Commit 8955f803 authored by Leonard Guetta's avatar Leonard Guetta
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slowly but surely

parent f08ffc43
......@@ -245,7 +245,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
Let $\begin{tikzcd} f : y \ar[r,shift left]&z :g\ar[l,shift left] \end{tikzcd}$ be an adjunction and $h : x \to y$ an equivalence with quasi-inverse $k : y \to x$. Then $fh$ is left adjoint to $kg$.
\end{lemma}
We can now state and prove the promised result.
\begin{proposition}
\begin{proposition}\label{prop:hmlgyderived}
Consider that $\oo\Cat$ is equipped with the Thomason weak equivalences. The abelianization functor $\lambda : \oo\Cat \to \Ch$ is strongly left derivable and the left derived morphism
\[
\LL \lambda : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)
......@@ -269,10 +269,10 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\[
\nu : (\Ch,\W_{\Ch}) \to (\oo\Cat,\W^{\Th}),
\]
\todo{Uniformiser les notations pour les é.f des cmplxs de chaines.} thanks to Lemma \ref{lemma:nuhomotopical}.
thanks to Lemma \ref{lemma:nuhomotopical}.\todo{Uniformiser les notations pour les é.f des cmplxs de chaines.}
\item The functor $N_{\omega}$ induces a morphism of localizers
\[
N_{\omega} : (\oo\Cat,\W^{\Th}) \to (\Psh{\Delta},\W_{\Delta})
N_{\omega} : (\oo\Cat,\W^{\Th}) \to (\Psh{\Delta},\W_{\Delta}),
\]
by definition of Thomason weak equivalences.
\item There is an isomorphism of functors $\lambda c_{\omega} \simeq \kappa$ (Lemma \ref{lemma:abelor}), hence an induced morphism of localizers
......@@ -293,9 +293,75 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\overline{\kappa} \overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \ar[r,shift left]& \ar[l,shift left] \Ho(\Ch) : M \overline{N_{\oo}} \overline{\nu} \simeq \overline{\nu}.
\end{tikzcd}
\]
From \todo{Critère de Gonzalez version dérivateur à écrire !}, we conclude that $\lambda : \oo\Cat \to \Ch$ is strongly left derivable and that $ \LL \lambda \simeq \overline{\kappa} \overline{N_{\oo}}$, which is, by definition, the ``true'' homology.
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.
\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
\[
\begin{tikzcd}
\ho(\oo\Cat^{\folk}) \ar[r] \ar[rd,"\sH^{\pol}"'] & \ho(\oo\Cat^{\Th}) \ar[d,"\sH"] \\
& \ho(\Ch).
\end{tikzcd}
\]
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
\[
\begin{tikzcd}
\oo\Cat \ar[r,"\mathrm{id}_{\oo\Cat}"]\ar[d] & \oo\Cat \ar[d] \ar[r,"\lambda"] & \Ch \ar[d] \\
\ho(\oo\Cat^{\Th}) \ar[r] &\ho(\oo\Cat^{can}) \ar[r,"\sH^{\pol}"] & \ho(\Ch),
\ar[from=2-2,to=1-3,"\alpha^{\folk}",Rightarrow]
\end{tikzcd}
\]
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
\[
\begin{tikzcd}
\ho(\oo\Cat^{\folk}) \ar[r] \ar[rd,"\sH^{\pol}"',""{name=A,above}] & \ho(\oo\Cat^{\Th}) \ar[d,"\sH"] \\
& \ho(\Ch)\ar[from=A,to=1-2,"\beta",Rightarrow]
\end{tikzcd}
\]
such that \todo{mettre diagrammes.}
For any $\oo$\nbd-category $X$, we shall refer to the map
\[
\beta_X : \sH^{\pol}(X) \to \sH(X)
\]
as the \emph{canonical comparison map.}
\end{paragr}
\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
\[
\beta_X : \sH^{\pol}(X) \to \sH(X)
\]
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,
\[
B_k=\begin{cases}\{\star\} \text{ if } k=0,1 \\ \mathbb{N} \text{ if } k=2.\end{cases}
\]
% 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 $\ho(\Ch)$):
\[
\begin{tikzcd}[column sep=small]
\mathbb{Z} & 0 \ar[l] & \ar[l] \mathbb{Z} & \ar[l] 0 & \ar[l] 0 & \ar[l] \cdots
\end{tikzcd}
\]
Hence, the polygraphic homology groups of $B$ are given by
\[
\sH^{\pol}_k(B)=\begin{cases} \mathbb{Z} \text{ if } k=0,2\\ 0 \text{ in other cases.}\end{cases}
\]
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.
\end{paragr}
......@@ -564,7 +564,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
Since $\Or_n$ is free, this last morphism is by definition a push-out of a coproduct of folk cofibrations, hence a folk cofibration.
\end{proof}
\begin{paragr}
\begin{paragr}\label{paragr:compweakeq}
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}
\[
......
......@@ -114,6 +114,7 @@
\newcommand{\ii}{\mathbf{i}} % a boldfont i
\newcommand{\bs}[1]{\ensuremath{\boldsymbol{#1}}} % a shortcut for \boldsymbol
\newcommand{\nbd}{\nobreakdash} % A shortcut for \nobreakdash
\newcommand{\nbar}{\mathbb{N}\cup\{ \oo \}} % Integers N with infinity
\newcommand{\Th}{\mathrm{Th}} %For Thomason equivalences related stuff
......@@ -153,6 +154,7 @@
\newcommand{\homoplax}{\ensuremath{\underline{\hom}_{\mathrm{oplax}}}}
\newcommand{\homlax}{\ensuremath{\underline{\hom}_{\mathrm{lax}}}}
% commentaires
\newcommand\remtt[1]{\texttt{[#1]}}
\newcommand\todo[1]{\remtt{TODO : #1}}
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