Commit 851f18d2 authored by Leonard Guetta's avatar Leonard Guetta
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edited typos

parent 6b0c8526
......@@ -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
$n$-simplices of $X$ are degenerated for $n\geq 2$, then $f$ is a
monomorphism. A proof of this assertion is contained in \cite[Paragraph
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.).
\end{proof}
\begin{paragr}
......@@ -153,7 +153,7 @@ In this section, we review some homotopical results on free
\]
where $\eta$ is the unit of the adjunction $c \dashv N$.
\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
\[
N^{k}(G) \to N^{k+1}(G)
......@@ -1617,7 +1617,7 @@ Now let $\sS_2$ be labelled as
$k=0$. Similarly, for $k>0$, we have already seen that $V_k(\sS_2)$
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
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
the $k$\nbd{}tuples of one of the following forms:
\begin{itemize}[label=-]
......
\chapter{Homology and abelianization of \texorpdfstring{$\oo$}{ω}-categories}
\section{Homology via the nerve}
\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}
\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,
......@@ -1086,7 +1086,7 @@ We now turn to truncations of chain complexes.
\end{itemize}
\end{proposition}
\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}
\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
\end{example}
\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}
\begin{proof}
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
%% 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.
\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}
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:
\[
......@@ -897,7 +897,6 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen
whose image by $u/c_0$ is equivalent for the relation $\sim_{\oo}$ to the above $(n+1)$\nbd{}cell of $B/c_0$. In particular, the source and target of $\alpha$ are respectively $f$ and $f'$. Finally, we obtain that $\alpha \sim_{\oo} \beta$ by applying the canonical $\oo$\nbd{}functor $A/b_0 \to A$, .
\end{enumerate}
\end{proof}
%\todo{Il faudrait vérifier que je n'ai pas écrit de bêtises dans la preuve précédente.}
\begin{paragr} The name ``folk Theorem A'' is an explicit reference of Quillen's Theorem A \cite[Theorem A]{quillen1973higher} and its $\oo$\nbd{}categorical generalization by Ara and Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}. For the sake of comparison we recall below the latter one.
\end{paragr}
......
......@@ -967,7 +967,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
$(\M,\W,\Cof,\Fib)$ only depends on its underlying localizer, the existence
of the classes $\Cof$ and $\Fib$ with the usual properties defining model
structure ought to be thought as a \emph{property} of the localizer
$(\M,\W)$, which is sufficcient to define a ``homotopy theory''. For
$(\M,\W)$, which is sufficient to define a ``homotopy theory''. For
example, Theorem \ref{thm:cisinskiI} should have been stated by saying that
if a localizer $(\M,\W)$ can be extended to a model category
$(\M,\W,\Cof,\Fib)$, then it is homotopy cocomplete.
......
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......@@ -118,7 +118,7 @@ x\comp_n y \comp_m z = (x \comp_n y) \comp_m z \text{ and } x \comp_m y \comp_n
\]
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
\[
f(1_x)=1_{f(x)},
\]
......@@ -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}
Recall that we have seen in Proposition \ref{prop:smallestcategoricalcongruence}
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
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
......
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