### 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 $m  ... ... @@ -514,21 +514,25 @@ From now on, we will consider that the category$\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$A/b_0 \to Ais simply expressed as \begin{align*} A/b_0 &\to A\\ (x,b) &\mapsto x_n, \end{align*} and the\oo$\nbd-functor$u/b_0as \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$u/c_0 : A/c_0 \to B/c_0defined as \begin{align*} u/c_0 : A/c_0 &\to B/c_0 \\ (x,c) &\mapsto (u(x),c). \end{align*} \end{paragr} \begin{proposition}(Folk TheoremA$) 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
 ... ... @@ -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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