is both injective on objects and
quasi-injective on arrows is Corollary \ref{cor:hmtpysquaregraph}. Hence, we
only have to treat the case when $\alpha$ is injective on objects and $\beta$
is quasi-injective on arrows; the remaining case being symmetric.
Let use denote by $E$ the set of objects of $B$ that are in the image of
$\beta$. We consider this set as well as the set $A_0$ of objects of $A$ as discrete reflexive graphs, i.e.\ reflexive graphs
with no non-unit arrows. Now, let $G$ be the reflexive graph defined by the
following cocartesian square
\[
\begin{tikzcd}
A_0\ar[r] \ar[d] & E \ar[d]\\
A \ar[r] & G, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd}
\]
where the morphism \[ A_0 \to A\] is the canonical inclusion, and the
morphism \[A_0 \to E\] is induced by the restriction of $\beta$ on objects. In other words, $G$ is
obtained from $A$ by collapsing the objects that are identified through
$\beta$. It admits the following explicit description: $G_0$ is (isomorphic
to) $E$ and the set of non-unit arrows of $G$ is (isomorphic to) the set of
non-unit arrows of $A$; the source (resp.\ target) of a non-unit arrow $f$ of
$G$ is the source (resp.\ target) of $\beta(f)$. This completely describes $G$.
% Notice also for later reference that the morphism \[ \coprod_{x \in E}F_x
% \to A\] is a monomorphism, i.e. injective on objects and arrows.
Now, we have the following solid arrow commutative diagram
\[
\begin{tikzcd}
A_0 \ar[r] \ar[d] & E \ar[ddr,bend left]\ar[d]&\\
A \ar[drr,bend right,"\beta"'] \ar[r] & G \ar[dr, dotted]&\\
&&B, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd}
\]
where the arrow $E \to B$ is the canonical inclusion. Hence, by universal
property, the dotted arrow exists and makes the whole diagram commute. A
thorough verification easily shows that the morphism $G \to B$ is a
monomorphism of $\Rgrph$.
By forming successive cocartesian squares and combining with the square
obtained earlier, we obtain a diagram of three cocartesian squares:
\[
\begin{tikzcd}[row sep = large]
A_0\ar[r] \ar[d] & E \ar[d]&\\
A \ar[d,"\alpha"] \ar[r] & G \ar[d] \ar[r] & B \ar[d,"\delta"]\\
C \ar[r] & H \ar[r] & D. \ar[from=1-1,to=2-2,phantom,"\ulcorner" very near
end,"\text{\textcircled{\tiny \textbf{1}}}" near start, description]
\ar[from=2-1,to=3-2,phantom,"\ulcorner" very near
end,"\text{\textcircled{\tiny \textbf{2}}}", description]
\ar[from=2-2,to=3-3,phantom,"\ulcorner" very near
end,"\text{\textcircled{\tiny \textbf{3}}}", description]
\end{tikzcd}
\]
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,
in virtue of Lemma \ref{lemma:pastinghmtpycocartesian}, all we have to show is
that the image by $L$ of square \textcircled{\tiny \textbf{2}} is homotopy
cocartesian. On the other hand, the morphisms
\[
A_0 \to A
\]
and
\[
A_0 \to C
\]
are monomorphisms and thus, using Corollary
\ref{cor:hmtpysquaregraph}, we deduce that the image by $L$ of square
\textcircled{\tiny \textbf{1}} and of the pasting of squares
\textcircled{\tiny \textbf{1}} and \textcircled{\tiny \textbf{2}}
are homotopy cocartesian. By Lemma \ref{lemma:pastinghmtpycocartesian} again, this proves that the image by $L$ of
square \textcircled{\tiny \textbf{2}} is homotopy cocartesian.
\end{proof}
We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\ref{prop:hmtpysquaregraphbetter} to a few examples.
\begin{example}[Identifying two objects]\label{example:identifyingobjects}
Let $C$ be a free category, $A$ and $B$ be two objects of $C$ with $A\neq B$ and let $C'$ be
the category obtained from $C$ by identifying $A$ and $B$, i.e.\ defined by
the following cocartesian square
\[
\begin{tikzcd}
\sS_0 \ar[d] \ar[r,"{\langle A,B \rangle}"] & C \ar[d] \\
\sD_0 \ar[r] & C'.
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
Then, this square is Thomason homotopy cocartesian. Indeed, it is obviously
the image by the functor $L$ of a cocartesian square of $\Rgrph$ and the top
morphism is a monomorphism. Hence, we can apply Corollary \ref{cor:hmtpysquaregraph}.
\end{example}
\begin{example}[Adding a generator]
Let $C$ be a free category, $A$ and $B$ two objects of $C$ (possibly equal)
and let $C'$ be the category obtained from $C$ by adding a generator $A \to
B$, i.e.\ defined by the following cocartesian square:
\[
\begin{tikzcd}
\sS_0 \ar[d,"i_1"] \ar[r,"{\langle A, B \rangle}"] & C \ar[d] \\
\sD_1 \ar[r] & C'.
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
Then, this square is Thomason homotopy cocartesian. Indeed, it obviously is the image of a square of $\Rgrph$ by
the functor $L$ and the morphism $i_1 : \sS_0 \to \sD_1$ comes from
a monomorphism of $\Rgrph$. Hence, we can apply Corollary
\ref{cor:hmtpysquaregraph}.
\end{example}
\begin{remark}
Since $i_1 : \sS_0 \to \sD_1$ is a folk cofibration% , since a Thomason homotopy
% cocartesian square in $\Cat$ is also so in $\oo\Cat$
and since every free category is obtained by recursively adding generators
starting from a set of objects (seen as a $0$-category), the previous example
yields another proof that \emph{free} (1\nbd{})categories are \good{} (which we
already knew since we have seen that \emph{all} (1-)categories are \good{}).
\end{remark}
\begin{example}[Identifying two generators]
Let $C$ be a free category and let $f,g : A \to B$ be parallel generating arrows of
$C$ such that $f\neq g$. Now consider the category $C'$ obtained from $C$ by
``identifying'' $f$ and $g$, i.e. defined by the following cocartesian
square
\[
\begin{tikzcd}
\sS_1\ar[d] \ar[r,"{\langle f, g \rangle}"] &C \ar[d] \\
\sD_1 \ar[r] & C',
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
where the morphism $\sS_1 \to \sD_1$ is the one that sends the two generating
arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square
is Thomason homotopy cocartesian.
Indeed, it is the image by the functor $L$ of a cocartesian square in
$\Rgrph$, the morphism $\sS_1 \to \sD_1$ is injective on objects and the
morphism $\sS_1 \to C$ is quasi-injective on arrows. Hence, we can apply
Proposition \ref{prop:hmtpysquaregraphbetter}. Note that since we did
\emph{not} suppose that $A\neq B$, the top morphism of the previous square is
not necessarily a monomorphism and we cannot always apply Corollary
\ref{cor:hmtpysquaregraph}.
\end{example}
\begin{example}[Killing a generator]\label{example:killinggenerator}
Let $C$ be a free category and let $f : A \to B$ be one of its generating arrows
such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by
``killing'' $f$, i.e. defined by the following cocartesian square:
\[
\begin{tikzcd}
\sD_1 \ar[d] \ar[r,"\langle f \rangle"] & C \ar[d] \\
\sD_0 \ar[r] & C'.
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
Then, this above square is Thomason homotopy cocartesian. Indeed, it
obviously is the image of a cocartesian square in $\Rgrph$ by the
functor $L$ and since the source and target of $f$ are different,
the top map comes from a monomorphism of $\Rgrph$. Hence, we can
apply Corollary \ref{cor:hmtpysquaregraph}.
\end{example}
\begin{remark}
Note that in the previous example, we see that it was useful to consider the
category of reflexive graphs and not only the category of graphs because the
map $\sD_1 \to \sD_0$ does not come from a morphism in the category of graphs.
Note also that the hypothesis that $A\neq B$ was fundamental in the previous
example as for $A=B$ the square is \emph{not} Thomason homotopy cocartesian.
\end{remark}
\section{Preliminaries: bisimplicial sets}
\begin{paragr}
A \emph{bisimplicial set} is a presheaf over the category $\Delta \times
\Delta$,
\[
X : \Delta^{\op} \times \Delta^{\op} \to \Set.
\]
In a similar fashion as for simplicial sets (\ref{paragr:simpset}), for $n,m
\geq 0$, we use the notations
\begin{align*}
X_{n,m} &:= X([n],[m]) \\
\partial_i^h &:=X(\delta^i,\mathrm{id}) : X_{n+1,m} \to X_{n,m}\\
\partial_j^v &:=X(\mathrm{id},\delta^j) : X_{n,m+1} \to X_{n,m}\\
s_i^h &:=X(\sigma^i,\mathrm{id}): X_{n,m} \to X_{n+1,m}\\
s_j^v&:=X(\mathrm{id},\sigma^j) : X_{n,m} \to X_{n,m+1}.
\end{align*}
The maps $\partial_i^h$ and $s_i^h$ will be referred to as the
\emph{horizontal} face and degeneracy operators; and $\partial_i^v$ and
$s_i^v$ as the \emph{vertical} face and degeneracy operators.
Note that for every $n\geq 0$, we have simplicial sets
\begin{align*}
X_{\bullet,n} : \Delta^{\op} &\to \Set \\
[k] &\mapsto X_{k,n}
\end{align*}
and
\begin{align*}
X_{n,\bullet} : \Delta^{\op} &\to \Set \\
[k] &\mapsto X_{n,k}.
\end{align*}
The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$.
\iffalse Moreover for every $n \geq 0$, if we fix the first variable to $n$,
we obtain a simplicial set
\begin{align*}
X_{n,\bullet} : \Delta^{\op} &\to \Set \\
[m] &\mapsto X_{n,m}.
\end{align*}
Similarly, if we fix the second variable to $n$, we obtain a simplicial
\begin{align*}
X_{\bullet,n} : \Delta^{\op} &\to \Set \\
[m] &\mapsto X_{m,n}.
\end{align*}
The correspondences
\[
n \mapsto X_{n,\bullet} \,\text{ and }\, n\mapsto X_{\bullet,n}
\]
actually define functors $\Delta \to \Psh{\Delta}$. They correspond to the two
``currying'' operations
\[
\Psh{\Delta\times\Delta} \to \underline{\Hom}(\Delta^{\op},\Psh{\Delta}),
\]
which are isomorphisms of categories. In other words, the category of
bisimplicial sets can be identified with the category of functors
$\underline{\Hom}(\Delta^{\op},\Psh{\Delta})$ in two canonical ways. \fi
\end{paragr}
\begin{paragr}
The functor
\begin{align*}
\delta : \Delta &\to \Delta\times\Delta \\
[n] &\mapsto ([n],[n])
\end{align*}
induces by pre-composition a functor
\[
\delta^* : \Psh{\Delta\times\Delta} \to \Psh{\Delta}.
\]
By the usual calculus of Kan extensions, $\delta^*$ admits a left adjoint
$\delta_!$ and a right adjoint $\delta_*$
\[
\delta_! \dashv \delta^* \dashv \delta_*.
\]
We say that a morphism $f : X \to Y$ of bisimplicial sets is a \emph{diagonal
weak equivalence} (resp.\ \emph{diagonal fibration}) when $\delta^*(f)$ is a
weak equivalence (resp.\ fibration) of simplicial sets. By
definition, $\delta^*$ induces a morphism of op-prederivators
\[
\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to
\Ho(\Psh{\Delta}).
\]
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. We shall refer to this model
structure as the \emph{diagonal model structure}.
\end{paragr}
\begin{proposition}\label{prop:diageqderivator}
Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model
structure. Then, the adjunction
\[
\begin{tikzcd}
\delta_! : \Psh{\Delta} \ar[r,shift left] & \Psh{\Delta\times\Delta}
\ar[l,shift left]: \delta^*,
\end{tikzcd}
\]
is a Quillen equivalence.
\end{proposition}
\begin{proof}
By definition $\delta^*$ preserves weak equivalences and fibrations and thus,
the adjunction is a Quillen adjunction. The fact that $\delta^*$ induces an
equivalence at the level of homotopy categories is \cite[Proposition
1.2]{moerdijk1989bisimplicial}.
\end{proof}
\begin{paragr}
In particular, the morphism of op-prederivators
\[
\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to
\Ho(\Psh{\Delta})
\]
is actually an equivalence of op-prederivators.
\end{paragr}
Diagonal weak equivalences are not the only interesting weak equivalences for
bisimplicial sets.
\begin{paragr}
A morphism $f : X \to Y$ of bisimplicial sets is a \emph{vertical (resp.\
horizontal) weak equivalence} when for every $n \geq 0$, the induced
morphism of simplicial sets
\[
f_{\bullet,n} : X_{\bullet,n} \to Y_{\bullet,n}
\]
(resp.
\[
f_{n,\bullet} : X_{n,\bullet} \to Y_{n,\bullet})
\]
is a weak equivalence of simplicial sets. Recall now a very useful lemma.
\end{paragr}
\begin{lemma}\label{bisimpliciallemma}
Let $f : X \to Y$ be a morphism of bisimplicial sets. If $f$ is a vertical or
horizontal weak equivalence then it is a diagonal weak equivalence.
\end{lemma}
\begin{proof}
See for example \cite[Chapter XII,4.3]{bousfield1972homotopy} or
\cite[Proposition 2.1.7]{cisinski2004localisateur}.
\end{proof}
\begin{paragr}
In particular, the identity functor of the category of bisimplicial sets
induces the morphisms of op-prederivators:
\[
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\]
and
\[
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}).
\]
\end{paragr}
\begin{proposition}\label{prop:bisimplicialcocontinuous}
The morphisms of op-prederivators
\[
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\]
and
\[
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\]
are homotopy cocontinuous.
\end{proposition}
\begin{proof}
Recall that the category of bisimplicial sets can be equipped with a model
structure where the weak equivalences are the vertical (resp.\ horizontal)
weak equivalences and the cofibrations are the monomorphisms (see for example
\cite[Chapter IV]{goerss2009simplicial} or \cite{cisinski2004localisateur}).
We respectively refer to these model structures as the \emph{vertical model
structure} and \emph{horizontal model structure}. Since the functor
$\delta^* : \Psh{\Delta\times\Delta} \to \Psh{\Delta}$ preserves
monomorphisms, it follows from Lemma \ref{bisimpliciallemma} that the
adjunction
\[
\begin{tikzcd}
\delta^* : \Psh{\Delta\times\Delta} \ar[r,shift left] & \ar[l,shift left]
\Psh{\Delta} : \delta_*
\end{tikzcd}
\]
is a Quillen adjunction when $\Psh{\Delta\times\Delta}$ is equipped with
either the vertical model structure or the horizontal model structure. In
particular, the induced morphisms of op-prederivators
\[
\overline{\delta^*} : \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to
\Ho(\Psh{\Delta})
\]
and
\[
\overline{\delta^*} : \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to
\Ho(\Psh{\Delta})
\]
are homotopy cocontinuous. Now, the obvious identity
$\delta^*=\delta^* \circ \mathrm{id}_{\Psh{\Delta\times\Delta}}$ implies that
we have commutative triangles
\[
\begin{tikzcd}
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \ar[r]
\ar[rd,"\overline{\delta^*}"']&
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\ar[d,"\overline{\delta^*}"] \\
&\Ho(\Psh{\Delta})
\end{tikzcd}
\]
and
\[
\begin{tikzcd}
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \ar[r]
\ar[rd,"\overline{\delta^*}"']&
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\ar[d,"\overline{\delta^*}"] \\
&\Ho(\Psh{\Delta}).
\end{tikzcd}
\]
The result follows then from the fact that $\overline{\delta^*} :
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \to \Ho(\Psh{\Delta})$ is an
equivalence of op-prederivators.
\end{proof}
In practice, we will use the following corollary.
\begin{corollary}\label{cor:bisimplicialsquare}
Let
\[
\begin{tikzcd}
A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\
C \ar[r,"v"] & D
\end{tikzcd}
\]
be a commutative square in the category of bisimplicial sets satisfying at least one of the two
following conditions:
\begin{enumerate}[label=(\alph*)]
\item For every $n\geq 0$, the square of simplicial sets
\[
\begin{tikzcd}
A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\
C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n}
\end{tikzcd}
\]
is homotopy cocartesian.
\item For every $n\geq 0$, the square of simplicial sets
\[
\begin{tikzcd}
A_{n,\bullet} \ar[r,"{u_{n,\bullet}}"]\ar[d,"{f_{n,\bullet}}"] & B_{n,\bullet} \ar[d,"{g_{n,\bullet}}"] \\
C_{n,\bullet} \ar[r,"{v_{n,\bullet}}"] & D_{n,\bullet}
\end{tikzcd}
\]
is homotopy cocartesian.
\end{enumerate}
Then, the square
\[
\begin{tikzcd}
\delta^*(A) \ar[r,"\delta^*(u)"]\ar[d,"\delta^*(f)"] & \delta^*(B) \ar[d,"\delta^*(g)"] \\
\delta^*(C) \ar[r,"\delta^*(v)"] & \delta^*(D)
\end{tikzcd}
\]
is a homotopy cocartesian square of simplicial sets.
\end{corollary}
\begin{proof}
From \cite[Corollary 10.3.10(i)]{groth2013book} we know that the square of
bisimplicial sets
\[
\begin{tikzcd}
A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\
C \ar[r,"v"] & D
\end{tikzcd}
\]
is homotopy cocartesian with respect to the vertical weak equivalences if and
only if for every $n\geq 0$, the square
\[
\begin{tikzcd}
A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\
C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n}
\end{tikzcd}
\]
is a homotopy cocartesian square of simplicial sets and similarly for
horizontal weak equivalences. The result follows then from Proposition
\ref{prop:bisimplicialcocontinuous}.
\end{proof}
\section{Bisimplicial nerve for 2-categories}\label{section:bisimplicialnerve}
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 nerve defined in \ref{paragr:nerve}.
\begin{notation}
\begin{itemize}
\item[-] Once again, we write $N : \Cat \to \Psh{\Delta}$ instead of $N_1$ for
the usual nerve of categories. Moreover, using the usual notation for the
set of $k$-simplices of a simplicial set, if $C$ is a (small) category, then
\[
N(C)_k
\]
is the set of $k$-simplices of the nerve of $C$.
\item[-] Similarly, we write $N : 2\Cat \to \Psh{\Delta}$ instead of $N_2$ for
the nerve of $2$-categories. This makes sense since the nerve for categories
is the restriction of the nerve for $2$-categories.
\item[-] For $2$-categories, we refer to the $\comp_0$-composition of
$2$-cells as the \emph{horizontal composition} and the $\comp_1$-composition
of $2$-cells as the \emph{vertical composition}.
\item[-] For a $2$-category $C$ and $x$ and $y$ objects of $C$, we denote by
\[
C(x,y)
\]
the category whose objects are the $1$-cells of $C$ with $x$ as source and
$y$ as target, and whose arrows are the $2$-cells of $C$ with $x$ as
$0$-source and $y$ as $0$-target. Composition is induced by vertical
composition in $C$.
\end{itemize}
\end{notation}
\begin{paragr}
Every $2$-category $C$ defines a simplicial object in $\Cat$,
\[S(C): \Delta^{\op} \to \Cat,\] where, for each $n \geq 0$,
\[
S_n(C):= \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}C(x_0,x_1)
\times \cdots \times C(x_{n-1},x_n).
\]
Note that for $n=0$, the above formula reads $S_0(C)=C_0$. For $n>0$, the face operators $\partial_i : S_{n}(C) \to S_{n-1}(C)$ are induced by horizontal
composition for $0 < i =
2em,Rightarrow]\end{tikzcd}}^{m}}_{n}\] More formally, $A_{(m,n)}$ is
described in the following way:
\begin{itemize}[label=-]
\item generating $0$-cells: $A_0,\cdots, A_m$, $B_1,\cdots,B_{n-1}$
\end{itemize}
(and for convenience, we also set $B_0:=A_0$ and $B_n:=A_m$)
\begin{itemize}[label=-]
\item generating $1$-cells: $\begin{cases} f_{i+1} : A_i \to A_{i+1} & \text{
for } 0\leq i \leq m-1 \\ g_{j+1} : B_j \to B_{j+1} &\text{ for } 0 \leq
j \leq n-1 %\\g_1 : A_0 \to B_1 & \\g_{n} : B_{n-1} \to A_m &
\end{cases}$
\item generating $2$-cell: $ \alpha : f_{m}\circ \cdots \circ f_1 \Rightarrow
g_n \circ \cdots \circ g_1$.
\end{itemize}
Notice that $A_{(1,1)}$ is nothing but $\sD_2$. We are going to prove that if
$n\neq 0$ or $m\neq 0$, 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
\[
%% \begin{tikzcd}
%% A \ar[r, bend left=70, "f",""{name=A,below}] \ar[r,bend
%% right=70,"1_A"',""{name=B,above}] & A,
%% \ar[from=A,to=B,Rightarrow,"\alpha"]
%% \end{tikzcd}
%% \text{ or }
\begin{tikzcd}
A \ar[loop,in=50,out=130,distance=1.5cm,"f",""{name=A,below}]
\ar[from=A,to=1-1,Rightarrow,"\alpha"]
\end{tikzcd}
\]
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 $2$-category $B^2\mathbb{N}$ and we have already seen that it
is \emph{not} \good{} (see \ref{paragr:bubble}).
\end{paragr}
\begin{paragr}
For $n\geq 0$, we write $\Delta_n$ for 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}\label{lemma:istrngdefrtract}
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
\[
r(0)=0\text{ and } r(k)=1 \text{ for } k>0.
\]
By definition we have $r \circ i = 1_{\Delta_1}$. Now, the natural order on
$\Delta_n$ induces a natural transformation
\[
\alpha : \mathrm{id}_{\Delta_n} \Rightarrow i\circ r,
\]
and it is straightforward to check that $\alpha \ast i = \mathrm{id}_i$.
\end{proof}
\begin{paragr}\label{paragr:Amn}
For every $n \geq 0$, 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 $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique
non-trivial $1$\nbd{}cell of $\Delta_1$ to the target of the generating
$2$-cell of $A_{(1,1)}$. It is not hard to check that $\tau$ is strong
deformation retract and thus, a co-universal Thomason equivalence (Lemma
\ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $\Delta_n \to
A_{(1,n)}$ is also a (co-universal) Thomason equivalence and the square is
Thomason homotopy cocartesian (Lemma \ref{lemma:hmtpycocartsquarewe}). Now,
the morphism $\tau : \Delta_1 \to A_{(1,1)}$ is also a folk cofibration and
since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ are \good{}, it follows from what
we said in \ref{paragr:criterion2cat} that $A_{(1,n)}$ is \good{}. Finally,
since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ have the homotopy type of a
point, the fact that the previous square is Thomason homotopy cocartesian
implies that $A_{(1,n)}$ has the homotopy type of a point.
Similarly, for every $m \geq 0$, by considering the cocartesian square
\[
\begin{tikzcd}
\Delta_1 \ar[r,"i"] \ar[d,"\sigma"] & \Delta_m \ar[d] \\
A_{(1,1)} \ar[r] & A_{(m,1)}, \ar[from=1-1,to=2-2,phantom,very near
end,"\ulcorner"]
\end{tikzcd}
\]
where $\sigma : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the
unique non trivial $1$\nbd{}cell of $\Delta_1$ to the source of the generating
$2$\nbd{}cell of $A_{(1,1)}$, we can prove that $A_{(m,1)}$ is \good{} and has
the homotopy type of a point.
Now, let $m\geq 0$ and $n > 0$ and consider the cocartesian square
\[
\begin{tikzcd}
\Delta_1 \ar[r,"i"] \ar[d,"\tau"] & \Delta_n \ar[d] \\
A_{(m,1)} \ar[r] & A_{(m,n)},
\ar[from=1-1,to=2-2,phantom,very near
end,"\ulcorner"]
\end{tikzcd}
\]
where $\tau$ is the $2$-functor that sends the unique non-trivial $1$-cell of
$\Delta_1$ to the target of the generating $2$-cell of $A_{(m,1)}$. This
$2$-functor is once again a folk cofibration, but it is \emph{not} in general
a co-universal Thomason equivalence (it would be if we had made the hypothesis that
$m\neq 0$, but we did not). However, since we made the hypothesis that $n\neq
0$, it follows from Lemma \ref{lemma:istrngdefrtract} that $i : \Delta_1 \to
\Delta_n$ is a co-universal Thomason equivalence. Hence, the previous square
is Thomason homotopy cocartesian and $A_{(m,n)}$ has the homotopy type of a
point. Since $A_{(m,1)}$, $\Delta_1$ and $\Delta_n$ are \good{}, this shows
that for $m \geq 0$ and $n >0$, $A_{(m,n)}$ is \good{}.
Similarly, if $m >0$ and $ n\geq 0$, then $A_{(m,n)}$ has the homotopy type of
a point and is \good{}.
\end{paragr}
Combined with the result of Paragraph \ref{paragr:bubble}, we have proved the
following proposition.
\begin{proposition}\label{prop:classificationAmn}
Let $m,n \geq 0$ and consider the $2$-category $A_{(m,n)}$. If $m\neq 0$ or
$n\neq 0$, then $A_{(m,n)}$ is \good{} and has the homotopy type of a point.
If $n=m=0$, then $A_{(0,0)}$ is not \good{} and has the homotopy type of a
$K(\mathbb{Z},2)$.
\end{proposition}
\section{Zoology of 2-categories: more examples}
As a warm-up, let us begin with an example which is a direct consequence of the
results at the end of the previous section.
\begin{paragr}
Let $P$ the free $2$\nbd{}category defined as follows:
\begin{itemize}[label=-]
\item generating $0$\nbd{}cell: $A$,
\item generating $1$\nbd{}cells: $f,g : A \to A$,
\item generating $2$\nbd{}cells: $\alpha : f \Rightarrow 1_A$, $\beta : g
\Rightarrow 1_A$.
\end{itemize}
In pictures, this gives
\[
\begin{tikzcd}[column sep=huge]
A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend
right=75,"g"',""{name=B,above}] \ar[r,"1_A"
pos=1/3,""{name=C,above},""{name=D,below}]& A
\ar[from=A,to=C,Rightarrow,"\alpha"] \ar[from=B,to=D,"\beta"
pos=9/20,Rightarrow]
\end{tikzcd}
\text{ or }
\begin{tikzcd}
A. \ar[loop,in=50,out=130,distance=1.5cm,"f",""{name=A,below}]
\ar[loop,in=-50,out=-130,distance=1.5cm,"g"',""{name=B,above}]
\ar[from=A,to=1-1,Rightarrow,"\alpha"]
\ar[from=B,to=1-1,Rightarrow,"\beta"]
\end{tikzcd}
\]
Notice that this category has many non-trivial $2$\nbd{}cells and it is not
\emph{a priori} clear what its homotopy type is and whether or not it is
\good{}. Observe that $P$ is obtained as the following amalgamated sum
\begin{equation}\label{square:lemniscate}
\begin{tikzcd}
\sD_0 \ar[r] \ar[d] & A_{(1,0)} \ar[d]\\
A_{(1,0)} \ar[r] & P. \ar[from=1-1,to=2-2,very near
end,"\ulcorner",phantom]
\end{tikzcd}
\end{equation}
Since $\sD_0$, $A_{(1,0)}$ are free and \good{} and since
$\sD_0 \to A_{(1,0)}$ is a folk cofibration, all we
have to show to prove that $P$ is \good{} is that the above square is Thomason
homotopy cocartesian. Notice that the $2$\nbd{}category $A_{(1,0)}$ is
obtained as the following amalgamated sum
\[
\begin{tikzcd}
\Delta_1 \ar[r,"\tau"] \ar[d]& A_{(1,1)} \ar[d] \\
\Delta_0 \ar[r] &A_{(1,0)}, \ar[from=1-1,to=2-2,very near
end,"\ulcorner",phantom]
\end{tikzcd}
\]
where $\tau : \Delta_1 \to A_{(1,1)}$ has already been defined in
\ref{paragr:Amn}. We have seen that $\tau$ is a co\nbd{}universal Thomason
equivalence and thus, so is $\sD_0 \to A_{(0,1)}$ (as $\sD_0$ and
$\Delta_0$ are two different names for the same category). Hence, square
\eqref{square:lemniscate} is Thomason homotopy cocartesian and this proves that
$P$ is \good{} and has the homotopy type of a point.
All the variations by reversing the direction of $\alpha$ or $\beta$ work
exactly the same way.
\end{paragr}
Let us now get into more sophisticated examples.
\begin{paragr}[Variations of spheres]\label{paragr:variationsphere}
Let $P$ the free $2$\nbd{}category defined as follows:
\begin{itemize}[label=-]
\item generating $0$\nbd{}cells: $A,B$,
\item generating $1$\nbd{}cells: $f,g : A \to B$,
\item generating $2$\nbd{}cells: $\alpha : f \Rightarrow g$, $\beta: g
\Rightarrow f$.
\end{itemize}
In pictures, this gives
\[
\begin{tikzcd}
A \ar[r,bend left=75,"f",""{name=A,below}] \ar[r,bend
right=75,"g"',""{name=B,above}] & B. \ar[from=A,to=B,bend
right,Rightarrow,"\alpha"'] \ar[from=B,to=A,bend
right,Rightarrow,"\beta"']
\end{tikzcd}
\]
Let $P'$ be the free $2$\nbd{}category defined as follows:
\begin{itemize}[label=-]
\item generating $0$\nbd{}cells: $A',B'$,
\item generating $1$\nbd{}cell: $h:A'\to B'$,
\item generating $2$\nbd{}cell: $\gamma : h \Rightarrow h$,
\end{itemize}
which can be pictured as
\[
\begin{tikzcd}
A' \ar[r,bend left=75,"h",""{name=A,below}] \ar[r,bend right=75,"h"']& B'
\ar[from=A,to=B,Rightarrow,"\gamma"]
\end{tikzcd}
\text{ or }
\begin{tikzcd}
A' \ar[r,"h"',""{name=A,above}] & B' \ar[from=A,to=A,loop, in=130,
out=50,distance=1cm, Rightarrow,"\gamma"']
\end{tikzcd}
\]
and let $F : P \to P'$ be the unique $2$\nbd{}functor such that
\begin{itemize}[label=-]
\item $F(A)=A'$ and $F(B)=B'$,
\item $F(f)=F(g)=h$,
\item $F(\alpha)=\gamma$ and $F(\beta)=1_h$.
\end{itemize}
We wish to prove that this $2$\nbd{}functor is a Thomason equivalence. Since
it is an isomorphism on objects, it suffices to prove that the functors induced
by $F$
\[
F_{A,A} : P(A,A) \to P'(F(A),F(A)),
\]
\[
F_{B,B} : P(B,B) \to P'(F(B),F(B)),
\]
\[
F_{B,A} : P(B,A) \to P'(F(B),F(A))
\]
and
\[
F_{A,B} : P(A,B) \to P'(F(A),F(B))
\]
are Thomason equivalences of categories (Corollary
\ref{cor:criterionThomeqI}). For the first two ones, this follows trivially
from the fact that the categories $P(A,A)$, $P'(A',A')$, $P(B,B)$ and
$P'(B',B')$ are all isomorphic to $\sD_0$. For the third one, this follows
trivially from the fact that the categories $P(B,A)$ and $P'(B',A')$ are the
empty category. For the fourth one, this can be seen as follows. The category $P(A,B)$ is the free category on the
graph
\[
\begin{tikzcd}
f \ar[r,shift left,"\alpha"] & g \ar[l,shift left,"\beta"]
\end{tikzcd}
\]
($2$\nbd{}cells of $P$ become $1$\nbd{}cells of $P(A,B)$ and $1$\nbd{}cells of
$P$ become $0$\nbd{}cells of $P(A,B)$) and the category
$P'(F(A),F(B))=P'(A',B')$ is the free category on the graph
\[
\begin{tikzcd}
h. \ar[loop above,"\gamma"]
\end{tikzcd}
\]
The functor $F_{A,B}$ comes from a morphism of reflexive graphs and is
obtained by ``killing the generator $\beta$'' (see Example
\ref{example:killinggenerator}). In particular, the square
\[
\begin{tikzcd}
\sD_1 \ar[r,"\langle \beta \rangle"] \ar[d]& P(A,B) \ar[d,"F_{A,B}"] \\
\sD_0 \ar[r,"\langle h \rangle" ] & P'(A',B')
\end{tikzcd}
\]
is Thomason homotopy cocartesian and thus, $F_{(A,B)}$ is a Thomason
equivalence.
Now consider (a copy of) $\sS_2$ labelled as follows:
\[
\begin{tikzcd}
\overline{A} \ar[r,bend left=75,"i",""{name=A,below}] \ar[r,bend
right=75,"j"',""{name=B,above}] & \overline{B} \ar[from=A,to=B,bend
right,Rightarrow,"\delta"'] \ar[from=A,to=B,bend left
,Rightarrow,"\epsilon"]
\end{tikzcd}
\]
and let $G : \sS_2 \to P'$ be the unique $2$\nbd{}functor such that
\begin{itemize}[label=-]
\item $G(\overline{A})=A'$ and $G(\overline{B})=B'$,
\item $G(i)=G(j)=h$,
\item $G(\delta)=\gamma$ and $G(\epsilon)=1_h$.
\end{itemize}
For similar reasons as for $F$, the $2$\nbd{}functor $G$ is a Thomason
equivalence. This proves that both $P'$ and $P$ have the homotopy type of
$\sS_2$.
Now, let $P''$ be the free $2$\nbd{}category defined as follows:
\begin{itemize}[label=-]
\item generating $0$\nbd{}cell: $A''$,
\item generating $1$\nbd{}cell: $l : A'' \to A''$,
\item generating $2$\nbd{}cells: $\lambda : l \Rightarrow 1_{A''}$ and $\mu: l
\Rightarrow 1_{A''}$.
\end{itemize}
In pictures, this gives
\[
\begin{tikzcd}[column sep=huge]
A'' \ar[r,bend left=75,"l",""{name=A'',below}]\ar[r,bend
right=75,"l"',""{name=B,above}] \ar[r,"1_{A''}"
pos=1/3,""{name=C,above},""{name=D,below}]& A''
\ar[from=A'',to=C,Rightarrow,"\lambda"] \ar[from=B,to=D,"\mu"
pos=9/20,Rightarrow]
\end{tikzcd}
\qquad \quad \text{ or }
\begin{tikzcd}
A''. \ar[loop,in=30,out=150,distance=3cm,"l",""{name=A,below}]
\ar[from=A,to=1-1,bend right,Rightarrow,"\lambda"'] \ar[from=A,to=1-1,bend
left,Rightarrow,"\mu"]
\end{tikzcd}
\]
Let $H : \sS_2 \to P''$ be the unique $2$\nbd{}functor such that:
\begin{itemize}[label=-]
\item $H(\overline{A})=H(\overline{B})=A''$,
\item $H(i)=l$ and $H(j)=1_{A''}$,
\item $H(\delta)=\lambda$ and $H(\epsilon)=\mu$.
\end{itemize}
Let us prove that $H$ is a Thomason equivalence using Corollary
\ref{cor:criterionThomeqII}. In order to do so, we have to compute $V_k(H) :
V_k(\sS_2) \to V_k(P'')$ for every $k\geq 0$. For $k=0$, the category
$V_0(\sS_2)$ is the free category on the graph
\[
\begin{tikzcd}
\overline{A} \ar[r,"i",shift left] \ar[r,"j"',shift right] & \overline{B},
\end{tikzcd}
\]
the category $V_0(P'')$ is the free category on the graph
\[
\begin{tikzcd}
A \ar[loop above,"l"]
\end{tikzcd}
\]
and $V_0(H)$ comes from a morphism of reflexive graphs obtained by ``killing
the generator $j$''. Hence, it is a Thomason equivalence of categories. For
$k>0$, the category $V_k(\sS_2)$ has two objects $\overline{A}$ and
$\overline{B}$ and an arrow $\overline{A} \to \overline{B}$ is a
$k$\nbd{}tuple of one of the following forms:
\begin{itemize}[label=-]
\item $(1_i,\cdots,1_i,\delta,1_j,\cdots,1_j)$,
\item $(1_i,\cdots,1_i,\epsilon,1_j,\cdots,1_j)$,
\item $(1_i,\cdots,1_i)$,
\item $(1_j,\cdots,1_j)$,
\end{itemize}
and these are the only non-trivial arrows. In other words, $V_k(\sS_2)$ is the
free category on the graph with two objects and $2k+2$ parallel arrows between
these two objects. In order to compute $V_k(P'')$, let us first notice that
every $2$\nbd{}cell of $P''$ (except for $\1^2_{A''}$) is uniquely encoded as
a finite word on the alphabet that has three symbols : $1_l$, $\lambda$ and
$\mu$. Concatenation corresponding to the $0$\nbd{}composition of these cells.
This means exactly that $V_1(P'')$ is free on the graph that has one object
and three arrows. More generally, it is a tedious but harmless exercise to
prove that for every $k>0$, the category $V_k(P'')$ is the
free category on the graph that has one object $A''$ and $2k+1$
arrows which are of one of the following forms:
\begin{itemize}[label=-]
\item $(1_l,\cdots,1_l,\lambda,1^2_{A''},\cdots,1^2_{A''})$,
\item $(1_l,\cdots,1_l,\mu,1^2_{A''},\cdots,1^2_{A''})$,
\item $(1_l,\cdots,1_l)$.
\end{itemize}
Once again, the functor $V_k(H)$ comes from a morphism of reflexive graphs and
is obtained by ``killing the generator $(1_j,\cdots,1_j)$''. Hence, it is a
Thomason equivalence and thus, so is $H$. This proves that $P''$ has the
homotopy type of $\sS_2$.
% is of exactly one of the following form
% \begin{itemize}[label=-]
% \item $\1^2_{A''}$,
% \item $1_{l}\comp_0 cdots\comp_0 1_l}$ (including the case $1_{l}$),
% \item $1_{l}\comp_0 \cdots $
% \end{itemize}
Finally, consider the commutative diagram of $\ho(\Ch)$
\[
\begin{tikzcd}[column sep=huge]
\sH^{\sing}(P) \ar[r,"\sH^{\sing}(F)"] \ar[d,"\pi_P"] & \sH^{\sing}(P')
\ar[d,"\pi_{P'}"] & \sH^{\sing}(\sS_2) \ar[l,"\sH^{\sing}(G)"']
\ar[d,"\pi_{\sS_2}"] \ar[r,"\sH^{\sing}(H)"]&\sH^{\sing}(P'') \ar[d,"\pi_{P''}"]\\
\sH^{\pol}(P) \ar[r,"\sH^{\pol}(F)"'] & \sH^{\pol}(P') & \sH^{\pol}(\sS_2)
\ar[l,"\sH^{\pol}(G)"] \ar[r,"\sH^{\pol}(H)"']&\sH^{\pol}(P'').
\end{tikzcd}
\]
Since $F$, $G$ and $H$ are Thomason equivalences, the three top horizontal
morphisms are isomorphisms. Besides, a simple computation using Proposition
\ref{prop:abelianizationfreeoocat}, which we leave to the reader, shows that
the three bottom horizontal morphisms are also isomorphisms. Since $\sS_2$ is
\good{} (Proposition \ref{prop:spheresaregood}), the morphism $\pi_{\sS_2}$ is
an isomorphism. This implies that $\pi_{P'}$, $\pi_{P'}$ and $\pi_{P''}$ are
isomorphisms, which means by definition that $P$, $P'$ and $P''$ are \good{}.
\end{paragr}
\begin{paragr}\label{paragr:anothercounterexample}
Let $P$ be the free $2$\nbd{}category defined as follows:
\begin{itemize}[label=-]
\item generating $0$\nbd{}cell: $A$,
\item generating $1$\nbd{}cell: $f : A \to A$,
\item generating $2$\nbd{}cells: $\alpha : f \Rightarrow 1_A$ and $\beta: 1_A
\Rightarrow f$.
\end{itemize}
In pictures, this gives
\[
\begin{tikzcd}[column sep=huge]
A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend
right=75,"f"',""{name=B,above}] \ar[r,"1_A"
pos=1/3,""{name=C,above},""{name=D,below}]& A
\ar[from=A,to=C,Rightarrow,"\alpha"] \ar[from=D,to=B,"\beta"
pos=9/20,Rightarrow]
\end{tikzcd}
\qquad \quad \text{ or }
\begin{tikzcd}
A. \ar[loop,in=30,out=150,distance=3cm,"f",""{name=A,below}]
\ar[from=A,to=1-1,bend right,Rightarrow,"\alpha"'] \ar[from=1-1,to=A,bend
right,Rightarrow,"\beta"']
\end{tikzcd}
\]
Now, let $P'$ be a copy of $B^2\mathbb{N}$ labelled as follows
\[
\begin{tikzcd}
A' \ar[r,bend left=75,"1_{A'}",""{name=A,below}] \ar[r,bend
right=75,"1_{A'}"',""{name=B,above}] & A',
\ar[from=A,to=B,"\gamma",Rightarrow]
\end{tikzcd}
\]
which can be also pictured as
\[
\begin{tikzcd}
A'. \ar[loop,in=120,out=60,distance=1.5cm,"\gamma"',Rightarrow]
\end{tikzcd}
\]
Let $F : P \to P'$ be the unique $2$\nbd{}functor such that:
\begin{itemize}[label=-]
\item $F(A)=A'$,
\item $F(f)=1_{A'}$
\item $F(\alpha)=\1^2_{A'}$ and $F(\beta)=\gamma$,
\end{itemize}
and $G : P' \to P$ be the unique $2$\nbd{}functor such that:
\begin{itemize}[label=-]
\item $G(A')=A$,
\item $G(\gamma)=\alpha\comp_1\beta$.
\end{itemize}
Notice that we have $F\circ G = \mathrm{id}_{P'}$, which means that $P'$ is a
retract of $P$. In particular, $\sH^{\sing}(P)$ is a retract of
$\sH^{\sing}(P')$ and since $P'$ has the homotopy type of a
$K(\mathbb{Z},2)$ (see \ref{paragr:bubble}), this proves that $P$ has
non-trivial singular homology groups in all even dimension. But since it is a
free $2$\nbd{}category, all its polygraphic homology groups are trivial strictly above
dimension $2$, which means that $P$ is \emph{not} \good{}.
\end{paragr}
\begin{paragr}\label{paragr:sumupsphere}
All the results from \ref{paragr:variationsphere} and
\ref{paragr:anothercounterexample} are summed up by the following table.
\begin{center}
\begin{tabular}{| l || c | c |}
\hline
$2$-category & \good{}? & homotopy type \\ \hline \hline
{
$\begin{tikzcd}
\bullet \ar[r,bend
left=75,""{name=A,below,pos=9/20},""{name=C,below,pos=11/20}]
\ar[r,bend
right=75,""{name=B,above,pos=9/20},""{name=D,above,pos=11/20}] & \bullet
\ar[from=C,to=D,bend left,Rightarrow] \ar[from=A,to=B,bend
right,Rightarrow]
\end{tikzcd}$
} & yes & $\sS_2$\\
\hline
{ $ \begin{tikzcd}
\bullet \ar[r,bend left=75,""{name=A,below}] \ar[r,bend
right=75,""{name=B,above}] & \bullet \ar[from=A,to=B,bend
right,Rightarrow] \ar[from=B,to=A,bend
right,Rightarrow]
\end{tikzcd}$} & yes & $\sS_2$ \\ \hline {$ \begin{tikzcd} \bullet
\ar[r,""{name=A,above}] & \bullet \ar[from=A,to=A,loop, in=130,
out=50,distance=1cm, Rightarrow] \end{tikzcd}$} & yes &$\sS_2$ \\
\hline
{
$\begin{tikzcd}
\bullet \ar[loop,in=30,out=150,distance=2cm,""{name=A,below}]
\ar[from=A,to=1-1,bend right,Rightarrow]
\ar[from=A,to=1-1,bend left,Rightarrow]
\end{tikzcd}$ } & yes & $\sS_2$ \\ \hline { $\begin{tikzcd} \bullet
\ar[loop,in=30,out=150,distance=2cm,""{name=A,below}]
\ar[from=A,to=1-1,bend right,Rightarrow] \ar[from=1-1,to=A,bend
right,Rightarrow]
\end{tikzcd}$ } & no & $K(\mathbb{Z},2)$ \\ \hline {$\begin{tikzcd}
\bullet \ar[loop,in=120,out=60,distance=1.2cm,Rightarrow]
\end{tikzcd}$} & no & $K(\mathbb{Z},2)$ \\ \hline
\end{tabular}
\end{center}
Notice that the fourth and fifth entries of this table only differ by the
direction of a generating $2$\nbd{}cell but the homotopy types are not the same.
\end{paragr}
Let us now move on to bouquets of spheres.
\begin{paragr}
Let $P$ be the free $2$\nbd{}category defined as follows:
\begin{itemize}[label=-]
\item generating $0$\nbd{}cell: $A$,
\item generating $1$\nbd{}cells: $f,g: A \to A$,
\item generating $2$\nbd{}cells: $\alpha,\beta : f \Rightarrow g$.
\end{itemize}
In pictures, this gives:
\[
\begin{tikzcd}[column sep=huge]
A \ar[r,bend
left=75,"f",""{name=A,below,pos=9/20},""{name=C,below,pos=11/20}]
\ar[r,bend
right=75,"g"',""{name=B,above,pos=9/20},""{name=D,above,pos=11/20}] & A.
\ar[from=C,to=D,bend left,"\alpha",Rightarrow] \ar[from=A,to=B,bend
right,"\beta"',Rightarrow]
\end{tikzcd}
\]
Now let $\sS_2$ be labelled as
follows:
\[
\begin{tikzcd}
C \ar[r,bend
left=75,"h",""{name=A,below,pos=9/20},""{name=C,below,pos=11/20}]
\ar[r,bend
right=75,"i"',""{name=B,above,pos=9/20},""{name=D,above,pos=11/20}] & D.
\ar[from=C,to=D,bend left,"\gamma",Rightarrow] \ar[from=A,to=B,bend
right,"\delta"',Rightarrow]
\end{tikzcd}
\]
Notice that $P$ is obtained as the following amalgamated sum:
\begin{equation}
\begin{tikzcd}
\sS_0 \ar[d,"p"] \ar[r,"{\langle C,D \rangle}"] & \sS_2 \ar[d]\\
\sD_0 \ar[r] & P.
\ar[from=1-1,to=2-2,"\ulcorner",phantom,very near end]
\end{tikzcd}\label{eq:squarebouquethybrid}
\end{equation}
Let us prove that this square is Thomason homotopy cocartesian using the first
part of Proposition \ref{prop:critverthorThomhmtpysquare}. This means that we
have to show that the induced square of $\Cat$
\begin{equation}
\begin{tikzcd}
V_k(\sS_0) \ar[r,"{V_k(\langle C,D \rangle)}"] \ar[d,"V_k(p)"] & V_k(\sS_2) \ar[d]\\
V_k(\sD_0) \ar[r] & V_k(P)
\end{tikzcd}\label{eq:squarebouquethybridvertical}
\end{equation}
is Thomason homotopy cocartesian for every $k \geq 0$. Notice first that we
trivially have $V_k(\sS_0)\simeq \sS_0$ and $V_k(\sD_0)\simeq \sD_0$ for every
$k\geq 0$ and that $V_0(\sS_2)$ is the free category on the
graph
\[
\begin{tikzcd}
C \ar[r,shift left,"f"] \ar[r,shift right,"g"'] & D.
\end{tikzcd}
\]
Besides, $V_0(P)$ is the free category on the
graph
\[
\begin{tikzcd}
A. \ar[loop above,"f"] \ar[loop below,"g"]
\end{tikzcd}
\]
In particular, square \eqref{eq:squarebouquethybridvertical} is
cocartesian for $k=0$ and we are in the situation of identification
of two objects of a free category (see Example
\ref{example:identifyingobjects}). Hence, square
\eqref{eq:squarebouquethybridvertical} is Thomason cocartesian for
$k=0$. Similarly, for $k>0$, we have already seen that $V_k(\sS_2)$
is the free category on the graph that has $2$ objects and $2k+2$
parallel arrows between these two objects and we leave as an easy
exercise to the reader to check that the category $V_k(P)$ is the
free category on the graph that has one object and $2k+2$ arrows, which are
the $k$\nbd{}tuples of one of the following forms:
\begin{itemize}[label=-]
\item $(1_f,\cdots,\alpha,\cdots,1_g)$,
\item $(1_f,\cdots,\beta,\cdots,1_g)$,
\item $(1_f,\cdots,1_f)$,
\item $(1_g,\cdots,1_g)$.
\end{itemize}
In particular, square \eqref{eq:squarebouquethybridvertical} is again a
cocartesian square of identification of two objects of a free category, and
thus, it is Thomason homotopy cocartesian. This implies that square
\eqref{eq:squarebouquethybrid} is Thomason homotopy cocartesian. Since $\sS_0$,
$\sD_0$ and $\sS_2$ are \good{} and since $\langle C, D \rangle : \sS_0 \to
\sS_2$ is a folk cofibration, this proves that $P$ is \good{}
and has the homotopy type of the bouquet of a $1$\nbd{}sphere with a $2$\nbd{}sphere.
\end{paragr}
\begin{paragr}
Let $P$ be the free $2$\nbd{}category defined as follows:
\begin{itemize}[label=-]
\item generating $0$\nbd{}cells: $A$ and $B$,
\item generating $1$\nbd{}cells: $f,g : A \to B$,
\item generating $2$\nbd{}cells: $\alpha,\beta,\gamma : f \Rightarrow g$.
\end{itemize}
In pictures, this gives
\[
\begin{tikzcd}[column sep=huge]
A \ar[r,bend
left=75,"f",""{name=A,below,pos=8/20},""{name=C,below,pos=1/2},""{name=E,below,pos=12/20}]
\ar[r,bend
right=75,"g"',""{name=B,above,pos=8/20},""{name=D,above,pos=1/2},""{name=F,above,pos=12/20}]&
B. \ar[from=A,to=B,Rightarrow,"\alpha"',bend right]
\ar[from=C,to=D,Rightarrow,"\beta"]
\ar[from=E,to=F,Rightarrow,"\gamma",bend left]
\end{tikzcd}
\]
Now let $P'$ be the sub-$2$\nbd{}category of $P$ spanned by $A$, $B$, $f$, $g$, $\alpha$
and $\beta$, and let $P''$ be the sub-$2$\nbd{}category of $P$ spanned by $A$,
$B$, $f$, $g$, $\beta$ and $\gamma$. These $2$\nbd{}categories are simply copies of
$\sS_2$. Notice that we have a cocartesian square
\begin{equation}\label{square:bouquet}
\begin{tikzcd}
\sD_1 \ar[r,"\langle \beta \rangle"] \ar[d,"\langle \beta \rangle"] &
P' \ar[d] \\
P'' \ar[r] & P, \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\end{equation}
and by reasoning as in the proof of Lemma \ref{lemma:squarenerve}, one can
show that the square induced by the nerve
\[
\begin{tikzcd}
N_{\oo}(\sD_1) \ar[r,"\langle \beta \rangle"] \ar[d,"\langle \beta
\rangle"] &
N_{\oo}(P') \ar[d] \\
N_{\oo}(P'') \ar[r] & N_{\oo}(P)
\end{tikzcd}
\]
is also cocartesian. Since $\langle \beta \rangle : \sD_1 \to P'$ and $\langle
\beta \rangle : \sD_1 \to P''$ are monomorphisms and $N_{\oo}$ preserves
monomorphisms, it follows from Lemma \ref{lemma:hmtpycocartesianreedy} that
square \eqref{square:bouquet} is Thomason homotopy cocartesian and in
particular that $P$ has the homotopy type of a bouquet of two
$2$\nbd{}spheres. Since $\sD_1$, $P'$ and $P''$ are free and \good{} and since
$\langle \beta \rangle : \sD_1 \to P'$ and $\langle \beta \rangle : \sD_1 \to
P''$ are folk cofibrations, this also proves that $P$ is \good{} (see \ref{paragr:criterion2cat}).
\end{paragr}
\begin{paragr}
Let $P$ be the free $2$\nbd{}category defined as follows:
\begin{itemize}[label=-]
\item generating $0$\nbd{}cells: $A$ and $B$,
\item generating $1$\nbd{}cells: $f,g,h : A \to B$,
\item generating $2$\nbd{}cells: $\alpha,\beta:f \Rightarrow g$ and
$\delta,\gamma:g \Rightarrow h$.
\end{itemize}
In pictures, this gives:
\[
\begin{tikzcd}[column sep=huge]
A \ar[r,bend
left=75,"f",""{name=A,below,pos=8/20},""{name=E,below,pos=12/20}]
\ar[r,"g",""{name=B,above,pos=8/20},""{name=C,below,pos=8/20},""{name=F,above,pos=12/20},""{name=G,below,pos=12/20}]
\ar[r,bend
right=75,"h"',""{name=D,above,pos=8/20},""{name=H,above,pos=12/20}] & B.
\ar[from=A,to=B,Rightarrow,"\alpha"',bend right]
\ar[from=C,to=D,Rightarrow,"\gamma"',bend right]
\ar[from=E,to=F,Rightarrow,"\beta",bend left]
\ar[from=G,to=H,Rightarrow,"\delta",bend left]
\end{tikzcd}
\]
Let us prove that this $2$\nbd{}category is \good{}. Let $P_0$ be the
sub-$1$\nbd{}category of $P$ spanned by $A$, $B$ and $g$, let $P_1$ be the
sub-$2$\nbd{}category of $P$ spanned by $A$, $B$, $g$, $h$, $\gamma$ and
$\delta$ and let $P_2$ be the sub-$2$\nbd{}category of $P$ spanned by $A$,
$B$, $f$, $g$, $\alpha$ and $\beta$. The $2$\nbd{}categories $P_1$ and $P_2$
are copies of $\sS_2$, and $P_0$ is a copy of $\sD_1$. Moreover, we have a
cocartesian square of inclusions
\begin{equation}\label{squarebouquet}
\begin{tikzcd}
P_0 \ar[d,hook] \ar[r,hook] & P_2 \ar[d,hook] \\\
P_1 \ar[r,hook] & P.
\end{tikzcd}
\end{equation}
Let us prove that this square is Thomason homotopy cocartesian using the
second part of Corollary \ref{prop:critverthorThomhmtpysquare}. This means
that we have to show that for every $k \geq 0$, the induced square of $\Cat$
\begin{equation}\label{squarebouquethorizontal}
\begin{tikzcd}
S_k(P_0) \ar[d] \ar[r] & S_k(P_2) \ar[d] \\\
S_k(P_1) \ar[r] & S_k(P)
\end{tikzcd}
\end{equation}
is Thomason homotopy cocartesian. For $k=0$, this is obvious since all of the
morphisms of square \eqref{squarebouquet} are isomorphisms at the level of
objects and the functor $S_0$ is the functor that sends a $2$\nbd{}category to
its set of objects (seen as a discrete category). Now, notice that the
categories $P_i(A,A)$ and $P_i(B,B)$ for $0 \leq i \leq 2$ are all isomorphic
to the terminal category $\sD_0$ and the categories $P_i(B,A)$ for $0 \leq i
\leq 2$ are all the empty category. It follows that for $k>0$, we have
\[
S_k(P_i)\simeq \sD_0\amalg \left( \coprod_{E_k}P_i(A,B) \right)\amalg \sD_0
\]
where $E_k$ is the set of all $k$\nbd{}tuples of the form
\[
(A,\cdots,A,B,\cdots,B).
\]
The set $E_1$ is empty and thus all of the morphisms of square
\eqref{squarebouquethorizontal} for the value $k=1$ are isomorphisms. This
makes square \eqref{squarebouquethorizontal} Thomason homotopy cocartesian for
$k=1$. For $k>1$, notice that the categories $P(A,B)$, $P_2(A,B)$ and
$P_1(A,B)$ are respectively free on the graphs
\[
\begin{tikzcd}
f \ar[r,shift left,"\alpha"] \ar[r, shift right, "\beta"'] & g \ar[r,shift
left,"\gamma"] \ar[r,shift right,"\delta"'] & h,
\end{tikzcd}
\]
\[
\begin{tikzcd}
f \ar[r,shift left,"\alpha"] \ar[r, shift right, "\beta"'] & g,
\end{tikzcd}
\]
and
\[
\begin{tikzcd}
g \ar[r,shift left,"\gamma"] \ar[r,shift right,"\delta"'] & h.
\end{tikzcd}
\]
It is then straightforward to check that we are in a situation where Corollary
\ref{cor:hmtpysquaregraph} applies and thus square
\eqref{squarebouquethorizontal} is Thomason homotopy cocartesian for $k\geq
1$. Altogether, this proves that square \eqref{squarebouquet} is Thomason homotopy
cocartesian. Since the inclusions $P_0 \hookrightarrow P_1$ and $P_0 \hookrightarrow P_2$ are folk cofibrations and since $P_0$, $P_1$ and $P_2$ are \good{}, this proves that $P$ is \good{} and has the homotopy type of a bouquet of
two $2$\nbd{}spheres.
\end{paragr}
\begin{paragr}
Let $P$ be the free $2$\nbd{}category defined as follows:
\begin{itemize}[label=-]
\item generating $0$\nbd{}cells: $A$, $B$ and $C$,
\item generating $1$\nbd{}cells: $f,g : A \to B$ and $h,i : B \to C$,
\item generating $2$\nbd{}cells: $\alpha,\beta : f \Rightarrow g$ and
$\gamma,\delta : h \Rightarrow i$.
\end{itemize}
In pictures, this gives:
\[
\begin{tikzcd}[column sep=huge]
A \ar[r,bend
left=75,"f",""{name=A,below,pos=8/20},""{name=C,below,pos=12/20}]
\ar[r,bend
right=75,"g"',""{name=B,above,pos=8/20},""{name=D,above,pos=12/20}] & B
\ar[r,bend
left=75,"h",""{name=E,below,pos=8/20},""{name=G,below,pos=12/20}]
\ar[r,bend
right=75,"i"',""{name=F,above,pos=8/20},""{name=H,above,pos=12/20}] & C.
\ar[from=A,to=B,bend right,"\alpha",Rightarrow] \ar[from=C,to=D,bend
left,"\beta",Rightarrow] \ar[from=E,to=F,bend right,"\gamma",Rightarrow]
\ar[from=G,to=H,bend left,"\delta",Rightarrow]
\end{tikzcd}
\]
Let us prove that $P$ is \good{}. Let $P'$ be the sub-$2$\nbd{}category of $P$
spanned by $A$, $B$, $f$, $g$, $\alpha$ and $\beta$ and let $P''$ be the
sub-$2$\nbd{}category of $P$ spanned by $B$, $C$, $h$, $i$, $\gamma$ and
$\delta$. These two $2$\nbd{}categories are copies of $\sS_2$ and we have a
cocartesian square
\begin{equation}\label{squarebouquetbis}
\begin{tikzcd}
\sD_0 \ar[r,"\langle B \rangle"] \ar[d,"\langle B \rangle"] & P' \ar[d,hook]\\
P'' \ar[r,hook] & P, \ar[from=1-1,to=2-2,phantom,very near end]
\end{tikzcd}
\end{equation}
where the anonymous arrows are the canonical inclusions. Let us prove that
this cocartesian square is Thomason homotopy cocartesian using the first part
of Proposition \ref{prop:critverthorThomhmtpysquare}. This means that we need
to prove that for every $k\geq 0$, the induced square of $\Cat$
\begin{equation}\label{squarebouquetvertical}
\begin{tikzcd}
V_k(\sD_0) \ar[r] \ar[d] & V_k(P') \ar[d]\\
V_k(P'') \ar[r] & V_k(P)
\end{tikzcd}
\end{equation}
is Thomason homotopy cocartesian. For every $k\geq 0$, we have $V_k(\sD_0) \simeq \sD_0$
and for $k=0$ the categories $V_0(P')$, $V_0(P'')$ and $V_0(P)$ are respectively free on
the graphs
\[
\begin{tikzcd}
A \ar[r,"f",shift left] \ar[r,"g"',shift right] & B,
\end{tikzcd}
\]
\[
\begin{tikzcd}
B \ar[r,"h",shift left] \ar[r,"i"',shift right] & C,
\end{tikzcd}
\]
and
\[
\begin{tikzcd}
A \ar[r,"f",shift left] \ar[r,"g"',shift right] & B \ar[r,"h",shift left] \ar[r,"i"',shift right] & C.
\end{tikzcd}
\]
This implies that square \eqref{squarebouquetvertical} is cocartesian for $k=0$ and in
virtue of Corollary \ref{cor:hmtpysquaregraph} it is also Thomason homotopy
cocartesian for this value of $k$. For $k>0$, since $P'$ and $P''$ are both
isomorphic to $\sS_2$, we have already seen in \ref{paragr:variationsphere} that $V_k(P')$ and $V_k(P'')$ are
(isomorphic) to the free category on the graph that has two objects and $2k+2$ parallel
arrows between these two objects.
% the category $V_k(P')$ has two objects $A$
% and $B$ and an arrow $A \to B$ is a $k$\nbd{}tuple of one of the following form
% \begin{itemize}[label=-]
% \item $(1_f,\cdots,1_f,\alpha,1_g,\cdots,1_g)$,
% \item $(1_f,\cdots,1_f,\beta,1_g,\cdots,1_g)$,
% \item $(1_f,\cdots,1_f)$,
% \item $(1_g,\cdots,1_g)$,
% \end{itemize}
% and these are the only non-trivial arrows of $V_k(P')$. This means that $V_k(P')$
% is the free category on the graph that has two objects and $2k+2$ parallel
% arrows between these two objects. The same goes for $V_k(P'')$ since $P'$ and
% $P''$ are isomorphic.
Similarly, the category $V_k(P)$ is free on the graph
that has three objects $A$, $B$, $C$, whose arrows from $A$ to $B$
are $k$\nbd{}tuples of one of the following form
\begin{itemize}[label=-]
\item $(1_f,\cdots,1_f,\alpha,1_g,\cdots,1_g)$,
\item $(1_f,\cdots,1_f,\beta,1_g,\cdots,1_g)$,
\item $(1_f,\cdots,1_f)$,
\item $(1_g,\cdots,1_g)$,
\end{itemize}
whose generating arrows from $B$ to $C$ are $k$\nbd{}tuple of one of the
following form
\begin{itemize}[label=-]
\item $(1_h,\cdots,1_h,\gamma,1_i,\cdots,1_i)$,
\item $(1_h,\cdots,1_h,\delta,1_i,\cdots,1_i)$,
\item $(1_h,\cdots,1_h)$,
\item $(1_i,\cdots,1_i)$,
\end{itemize}
and with no other arrows. This implies that square \eqref{squarebouquetvertical}
is cocartesian for every $k>0$ and in virtue of Corollary
\ref{cor:hmtpysquaregraph} it is also Thomason homotopy cocartesian for these values of $k$.
Altogether, this proves that square \eqref{squarebouquetbis} is Thomason
homotopy cocartesian. Hence, $P$ is \good{} and has the homotopy type of a
bouquet of two $2$\nbd{}spheres.
\end{paragr}
Let us end this section with an example of a $2$\nbd{}category that has the
homotopy type of the torus.
\begin{paragr}
Let $P$ be the free $2$\nbd{}category defined as follows:
\begin{itemize}[label=-]
\item generating $0$\nbd{}cell: $A$,
\item generating $1$\nbd{}cells: $f , g : A \to A$,
\item generating $2$\nbd{}cell: $\alpha : g\comp_0 f \Rightarrow f \comp_0
g$.
\end{itemize}
In pictures, this gives:
\[
\begin{tikzcd}
A \ar[r,"f"] \ar[d,"g"'] & A \ar[d,"g"] \\
A \ar[r,"f"'] & A. \ar[from=2-1,to=1-2,Rightarrow,"\alpha"]
\end{tikzcd}
\]
From now on, we will use concatenation instead of the symbol $\comp_0$ for the
$0$\nbd{}composition. For example, $fg$ will stand for $f \comp_0 g$. With
this notation, the set $1$\nbd{}cells of $P$ is canonically isomorphic to the
set of finite words in the alphabet $\{f,g\}$ and the set of $2$\nbd{}cells of
$P$ is canonically isomorphic to the set of finite words in the alphabet
$\{f,g,\alpha\}$. For a $1$\nbd{}cell $w$ such that $f$ appears $n$ times in
$w$ and $g$ appears $m$ times in $w$, it is a simple exercise left to the
reader to show that there exists a unique $2$\nbd{}cell of $P$ from $w$ to the
word
\[
f \cdots fg\cdots g
\]
where $f$ is repeated $n$ times and $g$ is repeated $m$ times.
Recall that the category $B^1(\mathbb{N}\times\mathbb{N})$ is nothing but the
monoid $\mathbb{N}\times\mathbb{N}$ considered as a category with only one
object, and let $F : P \to B^1(\mathbb{N}\times\mathbb{N})$ be the unique
$2$\nbd{}functor such that:
\begin{itemize}[label=-]
\item $F(f)=(1,0)$ and $F(g)=(0,1)$,
\item $F(\alpha)=1_{(1,1)}$.
\end{itemize}
This last equation makes sense since $(1,1)=(0,1)+(1,0)=(1,0)+(0,1)$. For every
$1$\nbd{}cell $w$ of $P$ (encoded as a finite words in the alphabet $\{f,g\}$)
such that $f$ appears $n$ times and $g$ appears $m$ times, we have
$F(w)=(n,m)$. Let us prove that $F$ is a Thomason equivalence using a dual of
\cite[Corollaire 5.26]{ara2020theoreme} (see Remark 5.20 of op.\ cit.). If we
write $\star$ for the only object of $B^1(\mathbb{N}\times\mathbb{N})$, what
we need to show is that the canonical $2$\nbd{}functor from $P/{\star}$ (see
\ref{paragr:comma}) to the terminal $2$\nbd{}category
\[
P/{\star} \to \sD_0
\]
is a Thomason equivalence. The $2$\nbd{}category $P/{\star}$ is described as
follows:
\begin{itemize}[label=-]
\item A $0$\nbd{}cell is a $1$\nbd{}cell of $B^1(\mathbb{N}\times
\mathbb{N})$.
\item For $(n,m)$ and $(n',m')$ two $0$\nbd{}cells of $P/{\star}$, a
$1$\nbd{}cell from $(n,m)$ to $(n',m')$ is a $1$\nbd{}cell $w$ of $P$ such
that the triangle
\[
\begin{tikzcd}[column sep=small,row sep=small]
\star \ar[rr,"F(w)"] \ar[rd,"{(n,m)}"']& & \star \ar[dl,"{(n',m')}"]\\
&\star&
\end{tikzcd}
\]
is commutative. More explicitly, if $F(w)=(n'',m'')$, the commutativity of
the previous triangle means
\[
n'+n''=n \text{ and } m'+m''=m.
\]
\item Given two parallel $1$\nbd{}cells $w$ and $w'$ of $P/\star$, a
$2$\nbd{}cell of $P/{\star}$ from $w$ to $w'$ is simply a $2$\nbd{}cell of
$P$ from $w$ to $w'$ seen as $1$\nbd{}cells of $P$.
\end{itemize}
From what we said earlier on the $1$\nbd{}cells and $2$\nbd{}cells of $P$, it
follows easily that for every $0$\nbd{}cell $(n,m)$ of $P/{\star}$, the
category
\[
P/{\star}((m,n),(0,0))
\]
has a terminal object, which is given by
\[
f\cdots fg\cdots g
\]
where $f$ is repeated $n$ times and $g$ is repeated $m$ times. Then, it
follows from \cite[Théorème 5.27 and Remarque 5.28]{ara2020theoreme} that
$P/{\star} \to \sD_0$ is a Thomason equivalence of $2$\nbd{}categories and
this proves that $F$ is a Thomason equivalence. Since
$B^1(\mathbb{N}\times\mathbb{N})\simeq B^1(\mathbb{N})\times B^1(\mathbb{N})$
and $B^1(\mathbb{N})$ has the homotopy type of $\sS_1$, we conclude that $P$
has the homotopy type of $\sS_1 \times \sS_1$, i.e.\ the homotopy type of the
torus.
Consider now the commutative square
\[
\begin{tikzcd}
\sH^{\sing}(P) \ar[r,"\sH^{\sing}(F)"] \ar[d,"\pi_{P}"] & \sH^{\sing}(B^1(\mathbb{N}\times\mathbb{N})) \ar[d,"\pi_{B^1(\mathbb{N}\times\mathbb{N})}"] \\
\sH^{\pol}(P) \ar[r,"\sH^{\pol}(F)"] &
\sH^{\pol}(B^1(\mathbb{N}\times\mathbb{N}))
\end{tikzcd}
\]
Since $F$ is a Thomason equivalence, the top horizontal arrow is an
isomorphism and since $B^1(\mathbb{N}\times\mathbb{N})$ is a
$1$\nbd{}category, it is \good{} (Theorem \ref{thm:categoriesaregood}), which
means that the right vertical arrow is an isomorphism. The $1$\nbd{}category
$B^1(\mathbb{N}\times \mathbb{N})$ is not free but since it has the homotopy
type of the torus, we have $H^{\sing}_k(B^1(\mathbb{N}\times \mathbb{N}))=0=H_k^{\pol}(B^1(\mathbb{N}\times \mathbb{N}))$
for $k\geq 2$ and it follows then from Corollary \ref{cor:polhmlgycofibrant}
and Paragraph \ref{paragr:polhmlgylowdimension} that the map canonical map
\[
\alpha^{\pol}_{B^{1}(\mathbb{N}\times\mathbb{N})} :
\sH^{\pol}(B^{1}(\mathbb{N}\times\mathbb{N})) \to \lambda(B^{1}(\mathbb{N}\times\mathbb{N}))
\]
is a quasi-isomorphism. Since $P$ is free, it follows that the map $\sH^{\pol}(F)$
can be identified with the image in $\ho(\Ch)$ of the map
\[
\lambda(F) : \lambda(P) \to \lambda(B^1(\mathbb{N}\times\mathbb{N})),
\]
which is easily checked to be a quasi-isomorphism.
By a 2-out-of-3 property, we deduce that $\pi_P : \sH^{\pol}(P) \to
\sH^{\sing}(P)$ is an isomorphism, which means by definition that $P$ is
\good{}.
\end{paragr}
% \begin{paragr}
% For every $k\geq 1$, let $P_k$ be the free $2$\nbd{}category defined as
% follows:
% \begin{itemize}[label=-]
% \item generating $0$\nbd{}cells: $A_0$, $A_1$, $A_2$, \dots, $A_{2k+1}$,
% \item generating $1$\nbd{}cells: $f,g : A_0 \to A_{2k+1}$ and $f_0,g_0 : A_0
% \to A_1$, $f_1,g_1 : A_1 \to A_2$, \dots, $f_{2k},g_{2k} : A_{2k} \to A_{2k+1}$,
% \item generating $2$\nbd{}cells: $\alpha : f \to $
% \end{itemize}
% For example, the case $k=2$ is pictured as
% \[
% \begin{tikzcd}
% A_0 \ar[r,bend left,"f_0"] \ar[r,bend right,dashed,"g_0"'] & A_1 \ar[r,bend left,"f_1"]
% \ar[r,bend right,"g_1"'] & A_2 \ar[r,bend right,dashed,""{name=C,above},""{name=F,below},"g_2"'] & A_3 \ar[r,bend left,"f_3"]
% \ar[r,bend right,"g_3"']& A_4 \ar[r,bend left,"f_4"] \ar[r,bend right,dashed,"g_5"'] & A_5
% \ar[from=1-1,to=1-6,bend left=75,""{name=A,below},"f"]
% \ar[from=A,to=C,"\beta"',bend right,Rightarrow,dashed]
% \ar[from=1-3,to=1-4,bend left,""{name=B,above},""{name=D,below},crossing over,"f_2"]
% \ar[from=A,to=B,"\alpha",bend left,Rightarrow,shorten >=2mm]
% \ar[from=1-1,to=1-6,bend right=75,""{name=E,above},"g"']
% \ar[from=E,to=D,"\gamma"',Rightarrow, bend right, crossing over]
% \ar[from=E,to=F,"\delta",Rightarrow, dashed,bend left,shorten >=2mm]
% \end{tikzcd}
% \]
% \end{paragr}
\section{The ``bubble-free'' conjecture}
\begin{definition}
Let $C$ be a $2$\nbd{}category. A \emph{bubble} (in $C$) is a $2$\nbd{}cell
$x$ of $C$ such that:
\begin{itemize}[label=-]
\item $x$ is not a unit,
\item $\src_0(x)=\trgt_0(x)$,
\item $\trgt(x)=\src(x)=1_{\src_0(x)}$.
\end{itemize}
\end{definition}
\begin{paragr}
In pictures, a bubble $x$ is represented as
\[
\begin{tikzcd}
A \ar[r,bend left=75,"1_A",""{name=A,below}] \ar[r,bend
right=75,"1_A"',pos=21/40,""{name=B,above}] &A,
\ar[from=A,to=B,"x",Rightarrow]
\end{tikzcd}
\text{ or }
\begin{tikzcd}
A \ar[loop,in=120,out=60,distance=1cm,"x"',Rightarrow]
\end{tikzcd}
\]
where $A=\src_0(x)=\trgt_0(x)$.
\end{paragr}
\begin{definition}\label{def:bubblefree2cat}
A $2$\nbd{}category is said to be \emph{bubble-free} if it has no bubbles.
\end{definition}
\begin{paragr}
The archetypal example of a $2$\nbd{}category that is \emph{not} bubble-free
is $B^2\mathbb{N}$. Another non-bubble $2$\nbd{}category is the one from Paragraph \ref{paragr:anothercounterexample}. It is
remarkable that of all the free $2$\nbd{}categories we have seen so far, these
are the only examples that are non-\good{}. This motivates the following conjecture.
\end{paragr}
\begin{conjecture}\label{conjecture:bubblefree}
A free $2$\nbd{}category is \good{} if and only if it is bubble-free.
\end{conjecture}
\begin{paragr}
At the time of writing, I do not have a real hint towards a proof of the above conjecture.
Yet, in light of all the examples seen in the previous section, it seems very
likely to be true. Note that we have also conjectured in Paragraph
\ref{paragr:conjectureH2} that for every $\oo$\nbd{}category $C$, we have
\[
H_2^{\pol}(C)\simeq H_2^{\sing}(C).
\]
If this conjecture on the second homology group is true, then conjecture \ref{conjecture:bubblefree} may be reformulated as: A free $2$\nbd{}category $P$ has trivial
singular homology groups strictly above dimension $2$ if and only if it is bubble-free.
\end{paragr}
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