Commit d12aa0fd authored by Leonard Guetta's avatar Leonard Guetta
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slowly but surely

parent efe5a564
......@@ -49,9 +49,9 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv
\end{proof}
In particular, this means that we have a morphism of localizers $\kappa : (\Psh{\Delta},\W_{\Delta}) \to (\Ch,\W_{\Ch})$, where $\W_{\Ch}$ is the class of quasi-isomorphisms.
\begin{definition}\label{def:hmlgycat}
The \emph{homology functor for $\oo$-categories} $\sH : \ho(\oo\Cat) \to \ho(\Ch)$ is defined as the following composition
The \emph{homology functor for $\oo$-categories} $\sH : \ho(\oo\Cat^{\Th}) \to \ho(\Ch)$ is defined as the following composition
\[
\sH : \ho(\oo\Cat) \overset{\overline{N_{\omega}}}{\longrightarrow} \ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \ho(\Ch).
\sH : \ho(\oo\Cat^{\Th}) \overset{\overline{N_{\omega}}}{\longrightarrow} \ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \ho(\Ch).
\]
For an $\oo$-category $X$, $\sH(X)$ is the \emph{homology of $X$}.
\end{definition}
......@@ -65,7 +65,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\begin{paragr}
We will also denote by $\sH$ the morphism of op-prederivators defined as the following composition
\[
\sH : \Ho(\oo\Cat) \overset{\overline{N_{\omega}}}{\longrightarrow} \Ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \Ho(\Ch).
\sH : \Ho(\oo\Cat^{\Th}) \overset{\overline{N_{\omega}}}{\longrightarrow} \Ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \Ho(\Ch).
\]
It follows from \todo{mettre références (interne)} that this morphism is cocontinuous. This property will be extremely important later.
\end{paragr}
......@@ -77,14 +77,14 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\]
for all $x,y \in X_n$ that are $k$-composable for some $k<n$. For $n=0$, we have $\lambda_0(X)=\mathbb{Z}X_0$. For $n>0$, consider the linear map
\begin{align*}
\mathbb{Z}X_n \to \mathbb{Z}X_{n-1}\\
\mathbb{Z}X_n &\to \mathbb{Z}X_{n-1}\\
x \in X_n &\mapsto t(x)-s(x).
\end{align*}
The axioms of $\oo$-categories implies that it induces a map
\[
\partial : \lambda_{n}(X) \to \lambda_{n-1}(X)
\]
which furtheremore satisfy the equation $\partial \circ \partial = 0$. Thus, all this data defines a chain complex $\lambda(X)$:
which furtheremore satisfy the equation $\partial \circ \partial = 0$. Thus, we have defined a chain complex $\lambda(X)$:
\[
\lambda_0(X) \overset{\partial}{\longleftarrow} \lambda_1(X) \overset{\partial}{\longleftarrow} \lambda_2(X) \overset{\partial}{\longleftarrow} \cdots
\]
......@@ -95,7 +95,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\end{align*}
induces a map
\[
\lambda(f)_n : \lambda_n(X) \to \lambda_n(Y).
\lambda_n(f) : \lambda_n(X) \to \lambda_n(Y).
\]
Since $f$ commutes with source and target, we obtain a morphism of chain complexes
\[
......@@ -114,11 +114,121 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\]
where the map on the left is induced by the canonical inclusion of $E$ in $X_n$ and the map on the right is the quotient map from the definition of $\lambda_n(X)$.
\end{paragr}
\begin{lemma}
\begin{lemma}\label{lemma:abelpol}
Let $X$ be a \emph{free} $\oo$-category and let $\Sigma=(\Sigma_n)_{n \in \mathbb{N}}$ be its basis. For every $n \in \mathbb{N}$, the map
\[
\mathbb{Z}\Sigma_n \to \lambda_n(X)
\]
from the previous paragraph is an isomorphism.
\end{lemma}
\section{``True'' homology as derived abelianization}
\begin{proof}
Let $G$ be an abelian group. For any $n \in \mathbb{N}$, we define an $n$-category $B^nG$ with:
\begin{itemize}
\item[-] $(B^nG)_{k}$ is a singleton set for every $k < n$,
\item[-] $(B^nG)_n = G$
\item[-] for all $x$ and $y$ in $G$ and $i<n$,
\[x \ast_i y := x +y.\]
\end{itemize}
It it straightforward to check that this defines an $n$-category. Note that the previous definition would still make sense with $G$ an abelian \emph{monoid}. Moreover, when $n=1$, we didn't even need it to be abelian, but for $n\geq 2$ this hypothesis is necessary because of the Eckmann-Hilton argument. For $n=0$, we only needed that $G$ was a set.
This defines a functor
\[
\begin{aligned}
B^n : \Ab &\to n\Cat\\
G &\mapsto B^nG,
\end{aligned}
\]
which is easily seen to be right adjoint to the functor
\[
\begin{aligned}
n\Cat &\to \Ab\\
X &\mapsto \lambda_n(X).
\end{aligned}
\]
Now if $X$ is an $\oo$-category, then $\lambda_n(X_{\leq n})=\lambda_n(X)$ \todo{Uniformiser les notations} and if $X$ is free with basis $\Sigma=(\Sigma_n)_{n \in \mathbb{N}}$, then for any abelian group $G$, the canonical inclusion $\Sigma_n \hookrightarrow X_n$ induces a natural isomorphism
\[
\Hom_{n\Cat}(X_{\leq n},B^nG) \simeq \Hom_{\Set}(\Sigma_n,\vert G \vert),
\]
where $\vert G \vert$ is the underlying set of $G$. Altogether, we have
\begin{align*}
\Hom_{\Ab}(\lambda_n(C),G) &\simeq \Hom_{\Ab}(\lambda_n(C_{\leq n}),G)\\
&\simeq \Hom_{n\Cat}(C_{\leq n},B^nG)\\
&\simeq \Hom_{\Set}(\Sigma_n,\vert G \vert)\\
&\simeq \Hom_{\Ab}(\mathbb{Z}\Sigma_n,G).
\end{align*}
\end{proof}
\begin{lemma}\label{lemma:adjlambda}
The functor $\lambda$ is a left ajdoint.
\end{lemma}
\begin{proof}
The category of chain complexes is equivalent to the category $\omega\Cat(\Ab)$ of $\omega$-categories internal to abelian groups (see \cite[Theorem 3.3]{bourn1990another}) and with this identification, the functor $\lambda : \omega\Cat \to \omega\Cat(\Ab)$ is nothing but the left adjoint of the canonical forgetful functor $\omega\Cat(\Ab) \to \omega\Cat$.
\end{proof}
\section{Polygraphic homology}
\begin{lemma}\label{lemma:abeloplax}
Let $u, v : X \to Y$ be two $\oo$-functors. If there is an oplax transformation $\alpha : u \Rightarrow v$, then there is a homotopy of chain complexes from $\lambda(u)$ to $\lambda(v)$.
\end{lemma}
\begin{proof}
For any $n$-cell $x$ of $X$ (resp. $Y$), let us use the notation $[x]$ for the image of $x$ in $\lambda_n(X)$ (resp. $\lambda_n(Y)$).
Let $h_n$ be the map
\[
\begin{aligned}
h_n : \lambda_n(X) &\to \lambda_{n+1}(Y)\\
[x] & \mapsto [\alpha_x].
\end{aligned}
\]
The formulas for oplax transformations from Paragraph \ref{paragr:formulasoplax} imply that $h_n$ is linear and that for every $n$-cell $x$ of $X$,
\[
\partial (h_n(x)) + h_{n-1}(\partial(x)) = [v(x)] - [u(x)].
\]
Details are left to the reader.
\end{proof}
\begin{proposition}
The abelianization functor $\lambda : \oo\Cat \to \Ch$ is left Quillen with respect to the canonical model structure on $\oo\Cat$.
\end{proposition}
\begin{proof}
The fact that $\lambda$ is a left adjoint is Lemma \ref{lemma:adjlambda}.
A simple computation using Lemma \ref{lemma:abelpol} shows that for every $n\in \mathbb{N}$,
\[
\lambda(i_n) : \lambda(\sS_{n-1}) \to \lambda(\sD_{n})
\]
\todo{Mettre réference où sont définies ces inclusisons} is a monomorphism with projective cokernel. Hence $\lambda$ preserves cofibrations.
Then, we know from \cite[Sections 4.6 and 4.7]{lafont2010folk} and \cite[Remarque B.1.16]{ara2016joint} (see also \cite[Paragraph 3.11]{ara2019folk}) that there exists a set of generating trivial cofibrations $J$ of the canonical model structure on $\omega\Cat$ such that any $j : X \to Y$ in $J$ is a deformation retract (see Paragraph \ref{paragr:defrtract}).
From Lemma \ref{lemma:abeloplax}, we conclude that $\lambda$ preserves trivial cofibrations.
\end{proof}
The previous lemma motivates the following definition.
\begin{definition}\label{de:polhom}
The \emph{polygraphic homology functor}
\[
\sH^{\pol} : \ho(\oo\Cat^{\Th}) \to \ho(\Ch)
\]
is the left derived functor of $\lambda : \oo\Cat \to \Ch$ with respect to the canonical model structure.
\end{definition}
\begin{paragr}
The functor $\lambda$ being left Quillen, it is strongly derivable (Definition \ref{def:strnglyder}) and hence induces a morphism of op-prederivators still denoted $\sH^{\pol}$:
\[
\sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch).
\]
From Theorem \cite{thm:cisinskiII}, this morphism of op-prederivators is cocontinuous. This property will be extremely important in the sequel.
\end{paragr}
\todo{Expliquer concrètement comment calculer l'homologie polygraphique ?}
\section{``True'' homology as derived abelianization}
We have seen in the previous section that the polygraphic homology is the left derived of the abelianization functor with respect to the canonical weak equivalences on $\oo\Cat$.\todo{Uniformiser appellations} As it turns out, the abelianization functor is also left derivable with respect to the Thomason weak equivalences and the left derived functor is the ``true'' homology functor for $\oo$-categories. In order to prove this result, we need a few lemmas.
\begin{lemma}
Let $\nu : \Ch \to \oo\Cat$ be a right adjoint to the abelianization functor (see Lemma \ref{lemma:adjlambda}). This functor sends weak equivalences of chain complexes to Thomason weak equivalences.
\end{lemma}
\begin{proof}
We have already seen that $\lambda : \oo\Cat \to \Ch$ is left Quillen with respect to the canonical model structure on $\oo\Cat$. By adjunction, this means that $\nu$ is right Quillen for this model structure. In particular, it sends trivial fibrations of chain complexes to trivial fibrations for the canonical model structure. From Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} and the fact that all chain complexes are fibrant, it follows that $\nu$ sends weak equivalences of chain complexes to weak equivalences of the canonical model structure, which are in particular Thomason weak equivalences (see Lemma \ref{lemma:nervehomotopical}).
\end{proof}
\begin{remark}
The proof of the previous lemma shows the stronger result that $\nu$ sends weak equivalences of chain complexes to weak equivalences for the canonical model structure on $\oo\Cat$. This will be of no use in the sequel.
\end{remark}
Recall now that the notion of adjunction is valid in any $2$-category. \todo{Peut-être que je l'aurai déjà dit quelque part ça.} We omit the proof of the following lemma, which is the same as when the ambient $2$-category is the $2$-category of categories.
\begin{lemma}
Let
\end{lemma}
\begin{proposition}
The abelianization functor $\lambda : \oo\Cat \to $
\end{proposition}
......@@ -250,7 +250,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\end{paragr}
\begin{paragr}[Formulas for oplax tranformations] We now give a third way of describing oplax transformations based on explicit formulas. The proof that this description is equivalent to the one given in the previous paragraph can be found in \cite[Appendice B.2]{ara2016joint}.
\begin{paragr}\label{paragr:formulasoplax}[Formulas for oplax tranformations] We now give a third way of describing oplax transformations based on explicit formulas. The proof that this description is equivalent to the one given in the previous paragraph can be found in \cite[Appendice B.2]{ara2016joint}.
Let $u, v : X \to Y$ two $\oo$-functors. An oplax transformation $\alpha : u \Rightarrow v$ is given by the data of:
\begin{itemize}[label=-]
......@@ -310,7 +310,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\begin{corollary}
A homotopy equivalence is a Thomason weak equivalence.
\end{corollary}
\begin{paragr}
\begin{paragr}\label{paragr:defrtract}
An $\oo$-functor $i : X \to Y$ is a \emph{deformation retract} if there exists an $\oo$-functor $r : Y \to X$ such that
\begin{enumerate}[label=(\alph*)]
\item $ri=\mathrm{id}_X$,
......
......@@ -119,6 +119,9 @@
\newcommand{\Th}{\mathrm{Th}} %For Thomason equivalences related stuff
\newcommand{\folk}{\mathrm{can}} %For folk related stuff
\newcommand{\pol}{\mathrm{pol}} %For polygraphic related stuff
% squelette
\newcommand{\sk}{\mathrm{sk}}
......
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