Commit fae18601 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

idem

parent ba4e0f82
......@@ -231,14 +231,14 @@ From the previous proposition, we deduce the following very useful corollary.
\end{tikzcd}
\]
be a cocartesian square in $\Rgrph$. If either $\alpha$ or $\beta$ is a
monomorphism, then the induced square
monomorphism, then the induced square of $\Cat$
\[
\begin{tikzcd}
L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"]& L(B) \ar[d,"L(\delta)"] \\
L(C) \ar[r,"L(\gamma)"]& L(D)
\end{tikzcd}
\]
is a Thomason homotopy cocartesian square of $\Cat$.
is a Thomason homotopy cocartesian.
\end{corollary}
\begin{proof}
Since the nerve $N$ induces an equivalence of op-prederivators
......@@ -1017,19 +1017,16 @@ of $2$-categories.
% \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
% \]
% is an \emph{equivalence} of op-prederivators.
\begin{paragr}
\todo{enlever l'environnement paragr}
From Proposition \ref{prop:streetvsbisimplicial}, we deduce the proposition
below which contains two useful criteria to detect Thomason homotopy
cocartesian squares of $2\Cat$.
\end{paragr}
below which contains two useful criteria to detect when a commutative square
of $2\Cat$ is Thomason homotopy cocartesian.
\begin{proposition}\label{prop:critverthorThomhmtpysquare}
Let
\begin{equation}\tag{$\ast$}\label{coucou}\begin{tikzcd}
A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\
C \ar[r,"v"] & D
\end{tikzcd}\end{equation}
be a square in $2\Cat$ satisfying at least one of the two following conditions:
be a commutative square in $2\Cat$ satisfying at least one of the two following conditions:
\begin{enumerate}[label=(\alph*)]
\item For every $n\geq 0$, the commutative square of $\Cat$
\[
......
......@@ -141,7 +141,7 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \re
\end{tikzcd}
\]
is cartesian and all four morphisms are monomorphisms. Since the
functor $\Hom_{\oo\Cat}(\Or_k,-)$ preserves limits, the square
functor \[\Hom_{\oo\Cat}(\Or_k,-):\oo\Cat \to \Set \] preserves limits, the square
\eqref{squarenervesphere} is a cartesian square of $\Set$ all of
whose four morphisms are monomorphisms. Hence, in order to prove
that square \eqref{squarenervesphere} is cocartesian, we only need
......
......@@ -257,7 +257,8 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c
&\simeq \Hom_{\Set}(\Sigma_n,\vert G \vert)\\
&\simeq \Hom_{\Ab}(\mathbb{Z}\Sigma_n,G),
\end{align*}
and it is easily checked that this isomorphism is induced by the map
and it is easily checked that this isomorphism is induced by
precomposition with the map
$\mathbb{Z}\Sigma_n \to \lambda_n(C)$ from the previous paragraph. The
result follows then from the Yoneda Lemma.
\end{proof}
......@@ -338,7 +339,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c
\]
Recall also that if there is a chain homotopy from $f$ to $g$, then the
localization functor $\gamma^{\Ch} : \Ch \to \ho(\Ch)$ identifies $f$ and
$g$, meaning that \[\gamma^{\Ch}(f)=\gamma^{\Ch}(g).\]
$g$, which means that \[\gamma^{\Ch}(f)=\gamma^{\Ch}(g).\]
\end{paragr}
%For the definition of \emph{homotopy of chain complexes} see for example \cite[Definition 1.4.4]{weibel1995introduction} (where it is called \emph{chain homotopy}).
\begin{lemma}\label{lemma:abeloplax}
......@@ -1114,12 +1115,12 @@ We now turn to truncations of chain complexes.
\begin{proposition}
There exists a model structure on $\Ch^{\leq n}$ such that:
\begin{itemize}[label=-]
\item weak equivalences are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a weak equivalence for the projective model structure on $\Ch$,
\item fibrations are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a fibration for the projective model structure on $\Ch$.
\item the weak equivalences are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a weak equivalence for the projective model structure on $\Ch$,
\item the fibrations are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a fibration for the projective model structure on $\Ch$.
\end{itemize}
\end{proposition}
\begin{proof}
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
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 $j \colon A \to B$ 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"] \\
......
......@@ -195,7 +195,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\end{theorem}
\begin{proof}
Recall from \ref{paragr:nerve} that $c_n : \Psh{\Delta} \to n\Cat$ denotes
the left adjoint to the nerve functor $N_n$. In \cite{gagna2018strict}, Gagna proves that there exists a functor $Q : \Psh{\Delta} \to \Psh{\Delta}$, as well as a zigzag of morphisms of functors
the left adjoint of the nerve functor $N_n$. In \cite{gagna2018strict}, Gagna proves that there exists a functor $Q : \Psh{\Delta} \to \Psh{\Delta}$, as well as a zigzag of morphisms of functors
\[
N_{n}c_{n}Q \overset{\alpha}{\longleftarrow} Q \overset{\gamma}{\longrightarrow} \mathrm{id}_{\Psh{\Delta}}
\]
......@@ -243,7 +243,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
This follows from Theorem \ref{thm:gagna} and the commutativity of the triangle
\[
\begin{tikzcd}[column sep=tiny]
\Ho(n\Cat^{\Th}) \ar[rr] \ar[rd,"\overline{N_n}"'] & & \Ho(m\Cat) \ar[dl,"\overline{N_m}"] \\
\Ho(n\Cat^{\Th}) \ar[rr] \ar[rd,"\overline{N_n}"'] & & \Ho(m\Cat^{\Th}) \ar[dl,"\overline{N_m}"] \\
&\Ho(\Psh{\Delta})&.
\end{tikzcd}
\]
......
......@@ -472,9 +472,9 @@ $\mathbf{Str}\oo\Cat$ pour la catégorie des $\oo$\nbd{}catégories (strictes).
groupes d'homologie polygraphique et singulière coincident toujours en basse
dimension.
Le cinquième chapitre a pour but de démontré le Théorème fondamental
Le cinquième chapitre a pour but de démontrer le Théorème fondamental
\ref{thm:categoriesaregood}, qui dit que toute catégorie est homologiquement
cohérente. Pour démontrer celui-ci, nous nous intéresserons en premier lieu
cohérente. Pour cela, nous nous intéresserons en premier lieu
à une classe particulière d'$\oo$\nbd{}catégories, dites
\emph{contractiles}, et nous montrerons que toute $\oo$\nbd{}catégorie
contractile est homologiquement cohérente (Proposition
......
......@@ -470,7 +470,7 @@ note={In preparation}
publisher={Springer}
}
@incollection{quillen1973higher,
title={Higher algebraic K-theory: I},
title={Higher algebraic {K}-theory: I},
author={Quillen, Daniel G.},
booktitle={Higher K-theories},
pages={85--147},
......
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