Commit a271e140 authored by Leonard Guetta's avatar Leonard Guetta
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I need to take up from 4.1.6

parent 0a8af9df
......@@ -11,15 +11,16 @@ We denote by $\Ch$ the category of chain complexes of abelian groups in non-nega
From now on, we will implicitly consider that the category $\Ch$ is equipped with this model structure.
\end{paragr}
\begin{paragr}
Let $X$ be a simplicial set. We denote by $K_n(X)$ the abelian group of $n$-chains of $X$, i.e. the free abelian group on the set $X_n$. For $n>0$, let $\partial : K_n(X) \to K_{n-1}(X)$ be the linear map defined for $x \in X_n$ by
Let $X$ be a simplicial set. We denote by $K_n(X)$ the abelian group of $n$-chains of $X$, i.e.\ the free abelian group on the set $X_n$. For $n>0$, let $\partial : K_n(X) \to K_{n-1}(X)$ be the linear map defined for $x \in X_n$ by
\[
\partial(x):=\sum_{i=0}^n(-1)^i\partial_i(x).
\]
Using the simplicial identities (see \cite[section 2.1]{gabriel1967calculus}), it can be shown that $\partial \circ \partial = 0$. Hence, the previous data defines a chain complex $K(X)$ and the correspondance $X \mapsto K(X)$ can canonically be extended to a functor
\[
K : \Psh{\Delta} \to \Ch.
\]
\todo{Bien dit?}
It follows from the simplicial identities (see \cite[section 2.1]{gabriel1967calculus}) that $\partial \circ \partial = 0$. Hence, the previous data defines a chain complex $K(X)$ and this defines a functor
\begin{align*}
K : \Psh{\Delta} &\to \Ch\\
X &\mapsto K(X)
\end{align*}
in the expected way.
\end{paragr}
\begin{paragr}
Recall that an $n$-simplex $x$ of a simplicial set $X$ is \emph{degenerate} if there exists an epimorphism $\varphi : [n] \to [m]$ with $m<n$ and an $m$-simplex $y$ such that $X(\varphi)(y)=x$. We denote by $D_n(X)$ the subgroup of $K_n(X)$ generated by the degenerate $n$-simplices. We denote by $\kappa_n(X)$ the abelian group of \emph{normalized chain complex},
......@@ -31,9 +32,10 @@ We denote by $\Ch$ the category of chain complexes of abelian groups in non-nega
\partial : \kappa_n(X) \to \kappa_{n-1}(X).
\]
This defines a chain complex $\kappa(X)$, which we call the \emph{normalized chain complex of $X$}. This yields a functor
\[
\kappa : \Psh{\Delta} \to \Ch
\]
\begin{align*}
\kappa : \Psh{\Delta} &\to \Ch \\
X &\mapsto \kappa(X).
\end{align*}
\end{paragr}
\begin{lemma}\label{lemma:normcompquil}
The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equivalences of simplicial sets to quasi-isomorphisms.
......@@ -47,14 +49,20 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv
The fact that $\kappa$ preserves weak equivalences follows from Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} and the fact that all simplicial sets are cofibrant.
\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{paragr}
In particular, $\kappa$ induces a morphism of localizers \[\kappa : (\Psh{\Delta},\W_{\Delta}) \to (\Ch,\W_{\Ch}),\]
where we wrote $\W_{\Ch}$ for the class of quasi-isomorphisms.
\end{paragr}
\begin{definition}\label{def:hmlgycat}
The \emph{homology functor for $\oo$-categories} $\sH : \ho(\oo\Cat^{\Th}) \to \ho(\Ch)$ is defined as the following composition
The \emph{singular homology functor for $\oo$-categories} $\sH^{\sing}$ is defined as the following composition
\[
\sH : \ho(\oo\Cat^{\Th}) \overset{\overline{N_{\omega}}}{\longrightarrow} \ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \ho(\Ch).
\sH^{\sing} : \ho(\oo\Cat^{\Th}) \overset{\overline{N_{\omega}}}{\longrightarrow} \ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \ho(\Ch).
\]
For an $\oo$-category $X$, $\sH(X)$ is the \emph{homology of $X$}.
For an $\oo$\nbd-category $X$, $\sH^{\sing}(X)$ is the \emph{singular homology of $X$}.
\end{definition}
\begin{paragr}
In other words, the singular homology of $X$ is the chain complex $\kappa(N_{\oo}(X))$ seen as an object of $\ho(\Ch)$ (see Remark \ref{remark:localizedfunctorobjects}).
\end{paragr}
\begin{paragr}
In simpler words, the homology of an $\oo$-category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{Street homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later.
Recall that we consider localization functors as identity on objects (see \ref{paragr:loc}). Hence, the homology of $X$ is simply $\kappa(N_{\oo}(X))$, only considered as a chain complex up to quasi-isomorphism, i.e. an object of $\ho(\Ch)$. The homology groups of $X$ are the homology groups of the chain complex $\kappa(N_{\oo}(X))$. However, with our definition, the \emph{homology of $X$} means something more precise than the mere sequence of homology groups. An alternative terminology would be to call $\sH(X)$ the \emph{homology type of $X$}, in reference to the homotopy type of a topological space.
......
......@@ -514,21 +514,25 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\item When $n=1$, $x \sim_{\omega} y$ means that $x$ and $y$ are isomorphic.
\item When $n=2$, $x \sim_{\omega} y$ means that $x$ and $y$ are equivalent, i.e.\ there exists $f : x \to y$ and $g : y \to x$ such that $fg$ is isomorphic to $1_y$ and $gf$ is isomorphic to $1_x$.
\end{itemize}
\end{example}
\end{example}
For later reference, we put here the following trivial but important lemma, whose proof is ommited.
\begin{lemma}
Let $F : C \to D$ be an $\oo$\nbd-functor and $x$,$y$ be $n$-cells of $C$ for some $n \geq 0$. If $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$.
\end{lemma}
\begin{definition}\label{def:eqomegacat}
An $\omega$-functor $f : C \to D$ is an \emph{equivalence of $\omega$-categories} when:
An $\omega$-functor $D : C \to D$ is an \emph{equivalence of $\oo$\nbd-categories} when:
\begin{itemize}
\item[-] for every $y \in D_0$, there exists a $x \in C_0$ such that
\[f(x)\sim_{\omega}y,\]
\item[-] for every $x,y \in C_n$ that are \emph{parallel} and every $\beta \in D_{n+1}$ such that \[\beta : f(x) \to f(y),\] there exists $\alpha \in C_{n+1}$ such that
\[F(x)\sim_{\omega}y,\]
\item[-] for every $x,y \in C_n$ that are \emph{parallel} and every $\beta \in D_{n+1}$ such that \[\beta : F(x) \to F(y),\] there exists $\alpha \in C_{n+1}$ such that
\[\alpha : x \to y
\]
and
\[f(\alpha)\sim_{\omega}\beta.\]
\[F(\alpha)\sim_{\omega}\beta.\]
\end{itemize}
\end{definition}
\begin{example}\label{example:equivalencecategories}
If $C$ and $D$ are (small) categories seen as $\oo$-categories, then a functor $F : C \to D$ is an equivalence of $\oo$-categories if and only if it is fully faithful, hence, an equivalence of categories.
If $C$ and $D$ are (small) categories seen as $\oo$-categories, then a functor $F : C \to D$ is an equivalence of $\oo$\nbd-categories if and only if it is fully faithful, hence, an equivalence of categories.
\end{example}
For the next theorem, recall that the canonical map $i_n : \sS_{n-1} \to \sD_n$ for $n \geq 0$ have been defined in \ref{paragr:defglobe}.
\begin{theorem}\label{thm:folkms}
......@@ -626,7 +630,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
\]
Let us now give an alternative definition of the $\oo$\nbd-category $A/a_0$ using explicit formulas. The equivalence with the previous definition follows from the dual of \cite[Proposition B.5.2]{ara2016joint}
\begin{itemize}[label=-]
\item An $n$-cell of $A/a_0$ is a matrix
\item An $n$-cell of $A/a_0$ is a matrix \todo{le mot ``matrix'' est-il maladroit ?}
\[
(x,a)=\begin{pmatrix}
\begin{matrix}
......@@ -758,12 +762,35 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
\end{pmatrix}
\]
is an $n$-cell of $B/b_0$.
The canonical $\oo$\nbd-functor $A/b_0 \to A$ is simply expressed as
\begin{align*}
A/b_0 &\to A\\
(x,b) &\mapsto x_n,
\end{align*}
and the $\oo$\nbd-functor $u/b_0$ as
\begin{align*}
u/b_0 : A/b_0 &\to B/b_0 \\
(x,b) &\mapsto (u(x),b).
\end{align*}
More generally, if we have a commutative triangle in $\oo\Cat$
\[
\begin{tikzcd}[column sep=small]
A \ar[rr,"u"] \ar[dr,"v"'] & &B \ar[dl,"w"] \\
&C&
\end{tikzcd},
\]
then for any object $c_0$ of $C$, we have a functor $u/c_0 : A/c_0 \to B/c_0$ defined as
\begin{align*}
u/c_0 : A/c_0 &\to B/c_0 \\
(x,c) &\mapsto (u(x),c).
\end{align*}
\end{paragr}
\begin{proposition}(Folk Theorem $A$) Let
\[
\begin{tikzcd}[column sep=small]
A \ar[rr,"u"] \ar[dr,"w"'] & &B \ar[dl,"v"] \\
A \ar[rr,"u"] \ar[dr,"v"'] & &B \ar[dl,"w"] \\
&C&
\end{tikzcd}
\]
......@@ -774,13 +801,69 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
is an equivalence of $\oo$-categories, then so is $u$.
\end{proposition}
\begin{proof}
Before anything else, let us note the following trivial but important fact: for any $\oo$\nbd-functor $F : X \to Y$ and any $n$-cells $x$ and $y$ of $X$, if $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$.
\begin{enumerate}[label=(\roman*)]
\item Let $b_0$ be $0$\nbd-cell of $B$ and set $c_0:=v(b_0)$. By definition, the pair $(b_0,1_{c_0})$ is a $0$-cell of $B/c_0$. By hypothesis, we know that there exists a $0$\nbd-cell $(a_0,c_1)$ of $A/c_0$ such that $(u(a_0),c_1)\sim_{\oo} (b_0,1_{c_0})$. Hence, by applying the canonical functor $B/c_0 \to B$, we obtain that $u(a_0) \sim_{\oo} b_0$.
\item Let $f$ and $f'$ be parallel $n$\nbd-cells of $A$ and $\beta : u(f) \to u(f')$ an $(n+1)$\nbd-cell of $B$. We need to show that there exists an $(n+1)$\nbd-cell $\alpha : f \to f'$ of $A$ such that $u(\alpha) \sim_{\oo} \beta$.
Let us use the notations:
\begin{itemize}[label=-]
\item $a_i := \src_i(f)=\src_i(f')$ for $0 \leq i <n$,
\item $a_i := \trgt_i(f)=\trgt_i(f')$ for $0 \leq i <n$,
\item $a_n:=f$ and $a_n'=f'$.
\end{itemize}
It is straightforward to check that
\[
\begin{pmatrix}
\begin{matrix}
(a_0,v(a_1')) & (a_1,v(a_2')) & \cdots & (a_{n-1},v(a_n')) \\[0.5em]
(a_0',1_{v(a_0')}) & (a_1',1_{v(a_1')}) & \cdots & (a_{n-1}',1_{(v(a_{n-1}'))})
\end{matrix}
& (a_n,w(\beta))
\end{pmatrix}
\]
and
\[
\begin{pmatrix}
\begin{matrix}
(a_0,v(a_1')) & (a_1,v(a_2')) & \cdots & (a_{n-1},v(a_n')) \\[0.5em]
(a_0',1_{v(a_0')}) & (a_1',1_{v(a_1')}) & \cdots & (a_{n-1}',1_{(v(a_{(n-1)}'))})
\end{matrix}
& (a_n',1_{v(a_n')})
\end{pmatrix}
\]
are parallel $n$\nbd-cells of $A/c_0$ where we set $c_0:=v(a_0')$. Similarly, we have an $(n+1)$\nbd-cell of $B/c_0$
\[
\begin{pmatrix}
\begin{matrix}
(u(a_0),v(a_1')) & \cdots & (u(a_{n-1}),v(a_n')) & (u(a_n),w(\beta)) \\[0.5em]
(u(a_0'),1_{v(a_0')}) & \cdots & (u(a_{n-1}'),1_{(v(a_{n-1}'))}) & (u(a_n'),1_{v(a_n')})
\end{matrix}
& (\beta,1_{w(\beta)})
\end{pmatrix}
\]
whose source and target respectively are the image by $u/c_0$ of the above two cells of $A/c_0$. By hypothesis, there exists an $(n+1)$\nbd-cell of $A/c_0$ of the form
\[
\begin{pmatrix}
\begin{matrix}
(a_0,v(a_1')) & \cdots & (a_{n-1},v(a_n')) & (a_n,w(\beta)) \\[0.5em]
(a_0',1_{v(a_0')}) & \cdots & (a_{n-1}',1_{(v(a_{n-1}'))}) & (a_n',1_{v(a_n')})
\end{matrix}
& (\alpha,\Lambda)
\end{pmatrix}
\]
whose image by $u/c_0$ is equivalent for the relation $\sim_{\oo}$ to the above $(n+1)$\nbd-cell of $B/c_0$ . In particular, the source and target of $\alpha$ are respectively $f$ and $f'$. Finally, we obtain that $\alpha \sim_{\oo} \beta$ by applying the canonical $\oo$\nbd-functor $A/b_0 \to A$, .
\end{enumerate}
\end{proof}
\todo{Il faudrait vérifier que je n'ai pas écrit de bêtises dans la preuve précédente.}
\begin{proposition} Let
\begin{paragr} The name ``folk Theorem A'' is an explicit reference of Quillen's Theorem A \cite[Theorem A]{quillen1973higher} and its $\oo$\nbd-categorical generalization by Ara and Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}. For the sake of comparison we recall this last theorem below.
\end{paragr}
\begin{proposition}[Ara and Maltsiniotis' Theorem A] Let
\[
\begin{tikzcd}[column sep=small]
A \ar[rr,"u"] \ar[dr,"w"'] & &B \ar[dl,"v"] \\
A \ar[rr,"u"] \ar[dr,"v"'] & &B \ar[dl,"w"] \\
&C&
\end{tikzcd}
\]
......
......@@ -67,7 +67,13 @@ A \emph{morphism of localizers} $F : (\C,\W) \to (\C',\W')$ is a functor $F~:~\C
\]
is commutative in an obvious sense.
\end{paragr}
\begin{remark}\label{remark:localizedfunctorobjects}
Since we always consider that for any localizer $(\C,\W)$ the categories $\C$ and $\ho(\C)$ have the same objects and the localization functor is the identity on objects, it follows that for a morphism of localizer ${F : (\C,\W) \to (\C',\W')}$, we tautologically have
\[
\overline{F}(X)=F(X)
\]
for every object $X$ of $\C$.
\end{remark}
\begin{paragr}\label{paragr:defleftderived}
Let $(\C,\W)$ and $(\C',\W')$ be two localizers. A functor $F : \C \to \C'$ is \emph{totally left derivable} when there exists a functor
\[
......
......@@ -154,6 +154,8 @@ headpunct=.]
\newcommand{\St}{\mathrm{St}} %For Street related stuff
\newcommand{\sing}{\mathrm{Sing}} %For ``singular'' homology
\newcommand{\folk}{\mathrm{folk}} %For folk related stuff
\newcommand{\pol}{\mathrm{pol}} %For polygraphic related stuff
......
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