Commit 851f18d2 authored by Leonard Guetta's avatar Leonard Guetta
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...@@ -98,7 +98,7 @@ In this section, we review some homotopical results on free ...@@ -98,7 +98,7 @@ In this section, we review some homotopical results on free
if $f_0 : X_0 \to Y_0$ and $f_1 : X_1 \to Y_1$ are monomorphisms and if all if $f_0 : X_0 \to Y_0$ and $f_1 : X_1 \to Y_1$ are monomorphisms and if all
$n$-simplices of $X$ are degenerated for $n\geq 2$, then $f$ is a $n$-simplices of $X$ are degenerated for $n\geq 2$, then $f$ is a
monomorphism. A proof of this assertion is contained in \cite[Paragraph monomorphism. A proof of this assertion is contained in \cite[Paragraph
3.4]{gabriel1967calculus}. The key argument is the Eilenberg-Zilber Lemma 3.4]{gabriel1967calculus}. The key argument is the Eilenberg--Zilber Lemma
(Proposition 3.1 of op. cit.). (Proposition 3.1 of op. cit.).
\end{proof} \end{proof}
\begin{paragr} \begin{paragr}
...@@ -153,7 +153,7 @@ In this section, we review some homotopical results on free ...@@ -153,7 +153,7 @@ In this section, we review some homotopical results on free
\] \]
where $\eta$ is the unit of the adjunction $c \dashv N$. where $\eta$ is the unit of the adjunction $c \dashv N$.
\end{paragr} \end{paragr}
\begin{lemma}[Dwyer-Kan]\label{lemma:dwyerkan} \begin{lemma}[Dwyer--Kan]\label{lemma:dwyerkan}
For every $k\geq 1$, the canonical inclusion map For every $k\geq 1$, the canonical inclusion map
\[ \[
N^{k}(G) \to N^{k+1}(G) N^{k}(G) \to N^{k+1}(G)
...@@ -1617,7 +1617,7 @@ Now let $\sS_2$ be labelled as ...@@ -1617,7 +1617,7 @@ Now let $\sS_2$ be labelled as
$k=0$. Similarly, for $k>0$, we have already seen that $V_k(\sS_2)$ $k=0$. Similarly, for $k>0$, we have already seen that $V_k(\sS_2)$
is the free category on the graph that has $2$ objects and $2k+2$ is the free category on the graph that has $2$ objects and $2k+2$
parallel arrows between these two objects and we leave as an easy parallel arrows between these two objects and we leave as an easy
exercice to the reader to check that the category $V_k(P)$ is the exercise to the reader to check that the category $V_k(P)$ is the
free category on the graph that has one object and $2k+2$ arrows, which are free category on the graph that has one object and $2k+2$ arrows, which are
the $k$\nbd{}tuples of one of the following forms: the $k$\nbd{}tuples of one of the following forms:
\begin{itemize}[label=-] \begin{itemize}[label=-]
......
\chapter{Homology and abelianization of \texorpdfstring{$\oo$}{ω}-categories} \chapter{Homology and abelianization of \texorpdfstring{$\oo$}{ω}-categories}
\section{Homology via the nerve} \section{Homology via the nerve}
\begin{paragr} \begin{paragr}
We denote by $\Ch$ the category of non-negatively graded chain complexes of abelian groups. Recall that $\Ch$ can be equipped with a cofibrantely generated model structure, known as the \emph{projective model structure on $\Ch$}, where: We denote by $\Ch$ the category of non-negatively graded chain complexes of abelian groups. Recall that $\Ch$ can be equipped with a cofibrantly generated model structure, known as the \emph{projective model structure on $\Ch$}, where:
\begin{itemize} \begin{itemize}
\item[-] the weak equivalences are the quasi-isomorphisms, i.e.\ morphisms of chain complexes that induce an isomorphism on homology groups, \item[-] the weak equivalences are the quasi-isomorphisms, i.e.\ morphisms of chain complexes that induce an isomorphism on homology groups,
\item[-] the cofibrations are the morphisms of chain complexes $f: X\to Y$ such that for every $n\geq 0$, $f_n : X_n \to Y_n$ is a monomorphism with projective cokernel, \item[-] the cofibrations are the morphisms of chain complexes $f: X\to Y$ such that for every $n\geq 0$, $f_n : X_n \to Y_n$ is a monomorphism with projective cokernel,
...@@ -1086,7 +1086,7 @@ We now turn to truncations of chain complexes. ...@@ -1086,7 +1086,7 @@ We now turn to truncations of chain complexes.
\end{itemize} \end{itemize}
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
This is a typical example of a transfer of a cofibrantely generated model structure along a right adjoint as in \cite[Proposition 2.3]{beke2001sheafifiableII}. The only \emph{a priori} non-trivial hypothesis to check is that there exists a set $J$ of generating trivial cofibrations of the projective model structure on $\Ch$ such that for every $j : A \to B$ in $J$ and every cocartesian square This is a typical example of a transfer of a cofibrantly generated model structure along a right adjoint as in \cite[Proposition 2.3]{beke2001sheafifiableII}. The only \emph{a priori} non-trivial hypothesis to check is that there exists a set $J$ of generating trivial cofibrations of the projective model structure on $\Ch$ such that for every $j : A \to B$ in $J$ and every cocartesian square
\[ \[
\begin{tikzcd} \begin{tikzcd}
\tau^{i}_{\leq n}(A) \ar[r] \ar[d,"\tau^{i}_{\leq n}(j)"'] & X \ar[d,"g"] \\ \tau^{i}_{\leq n}(A) \ar[r] \ar[d,"\tau^{i}_{\leq n}(j)"'] & X \ar[d,"g"] \\
......
...@@ -548,7 +548,7 @@ For later reference, we put here the following trivial but important lemma, whos ...@@ -548,7 +548,7 @@ For later reference, we put here the following trivial but important lemma, whos
\end{example} \end{example}
\begin{theorem}\label{thm:folkms} \begin{theorem}\label{thm:folkms}
There exists a cofibrantely generated model structure on $\omega\Cat$ such that the weak equivalences are the equivalences of $\omega$-categories, and the set $\{i_n : \sS_{n-1} \to \sD_n \vert n \in \mathbb{N}\}$ (see \ref{paragr:defglobe}) is a set of generating cofibrations. There exists a cofibrantly generated model structure on $\omega\Cat$ such that the weak equivalences are the equivalences of $\oo$\nbd{}categories, and the set $\{i_n : \sS_{n-1} \to \sD_n \vert n \in \mathbb{N}\}$ (see \ref{paragr:defglobe}) is a set of generating cofibrations.
\end{theorem} \end{theorem}
\begin{proof} \begin{proof}
This is the main result of \cite{lafont2010folk}. This is the main result of \cite{lafont2010folk}.
...@@ -655,7 +655,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen ...@@ -655,7 +655,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen
%% is the identity on objects. %% is the identity on objects.
This functor cannot be an equivalence since this would imply that every Thomason equivalence is an equivalence of $\oo$\nbd{}categories. This functor cannot be an equivalence since this would imply that every Thomason equivalence is an equivalence of $\oo$\nbd{}categories.
\end{paragr} \end{paragr}
\section{Slices of \texorpdfstring{$\oo$}{ω}-category and a folk Theorem A} \section{Slice \texorpdfstring{$\oo$}{ω}-categories and a folk Theorem A}
\begin{paragr}\label{paragr:slices} \begin{paragr}\label{paragr:slices}
Let $A$ be an $\oo$\nbd{}category and $a_0$ an object of $A$. We define the slice $\oo$\nbd{}category $A/a_0$ as the following fibred product: Let $A$ be an $\oo$\nbd{}category and $a_0$ an object of $A$. We define the slice $\oo$\nbd{}category $A/a_0$ as the following fibred product:
\[ \[
...@@ -897,7 +897,6 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen ...@@ -897,7 +897,6 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen
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$, . 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{enumerate}
\end{proof} \end{proof}
%\todo{Il faudrait vérifier que je n'ai pas écrit de bêtises dans la preuve précédente.}
\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 below the latter one. \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 below the latter one.
\end{paragr} \end{paragr}
......
...@@ -967,7 +967,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators. ...@@ -967,7 +967,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
$(\M,\W,\Cof,\Fib)$ only depends on its underlying localizer, the existence $(\M,\W,\Cof,\Fib)$ only depends on its underlying localizer, the existence
of the classes $\Cof$ and $\Fib$ with the usual properties defining model of the classes $\Cof$ and $\Fib$ with the usual properties defining model
structure ought to be thought as a \emph{property} of the localizer structure ought to be thought as a \emph{property} of the localizer
$(\M,\W)$, which is sufficcient to define a ``homotopy theory''. For $(\M,\W)$, which is sufficient to define a ``homotopy theory''. For
example, Theorem \ref{thm:cisinskiI} should have been stated by saying that example, Theorem \ref{thm:cisinskiI} should have been stated by saying that
if a localizer $(\M,\W)$ can be extended to a model category if a localizer $(\M,\W)$ can be extended to a model category
$(\M,\W,\Cof,\Fib)$, then it is homotopy cocomplete. $(\M,\W,\Cof,\Fib)$, then it is homotopy cocomplete.
......
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...@@ -118,7 +118,7 @@ x\comp_n y \comp_m z = (x \comp_n y) \comp_m z \text{ and } x \comp_m y \comp_n ...@@ -118,7 +118,7 @@ x\comp_n y \comp_m z = (x \comp_n y) \comp_m z \text{ and } x \comp_m y \comp_n
\] \]
whenever these equations make sense. whenever these equations make sense.
A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo$\nbd{}graphs that is compatible with units and compositions, i.e.\ for every $n$\nbd{}cell $x$, we have A \emph{morphism of $\oo$\nbd{}magmas} $f : X \to Y$ is a morphism of underlying $\oo$\nbd{}graphs that is compatible with units and compositions, i.e.\ for every $n$\nbd{}cell $x$, we have
\[ \[
f(1_x)=1_{f(x)}, f(1_x)=1_{f(x)},
\] \]
...@@ -1623,7 +1623,7 @@ with $w_1$ and $w_2$ well formed words and $0\leq k \leq n$. Then $\src_k(w_1)=\ ...@@ -1623,7 +1623,7 @@ with $w_1$ and $w_2$ well formed words and $0\leq k \leq n$. Then $\src_k(w_1)=\
\section{Proof of Theorem \ref{thm:conduche}: part II} \section{Proof of Theorem \ref{thm:conduche}: part II}
Recall that we have seen in Proposition \ref{prop:smallestcategoricalcongruence} Recall that we have seen in Proposition \ref{prop:smallestcategoricalcongruence}
that there exists a smallest categorical congruence on every $n$\nbd{}magma $X$ that there exists a smallest categorical congruence on every $n$\nbd{}magma $X$
with $n \geq 1$. However, the description of this congruence that we we used for with $n \geq 1$. However, the description of this congruence that we used for
proving its existence is rather abstract. The main goal of this section is to proving its existence is rather abstract. The main goal of this section is to
give a more concrete description of the smallest categorical congruence in the give a more concrete description of the smallest categorical congruence in the
case that $X= \E^+$ for an $n$\nbd{}cellular extension $\E$. This description case that $X= \E^+$ for an $n$\nbd{}cellular extension $\E$. This description
......
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