Commit 3c76aeef authored by Leonard Guetta's avatar Leonard Guetta
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I need to change the orientation of the canonical comparison map, which is the...

I need to change the orientation of the canonical comparison map, which is the wrong way. Moreover, I reaally need to write down Proposition 4.5.6, the most important technical result of the dissertation.
parent 8955f803
......@@ -299,19 +299,19 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
Beware that neither $c_{\oo} : \Psh{\Delta} \to \oo\Cat$ nor $\lambda : \oo\Cat \to \Ch$ preserve weak equivalences \todo{Uniformiser appellations}, but this doesn't contradict the fact that $\lambda c_{\oo} : \Psh{\Delta} \to \Ch$ does.
\end{remark}
\section{Polygraphic homology vs. ``true'' homology}
\begin{paragr}
Recall from Paragraph \ref{paragr:compweakeq} that the identity functor on $\oo\Cat$ induces a morphism of localizer $(\oo\Cat,\W^{\folk}) \to (\oo\Cat,\W^{Th})$. Hence, we have a triangle of functors
\[
\begin{paragr}\label{paragr:cmparisonmap}
Recall from Paragraph \ref{paragr:compweakeq} that the identity functor on $\oo\Cat$ induces a morphism of localizers $(\oo\Cat,\W^{\folk}) \to (\oo\Cat,\W^{\Th})$, and hence a functor $ \J : \ho(\oo\Cat^{\folk} \to \ho(\oo\Cat^{\Th})$. Together with the polygraphic and ``true'' homology functors, this yields a triangle of functors
\begin{equation}\label{cmprisontrngle}
\begin{tikzcd}
\ho(\oo\Cat^{\folk}) \ar[r] \ar[rd,"\sH^{\pol}"'] & \ho(\oo\Cat^{\Th}) \ar[d,"\sH"] \\
\ho(\oo\Cat^{\folk}) \ar[r,"\J"] \ar[rd,"\sH^{\pol}"'] & \ho(\oo\Cat^{\Th}) \ar[d,"\sH"] \\
& \ho(\Ch).
\end{tikzcd}
\]
\end{equation}
As we shall see later, this triangle is \emph{not} commutative, even up to an iso. However, it can be filled up with a natural transformation. Indeed, consider the following $2$-diagram
\[
\begin{tikzcd}
\oo\Cat \ar[r,"\mathrm{id}_{\oo\Cat}"]\ar[d] & \oo\Cat \ar[d] \ar[r,"\lambda"] & \Ch \ar[d] \\
\ho(\oo\Cat^{\Th}) \ar[r] &\ho(\oo\Cat^{can}) \ar[r,"\sH^{\pol}"] & \ho(\Ch),
\ho(\oo\Cat^{\Th}) \ar[r] &\ho(\oo\Cat^{\folk}) \ar[r,"\sH^{\pol}"] & \ho(\Ch),
\ar[from=2-2,to=1-3,"\alpha^{\folk}",Rightarrow]
\end{tikzcd}
\]
......@@ -325,35 +325,44 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\[
\begin{tikzcd}
\ho(\oo\Cat^{\folk}) \ar[r] \ar[rd,"\sH^{\pol}"',""{name=A,above}] & \ho(\oo\Cat^{\Th}) \ar[d,"\sH"] \\
& \ho(\Ch)\ar[from=A,to=1-2,"\beta",Rightarrow]
& \ho(\Ch)\ar[from=A,to=1-2,"\pi",Rightarrow]
\end{tikzcd}
\]
such that \todo{mettre diagrammes.}
For any $\oo$\nbd-category $X$, we shall refer to the map
\[
\beta_X : \sH^{\pol}(X) \to \sH(X)
\pi_X : \sH^{\pol}(X) \to \sH(X)
\]
as the \emph{canonical comparison map.}
\end{paragr}
\begin{paragr}
This motivates the following definition.
\begin{definition}
An $\oo$\nbd-category $X$ is said to be \emph{\good} when the canonical comparison map
\[
\pi_X : \sH^{\pol}(X) \to \sH(X)
\]
is an isomorphism of $\ho(\Ch)$.
\end{definition}
\iffalse \begin{paragr}
We can now finally properly state the question that was tried to be answered in this doctoral dissertation:
\begin{center}
For which $\oo$-categories $X$ is the canonical comparison map
\[
\beta_X : \sH^{\pol}(X) \to \sH(X)
\pi_X : \sH^{\pol}(X) \to \sH(X)
\]
an isomorphism ?
\end{center}
The rest of this document is devoted to (partially) answering this question. We start by giving in the next paragraph an example due to Ara and Maltsiniotis of an $\oo$-category for which the comparison map is \emph{not} an isomorphism.
\end{paragr}
\begin{paragr}
Let $B$ the commutative monoid $(\mathbb{N},+)$ considered as $2$-category.\footnote{The letter $B$ stands for ``bubble''.} That is,
\end{paragr}\fi
We begin by spelling out an example due to Ara and Maltsiniotis of an $\oo$-category which is \emph{not} \good. Examples of \good $\oo$-categories will be presented later. The rest of this dissertation is devoted to the study of \good $\oo$\nbd-categories.
\begin{paragr}
Let $B$ the commutative monoid $(\mathbb{N},+)$ considered as $2$-category.\footnote{The letter $B$ stands for ``bubble''.} That is,
\[
B_k=\begin{cases}\{\star\} \text{ if } k=0,1 \\ \mathbb{N} \text{ if } k=2.\end{cases}
B_k=\begin{cases}\{\star\} \text{ if } k=0,1 \\ \mathbb{N} \text{ if } k=2,\end{cases}
\]
and the two compositions laws on the set of $2$-cells coincide and are given by integral addition. Note that we have already encountered this contruction in the proof of Lemma \ref{lemma:abelpol}.
% as a unique $0$-cell, no non trivial $1$-cell, the set of non-negative integers $\mathbb{N}$ as the set of $2$-cells and the two composition laws on $2$-cells coincide and are given by the addition. All axioms of $2$-categories are trivial except maybe for the exchange law, which follows from the commutativity of the integral addition.
\todo{Faire le lien avec la preuve lemme abelpol?}
This $2$-category is free in the sense of \todo{ref}, namely it has a unique generating $0$-cell, no generating $1$-cell and the integer $1$ as the unique generating $2$-cell. It follows from Lemma \ref{lemma:abelpol} that $\sH^{\pol}(B)$ is given by the following chain complex (seen as an object of $\ho(\Ch)$):
This $2$-category is free in the sense of \todo{ref}, namely it has a unique generating $0$\nbd-cell, no generating $1$\nbd-cell and the integer $1$ as the unique generating $2$\nbd-cell. It follows from Lemma \ref{lemma:abelpol} that $\sH^{\pol}(B)$ is given by the following chain complex (seen as an object of $\ho(\Ch)$):
\[
\begin{tikzcd}[column sep=small]
\mathbb{Z} & 0 \ar[l] & \ar[l] \mathbb{Z} & \ar[l] 0 & \ar[l] 0 & \ar[l] \cdots
......@@ -363,5 +372,32 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\[
\sH^{\pol}_k(B)=\begin{cases} \mathbb{Z} \text{ if } k=0,2\\ 0 \text{ in other cases.}\end{cases}
\]
On the other hand, it was proven in \cite[Theorem 4.9 and Example 4.10]{ara2019quillen} that (the nerve of) $B$ is a $K(\mathbb{Z},2)$-space. In particular, it has non-trivial homology groups in every even dimension.
On the other hand, it was proven in \cite[Theorem 4.9 and Example 4.10]{ara2019quillen} that (the nerve of) $B$ is a $K(\mathbb{Z},2)$-space. In particular, it has non-trivial homology groups in every even dimension. This proves that $B$ is \emph{not} \good.
\end{paragr}
\iffalse\begin{remark}
The previous example of non \good $\oo$-category also proves that triangle \ref{cmprisontrngle} cannot be commutative up to an iso.
\end{remark}\fi
We shall now proceed to give an abstract criterion to find \good $\oo$-categories. In the rest of this dissertation, we will exploit this criterion to exhibit the largest classes possible of \good $\oo$-categories.
\begin{paragr}
Both the polygraphic homology
\[
\sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch)
\]
and the ``true'' homology
\[
\sH : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)
\]
are homotopy cocontinous (Definition \ref{def:cocontinuous}). In the first case, this follows from Theorem \ref{thm:cisinskiII} and the fact that $\lambda : \oo\Cat \to \Ch$ is left Quillen with respect to the canonical model structure on $\oo\Cat$. In the second case, this follows from the fact that $\overline{N_{\oo}} : \Ho(\oo\Cat^{\Th} \to \Ho(\Ch)$ induces an equivalence of op-prederivators and that $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen. Besides, the construction of the canonical comparison map from Paragraph \ref{paragr:cmparisonmap} can be reproduced \emph{mutatis mutandis} in the $2$\nbd-category of op-prederivators, yielding a $2$-morphism of op-prederivators
\[
\begin{tikzcd}
\Ho(\oo\Cat^{\folk}) \ar[r,"\J"] \ar[rd,"\sH^{\pol}"',""{name=A,above}] & \Ho(\oo\Cat^{\Th}) \ar[d,"\sH"] \\
& \Ho(\Ch).\ar[from=A,to=1-2,"\pi",Rightarrow]
\end{tikzcd}
\]
We will see later that the top arrow of the previous diagram, which is induced by the identity functor on $\oo\Cat$, cannot be homotopy cocontinuous as if it were every $\oo$\nbd-category would be \good. \todo{Mettre ref interne de où il sera montré que le morphisme n'est pas cocontinu.}
However, as for any morphism of op-prederivators \todo{ref}, for any diagram
\[
d : I \to \oo\Cat
\]
\end{paragr}
......@@ -568,7 +568,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
In particular, the previous lemma implies that the identity functor on $\oo\Cat$ induces a morphism of localizers $(\oo\Cat,\W^{\folk}) \to (\oo\Cat,\W^{\Th})$, which then induces a morphism of op-prederivators
% \begin{equation}\label{cantoTh}
\[
\Ho(\oo\Cat^{\folk}) \to \Ho(\oo\Cat^{\Th}).
\mathcal{J} : \Ho(\oo\Cat^{\folk}) \to \Ho(\oo\Cat^{\Th}).
\]
%\end{equation}
\iffalse Note that for every small category $A$, the functor
......
......@@ -70,6 +70,7 @@
\newcommand{\R}{\ensuremath{\mathcal{R}}} %Idem
\newcommand{\T}{\ensuremath{\mathcal{T}}} %Idem
\newcommand{\G}{\ensuremath{\mathcal{G}}} %Idem
\newcommand{\J}{\ensuremath{\mathcal{J}}} %Idem
%Small categories
......@@ -123,6 +124,8 @@
\newcommand{\pol}{\mathrm{pol}} %For polygraphic related stuff
\newcommand{\good}{homologically coherent} %This is provisional. I need to find a good terminology
% squelette
\newcommand{\sk}{\mathrm{sk}}
......
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