### 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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