### dodo

parent 0c571127
 ... ... @@ -297,7 +297,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp \] 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 homotopy cocartesian in $\Cat$ (when equipped with Thomason equivalences). 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] \begin{example}[Killing a generator]\label{example:killinggenerator} Let $C$ be a free category and let $f : A \to B$ one of its generating arrow such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by killing'' $f$, i.e. defined with the following cocartesian square: $\begin{tikzcd} ... ... @@ -688,7 +688,7 @@ From this lemma, we deduce two useful criteria to detect Thomason equivalences o \begin{corollary}\label{cor:criterionThomeqI} Let F : C \to D be a 2-functor. If \begin{enumerate}[label=\alph*)] \item F_0 : C_0 \to D_0 is an bijection, \item F_0 : C_0 \to D_0 is a bijection, \end{enumerate} and \begin{enumerate}[resume*] ... ... @@ -898,7 +898,7 @@ For any n \geq 0, consider the following cocartesian square \[ \begin{tikzcd}[column sep=huge] A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend right=75,"g"',""{name=B,above}] \ar[r,"1_A" pos=1/3,""{name=C,above},""{name=D,below}]& A. \ar[from=A,to=C,Rightarrow,"\alpha"] \ar[from=B,to=D,"\beta",Rightarrow] \ar[from=A,to=C,Rightarrow,"\alpha"] \ar[from=B,to=D,"\beta" pos=9/20,Rightarrow] \end{tikzcd}$ Notice that this category has many non-trivial $2$\nbd{}cells and it is not ... ... @@ -932,7 +932,7 @@ Observe now that $P$ is obtained as the following amalgamated sum All the variations by reversing the direction of $\alpha$ or $\beta$ work exactly the same way. \end{paragr} \begin{paragr}[Sphère tête-bêche''] \begin{paragr}[Variations of spheres] Let $P$ the free $2$\nbd{}category defined as follows: \begin{itemize}[label=-] \item generating $0$\nbd{}cells: $A,B$, ... ... @@ -948,20 +948,82 @@ Observe now that $P$ is obtained as the following amalgamated sum \ar[from=B,to=A,bend right,Rightarrow,"\beta"'] \end{tikzcd} \] \end{paragr} Let us prove that this $2$\nbd{}category is \good{}. First, let $P'$ be the free We shall see that this $2$\nbd{}category is \good{}. First, 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}[item=-] \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 functor induced by $F$ $F_{A,B} : P(A,B) \to P'(F(A),F(B))$ is a Thomason equivalence of categories (Corollary \ref{cor:criterionThomeqI}). 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 let $P''$ be a copy of $\sS_2$ labelled as follows: $\begin{tikzcd} A'' \ar[r,bend left=75,"i",""{name=A,below}] \ar[r,bend right=75,"j"',""{name=B,above}] & B'' \ar[from=A,to=B,bend right,Rightarrow,"\delta"'] \ar[from=A,to=B,bend left ,Rightarrow,"\epsilon"] \end{tikzcd}$ and let $G : P'' \to P'$ be the unique $2$\nbd{}functor such that \begin{itemize}[label=-] \item $G(A'')=A'$ and $G(B'')=B'$, \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 $\sS_2$. Finally, consider the commutative diagram of $\ho(\Ch)$ $\begin{tikzcd}[column sep=huge] \sH^{\sing}(P) \ar[r,"\sH^{\sing}(F)"] \ar[d,"\pi_P"] & \sH^{\sing}(P') \ar[d,"\pi_{P'}"] & \sH^{\sing}(P'') \ar[l,"\sH^{\sing}(G)"'] \ar[d,"\pi_{P''}"] \\ \sH^{\pol}(P) \ar[r,"\sH^{\pol}(F)"'] & \sH^{\pol}(P') & \sH^{\pol}(P''). \ar[l,"\sH^{\pol}(G)"] \end{tikzcd}$ Since $F$ and $G$ are Thomason equivalences, the two bottom horizontal morphisms are isomorphisms. Besides, a simple computation using Proposition \ref{prop:abelianizationfreeoocat}, which we left to the reader, shows that the two top horizontal morphisms are isomorphisms. Since $P''$ is \good{} (Proposition \ref{prop:spheresaregood}), the morphism is $\pi_{P''}$ is an isomorphism. This implies that $\pi_{P'}$ and $\pi_{P'}$ are isomorphisms, which means by definition that $P$ and $P'$ are \good{}. \end{paragr} \begin{paragr} \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 ... ...
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