### The definition of the N^1 of a graph is flawed. I need to change it.

parent 0fe53bbc
 ... ... @@ -12,7 +12,9 @@ In this section, we review some homotopical results concerning free ($1$-)catego $\src(1_{x}) = \trgt (1_{x}) = x.$ The vocabulary of categories is used : elements of $G_0$ are \emph{objects} or \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells}, arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A \emph{morphism of reflexive graphs} $f : G \to G'$ consists of maps $f_0 : G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and units in an obvious sense. This defines the category $\Rgrph$ of reflexive graphs. There is a underlying reflexive graph'' functor The vocabulary of categories is used : elements of $G_0$ are \emph{objects} or \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells}, arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A \emph{morphism of reflexive graphs} $f : G \to G'$ consists of maps $f_0 : G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and units in an obvious sense. This defines the category $\Rgrph$ of reflexive graphs. Later we will make use of monomorphisms in the category $\Rgrph$; they are the morphisms $f : G \to G'$ that are injective on objects and on arrows, i.e. such that $f_0 : G_0 \to G_0'$ and $f_1 : G_1 \to G'_1$ are injective. There is a underlying reflexive graph'' functor $U : \Cat \to \Rgrph,$ ... ... @@ -57,7 +59,11 @@ In this section, we review some homotopical results concerning free ($1$-)catego The functor $i_! : \Rgrph \to \Psh{\Delta}$ preserves monomorphism. \end{lemma} \begin{proof} What we need to show is that, given a morphism of presheaves $f : X \to Y,$ 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 (Proposition 3.1 of op. cit.). \end{proof} \begin{paragr} Let us denote by $N : \Psh{\Delta} \to \Cat$ (instead of $N_1$ as in Paragraph \todo{ref}) the usual nerve of categories and by $c : \Cat \to \Psh{\Delta}$ its left adjoint. Recall that for a (small) category $C$, an $n$-simplex of $N(C)$ is a chain ... ... @@ -102,10 +108,20 @@ In this section, we review some homotopical results concerning free ($1$-)catego $N^{k}(G) \to N^{k+1}(G)$ is a weak equivalence of simplicial sets. is a trivial cofibration of simplicial sets. \end{lemma} \begin{proof} Let $A_{k+1}=\mathrm{Im}(\partial_0)\cup\mathrm{Im}(\partial_{k+1})$ be the union of the first and last face of the standard $(k+1)$-simplex $\Delta_{k+1}$. Notice that the canonical inclusion $A_{k+1} \hookedrightarrow \Delta_{k+1}$ is a trivial cofibration. Now let $I_{k+1}$ be the set of chains $\begin{tikzcd} f = X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} \end{tikzcd}$ where each $f_i$ is a non-unit arrow of $G$. For every $f \in I_{k+1}$ \end{proof} From this lemma, we deduce the following propositon. \begin{proposition} ... ... @@ -113,10 +129,10 @@ From this lemma, we deduce the following propositon. $\eta_{i_!(G)} : i_!(G) \to Nci_!(G),$ where $\eta$ is the unit of the adjunction $c \dashv N$, is a weak equivalence of simplicial sets. where $\eta$ is the unit of the adjunction $c \dashv N$, is a trivial cofibration of simplicial sets. \end{proposition} \begin{proof} It is an immediate consequence of Lemma \ref{lemma:dwyerkan} and the fact that filtered colimits are homotopic in a model category whose objects are all cofibrants \todo{ref}. This follows from the fact that trivial cofibrations are stable by transfinite composition. \end{proof} From the previous proposition, we deduce the following very useful corollary. \begin{corollary}\label{cor:hmtpysquaregraph} ... ... @@ -237,7 +253,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp \sD_1 \ar[r] & C'. \end{tikzcd} \] Then, this square is homotopy cocartesian in $\Cat$ (when equipped with the Thomason equivalences). Indeed, it obviously is the image of a square of $\Rgrph$ by the functor $L$ and the morphism $i_1 : \partial\sD_1 \to \sD_1$ comes from a monomorphism of $\Rgrph$. Hence, we can apply Corollary \ref{cor:hmptysquaregraph}. Then, this square is homotopy cocartesian in $\Cat$ (when equipped with the Thomason equivalences). Indeed, it obviously is the image of a square of $\Rgrph$ by the functor $L$ and the morphism $i_1 : \partial\sD_1 \to \sD_1$ comes from a monomorphism of $\Rgrph$. Hence, we can apply Corollary \ref{cor:hmtpysquaregraph}. \end{example} \begin{remark} Since every free category is obtained by recursively adding generators starting from a set of objects (seen as a $0$-category), the previous example yields another proof that \emph{free} (1-)categories are \good{} (which we already knew since we have seen that \emph{all} (1-)categories are \good{}). ... ... @@ -250,7 +266,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp \sD_1 \ar[r] & C', \end{tikzcd} \] where the morphism $\sS_1 \to \sD_1$ is the one that sends the two generating arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square is homotopy cocartesian in $\Cat$ (when equipped with Thomason weak equivalences). Indeed, it is the image by the functor $L$ of a cocartesian square in $\Rgrph$, the morphism $\sS_1 \to \sD_1$ is injective on objects and the morphism $\sS_1 \to C$ is quasi-injective on arrows. Hence, we can apply Proposition \ref{prop:hmtpysquaregraphbetter}. Note that since we did \emph{not} suppose that $A\neq B$, the top morphism is not necessarily a monomorphism and we cannot always apply Corollary \ref{cor:hmtpysquaregraph}. where the morphism $\sS_1 \to \sD_1$ is the one that sends the two generating arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square is homotopy cocartesian in $\Cat$ (when equipped with Thomason weak equivalences). Indeed, it is the image by the functor $L$ of a cocartesian square in $\Rgrph$, the morphism $\sS_1 \to \sD_1$ is injective on objects and the morphism $\sS_1 \to C$ is quasi-injective on arrows. Hence, we can apply Proposition \ref{prop:hmtpysquaregraphbetter}. Note that since we did \emph{not} suppose that $A\neq B$, the top morphism of the previous square is not necessarily a monomorphism and we cannot always apply Corollary \ref{cor:hmtpysquaregraph}. \end{example} \begin{example}[Killing a generator] Let $C$ be a free category and let $f : A \to B$ one of its generating arrow such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by killing'' $f$, i.e. defined with the following cocartesian square: ... ...
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