Commit 18c9ef9b authored by Leonard Guetta's avatar Leonard Guetta
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Starting a new chapter

parent 3c76aeef
......@@ -296,50 +296,103 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
From \todo{Critère de Gonzalez version dérivateur à écrire !}, we conclude that $\lambda : \oo\Cat \to \Ch$ is strongly left derivable and that $ \LL^{\Th} \lambda \simeq \overline{\kappa} \overline{N_{\oo}}$, which is, by definition, the ``true'' homology.
\end{proof}
\begin{remark}
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.
Beware that neither $c_{\oo} : \Psh{\Delta} \to \oo\Cat$ nor $\lambda : \oo\Cat \to \Ch$ preserve weak equivalences \todo{Uniformiser appellations}, but this does not contradict the fact that $\lambda c_{\oo} : \Psh{\Delta} \to \Ch$ does.
\end{remark}
\section{Polygraphic homology vs. ``true'' homology}
\begin{paragr}\label{paragr:univmor}
Since $\sH : \Ho(\oo\Cat) \to \Ho(\Ch)$ is the left derived of the abelianization functor, it comes with a $2$-square
\[
\begin{tikzcd}
\oo\Cat \ar[d] \ar[r,"\lambda"] & \Ch \ar[d] \\
\Ho(\oo\Cat^{\Th}) \ar[r,"\sH"] & \Ho(\Ch).
\ar[from=2-1,to=1-2,"\alpha^{\Th}",Rightarrow]
\end{tikzcd}
\]
The $2$-morphism $\alpha^{\Th}$ admits the following description. By post-composing the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$
\iffalse
\begin{tikzcd}
c_{\oo} : \Psh{\Delta} \ar[r,shift left] &\oo\Cat : N_{\oo}\ar[l,shift left]
\end{tikzcd}
\fi
with the abelianization functor, we obtain $2$-morphism
\[
\lambda c_{\oo} N_{\oo} \Rightarrow \lambda,
\]
and thus the promised $2$-square:
\[
\begin{tikzcd}[column sep=huge]
\oo\Cat \ar[d]\ar[r,bend left,"\lambda",""{name=A,below}] \ar[r,"\lambda c_{\oo} N_{\oo}"',""{name=B,above}] & \Ch \ar[d] \\
\Ho(\oo\Cat^{\Th}) \ar[r,"\sH\,\simeq\,\overline{\lambda c_{\oo}} \overline{N_{\oo}}"'] & \Ho(\Ch).
\ar[from=B,to=A,Rightarrow]
\end{tikzcd}
\]
\end{paragr}
\section{Comparing homologies}
\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
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 then a morphism of op-prederivators
\[\J : \Ho(\oo\Cat^{\folk}) \to \Ho(\oo\Cat^{\Th}).\]
This yields the following triangle,
\begin{equation}\label{cmprisontrngle}
\begin{tikzcd}
\ho(\oo\Cat^{\folk}) \ar[r,"\J"] \ar[rd,"\sH^{\pol}"'] & \ho(\oo\Cat^{\Th}) \ar[d,"\sH"] \\
& \ho(\Ch).
\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
As we shall see later, this triangle is \emph{not} commutative, even up to an iso. However, it can be filled up with a $2$-morphism. Indeed, consider the following $2$-diagram
\[
\begin{tikzcd}
\oo\Cat \ar[r,"\mathrm{id}_{\oo\Cat}"]\ar[d] & \oo\Cat \ar[d] \ar[r,"\lambda"] & \Ch \ar[d] \\
\ho(\oo\Cat^{\Th}) \ar[r] &\ho(\oo\Cat^{\folk}) \ar[r,"\sH^{\pol}"] & \ho(\Ch),
\ar[from=2-2,to=1-3,"\alpha^{\folk}",Rightarrow]
\Ho(\oo\Cat^{\folk}) \ar[r] &\ho(\oo\Cat^{\Th}) \ar[r,"\sH"] & \Ho(\Ch),
\ar[from=2-2,to=1-3,"\alpha^{\Th}",Rightarrow]
\end{tikzcd}
\]
where the left square is commutative and the $2$-morphism on the right square is the one from Paragraph \ref{paragr:univmor}. Since the polygraphic homology is, by definition, the left derived of the abelianization functor when $\oo\Cat$ is equipped with canonical weak equivalences, the universal property of strongly derivable functor yields a canonical $2$-morphism
\begin{equation}\label{cmparisonmapdiag}
\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=1-2,to=A,"\pi",Rightarrow]
\end{tikzcd}
\end{equation}
such that the triangle
\[
\begin{tikzcd}
\lambda \ar[r] \ar[dr] & \sH \J \ar[d] \\
& \sH^{\pol}
\end{tikzcd}
\]
where the left square is commutative and the natural transformation in the right square is the canonical derivation morphism. \todo{Uniformiser appellations.}
From Proposition \ref{prop:hmlgyderived} we know that the ``true'' homology functor
\[\sH : \ho(\oo\Cat^{\Th}) \to \ho(\Ch)\]
is the left derived functor of the abelianization functor
\[\lambda : \oo\Cat \to \Ch.\]
Hence, the universal property of left derived functors yields a unique natural transformation
is commutative \todo{Compléter diagramme précédent.}
\iffalse
\[
\begin{tikzcd}
\oo\Cat \ar[r,"\mathrm{id}_{\oo\Cat}"]\ar[d] & \oo\Cat \ar[d] \ar[r,"\lambda"] & \Ch \ar[d] \\
\Ho(\oo\Cat^{\folk}) \ar[r]&\Ho(\oo\Cat^{\Th}) \ar[r,"\sH"] & \Ho(\Ch),
\ar[from=2-2,to=1-3,"\alpha^{\Th}",Rightarrow]
\end{tikzcd}
=
\begin{tikzcd}
\oo\Cat \ar[r,"\lambda"] \ar[d] & \Ch \ar[d] \\
\Ho(\oo\Cat) \ar[r,"\sH^{\pol}"]
\end{tikzcd}
\]\fi
The $2$-morphism \ref{cmparisonmapdiag} induces a natural transformation at the level of localized \emph{categories}
\[
\begin{tikzcd}
\ho(\oo\Cat^{\folk}) \ar[r] \ar[rd,"\sH^{\pol}"',""{name=A,above}] & \ho(\oo\Cat^{\Th}) \ar[d,"\sH"] \\
& \ho(\Ch)\ar[from=A,to=1-2,"\pi",Rightarrow]
& \ho(\Ch)\ar[from=1-2,to=A,"\pi",Rightarrow]
\end{tikzcd}
\]
such that \todo{mettre diagrammes.}
For any $\oo$\nbd-category $X$, we shall refer to the map
Since $\J$ is nothing but the identity on objects, for any $\oo$\nbd-category $X$, the previous natural transformation yields a map
\[
\pi_X : \sH^{\pol}(X) \to \sH(X)
\pi_X : \sH(X) \to \sH^{\pol}(X),
\]
as the \emph{canonical comparison map.}
which we shall refer to as the \emph{canonical comparison map.}
\end{paragr}
This motivates the following definition.
\begin{definition}
An $\oo$\nbd-category $X$ is said to be \emph{\good} when the canonical comparison map
An $\oo$\nbd-category $X$ is said to be \emph{\good{}} when the canonical comparison map
\[
\pi_X : \sH^{\pol}(X) \to \sH(X)
\pi_X : \sH(X) \to \sH^{\pol}(X)
\]
is an isomorphism of $\ho(\Ch)$.
\end{definition}
......@@ -354,7 +407,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\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}\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.
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,
\[
......@@ -372,12 +425,12 @@ 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. This proves that $B$ is \emph{not} \good.
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.
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.
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
......@@ -395,9 +448,31 @@ We shall now proceed to give an abstract criterion to find \good $\oo$-categorie
& \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
\[
d : I \to \oo\Cat
\]
We will see later that $\J$ 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.}
\end{paragr}
\begin{proposition}
Let $X$ be an $\oo$\nbd-category. Suppose that there exists a diagram
\[
d : I \to \oo\Cat
\]
and a co-cone
\[
(\varphi_i : d(i) \to \oo\Cat)_{i \in \Ob(I)}
\]
such that:
\begin{enumerate}[label=\roman*)]
\item For every $i \in \Ob(I)$, the $oo$\nbd-category $d(i)$ is \good{}.
\item The canonical morphism
\[
\hocolim^{\folk}d \to X
\]
is an isomorphism of $\ho(\oo\Cat^{\folk})$.
\item The canonical morphism
\[
\hocolim^{\Th}d \to X
\]
is an isomorphism of $\ho(\oo\Cat^{\Th})$.
\end{enumerate}
Then the $\oo$\nbd-category $X$ is \good{}.
\end{proposition}
\todo{Preuve à écrire. Section à reprendre un peu}
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