Commit cb36cb3c by Leonard Guetta

### OMG! 6.5.2 is complete

parent aa388927
 ... ... @@ -707,7 +707,7 @@ From this lemma, we deduce two useful criteria to detect Thomason equivalences o \] The result follows then from Lemma \ref{bisimpliciallemma} and the fact that weak equivalences of simplicial sets are stable by coproducts and finite products. \end{proof} \begin{corollary} \begin{corollary}\label{cor:criterionThomeqII} Let $F : C \to D$ be a $2$-functor. If for every $k \geq 0$, $V(F)_k : V(C)_k \to V(D)_k$ is a Thomason equivalence of $1$-categories, then $F$ is a Thomason equivalence of $2$-categories. \end{corollary} ... ... @@ -953,8 +953,9 @@ Observe now that $P$ is obtained as the following amalgamated sum \ar[from=A,to=B,bend right,Rightarrow,"\alpha"'] \ar[from=B,to=A,bend right,Rightarrow,"\beta"'] \end{tikzcd} \] We shall see that this $2$\nbd{}category is \good{}. First, let $P'$ be the free \] Let $P'$ be the free $2$\nbd{}category defined as follows: \begin{itemize}[label=-] \item generating $0$\nbd{}cells: $A',B'$, ... ... @@ -1003,29 +1004,120 @@ The functor $F_{A,B}$ comes from a morphism of reflexive graphs and is obtained \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: Now consider (a copy of) $\sS_2$ labelled as follows: $\begin{tikzcd} A'' \ar[r,bend left=75,"i",""{name=A,below}] \ar[r,bend right=75,"j"',""{name=B,above}] & B'' \overline{A} \ar[r,bend left=75,"i",""{name=A,below}] \ar[r,bend right=75,"j"',""{name=B,above}] & \overline{B} \ar[from=A,to=B,bend right,Rightarrow,"\delta"'] \ar[from=A,to=B,bend left ,Rightarrow,"\epsilon"] \end{tikzcd}$ and let $G : P'' \to P'$ be the unique $2$\nbd{}functor such that and let $G : \sS_2 \to P'$ be the unique $2$\nbd{}functor such that \begin{itemize}[label=-] \item $G(A'')=A'$ and $G(B'')=B'$, \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 $\sS_2$. Finally, consider the commutative diagram of $\ho(\Ch)$ 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 $\sS_2$. 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 picture, 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 $H(\overline{A})=(\overline{B})=A''$, \item $H(i)=l$ and $H(j)=1_{A''}$, \item $H(\delta)=\lambda$ and $H(\epsilon)=\lambda$. \end{itemize} Let us prove that $H$ is a Thomason equivalence using Corollary \ref{cor:criterionThomeqII}. To do that, we have to compute $V(H)_k : V(\sS_2)_k \to V(P'')_k$ for every $k\leq 0$. For $k=0$, the category $V(\sS_2)_0$ 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 $V(P'')$ is the free category on the graph $\begin{tikzcd} A \ar[loop above,"l"] \end{tikzcd}$ and $V(H)_0$ 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 $V(\sS_2)_k$ has two objects $\overline{A}$ and $\overline{B}$ and an arrow $\overline{A} \to \overline{B}$ is a $k$\nbd{}uple of either one of the following form \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, $V(\sS_2)_k$ is the free category on the graph with two objects and $2k+2$ parallel arrows between these two objects. In order to compute $V(P'')_k$, 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. From this observation, it is easily seen that the category $V(P'')_k$ is the free category on the graph that has one objects $A''$ and $2k+1$ generating arrows which are of either one of the following form \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 $V(H)_k$ comes from a morphism a 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 $\ho(\Ch)$ $\begin{tikzcd}[column sep=huge] \sH^{\sing}(P) \ar[r,"\sH^{\sing}(F)"] \ar[d,"\pi_P"] & \sH^{\sing}(P') \ar[d,"\pi_{P'}"] & \sH^{\sing}(P'') \ar[l,"\sH^{\sing}(G)"'] \ar[d,"\pi_{P''}"] \\ \sH^{\pol}(P) \ar[r,"\sH^{\pol}(F)"'] & \sH^{\pol}(P') & \sH^{\pol}(P''). \ar[l,"\sH^{\pol}(G)"] \sH^{\sing}(P) \ar[r,"\sH^{\sing}(F)"] \ar[d,"\pi_P"] & \sH^{\sing}(P') \ar[d,"\pi_{P'}"] & \sH^{\sing}(\sS_2) \ar[l,"\sH^{\sing}(G)"'] \ar[d,"\pi_{\sS_2}"] \ar[r,"\sH^{\sing}(H)"]&\sH^{\sing}(P'') \ar[d,"\pi_{P''}"]\\ \sH^{\pol}(P) \ar[r,"\sH^{\pol}(F)"'] & \sH^{\pol}(P') & \sH^{\pol}(\sS_2) \ar[l,"\sH^{\pol}(G)"] \ar[r,"\sH^{\pol}(H)"']&\sH^{\pol}(P''). \end{tikzcd}$ Since $F$ 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{}. 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 left 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} Let $P$ be the free $2$\nbd{}category defined as follows: ... ... @@ -1040,7 +1132,7 @@ Since $F$ and $G$ are Thomason equivalences, the two bottom horizontal morphisms 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} \text{ or } \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"'] ... ... @@ -1072,7 +1164,29 @@ Since $F$ and $G$ are Thomason equivalences, the two bottom horizontal morphisms \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 $K(\mathbb{Z},2)$-space (\ref{paragr:bubble}), this proves that $P$ have 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 above dimension $2$, which means that $P$ is \emph{not} \good{}. \end{paragr} \end{paragr} \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{}cell: $f : A \to A$, \item generating $2$\nbd{}cells: $\alpha : f \Rightarrow 1_A$ and $\beta: f \Rightarrow 1_A$. \end{itemize} In picture, this gives $\begin{tikzcd}[column sep=huge] A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend right=75,"f"',""{name=B,above}] \ar[r,"1_A" pos=1/3,""{name=C,above},""{name=D,below}]& A \ar[from=A,to=C,Rightarrow,"\alpha"] \ar[from=B,to=D,"\beta" pos=9/20,Rightarrow] \end{tikzcd} \qquad \quad \text{ or } \begin{tikzcd} A. \ar[loop,in=30,out=150,distance=3cm,"f",""{name=A,below}] \ar[from=A,to=1-1,bend right,Rightarrow,"\alpha"'] \ar[from=A,to=1-1,bend left,Rightarrow,"\beta"] \end{tikzcd}$ We shall now prove that $P$ is \good{} using the techniques introduced in \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 ... ...
No preview for this file type
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!