### security commit

parent 1b85bdae
 ... ... @@ -12,7 +12,7 @@ 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. 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. 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 $... ... @@ -31,7 +31,7 @@ In this section, we review some homotopical results concerning free (1-)catego of arrows of G, such that \emph{none} of the f_k are units. The integer n is referred to as the \emph{length} of f and is denoted by \ell(f). Composition is given by concatenation of chains. \end{paragr} \begin{lemma} A category C is free in the sense of \todo{ref} if and only if there exists a reflexive graph G such that A category C is free in the sense of \ref{def:freeoocat} if and only if there exists a reflexive graph G such that \[ C \simeq L(G).$ ... ... @@ -42,7 +42,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego Conversely, if $C \simeq L(G)$ for some reflexive graph $G$, then the description of the arrows of $L(G)$ given in the previous paragraph shows that $C$ is free and that its set of generating $1$-cells is (isomorphic to) the non unital $1$-cells of $G$. \end{proof} \begin{remark} Note that for a morphism of reflexive graphs $f : G \to G'$, the functor $L(f)$ is not necessarily rigid in the sense of \todo{ref} because generating $1$-cells may be sent to units. In other words, a category is free on a graph if and only if it is free on a reflexive graph. The difference between these two notions is at the level of morphisms: there are more morphisms of reflexive graphs because (generating) $1$\nbd{}cells may be sent to units. Hence, for a morphism of reflexing graphs $f : G \to G'$, the induced functor $L(f)$ is not necessarily rigid in the sense of Definition \ref{def:rigidmorphism}. \end{remark} \begin{paragr} There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq 1}$ the full subcategory of $\Delta$ spanned by $$ and $$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq 1}}$, the category of pre-sheaves on $\Delta_{\leq 1}$. In particular, the canonical inclusion $i : \Delta_{\leq 1} \rightarrow \Delta$ induces by pre-composition a functor ... ... @@ -66,7 +66,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego 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 Let us denote by $N : \Psh{\Delta} \to \Cat$ (instead of $N_1$ as in Paragraph \ref{paragr:nerve}) 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 $\begin{tikzcd} X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} ... ... @@ -200,7 +200,7 @@ From the previous proposition, we deduce the following very useful corollary. i_!(C) \ar[r,"i_!(\gamma)"]& i_!(D). \end{tikzcd}$ This square is cocartesian because $i_!$ is a left adjoint. Since $i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the result follows from the fact that monomorphisms are cofibrations of simplicial sets for the standard Quillen model structure and \todo{ref}. This square is cocartesian because $i_!$ is a left adjoint. Since $i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the result follows from the fact that monomorphisms are cofibrations of simplicial sets for the standard Quillen model structure and Lemma \ref{lemma:hmtpycocartesianreedy}. \end{proof} \begin{paragr} Actually, by working a little more, we obtain a more general result, which is stated in the propositon below. Let us say that a morphism of reflexive graphs, $\alpha : A \to B$, is \emph{quasi-injective on arrows} when for all $f$ and $g$ arrows of $A$, if ... ... @@ -267,7 +267,7 @@ From the previous proposition, we deduce the following very useful corollary. \ar[from=2-2,to=3-3,phantom,"\ulcorner" very near end,"\text{\textcircled{\tiny \textbf{3}}}", description] \end{tikzcd}. \] What we want to prove is that the image by the functor $L$ of the pasting of squares \textcircled{\tiny \textbf{2}} and \textcircled{\tiny \textbf{3}} is homotopy cocartesian. Since the morphism $G \to B$ is a monomorphism, we deduce from Corollary \ref{cor:hmtpysquaregraph} that the image by the functor $L$ of square \textcircled{\tiny \textbf{3}} is homotopy cocartesian. Hence, all we have to show is that the image by $L$ of square \textcircled{\tiny \textbf{2}} is homotopy cocartesian. \todo{Parler du pasting lemma ?} On the other hand, we know that both morphisms What we want to prove is that the image by the functor $L$ of the pasting of squares \textcircled{\tiny \textbf{2}} and \textcircled{\tiny \textbf{3}} is homotopy cocartesian. Since the morphism $G \to B$ is a monomorphism, we deduce from Corollary \ref{cor:hmtpysquaregraph} that the image by the functor $L$ of square \textcircled{\tiny \textbf{3}} is homotopy cocartesian. Hence, in virtue of Lemma \ref{lemma:pastinghmtpycocartesian}, all we have to show is that the image by $L$ of square \textcircled{\tiny \textbf{2}} is homotopy cocartesian. On the other hand, we know that both morphisms $\coprod_{x \in E}F_x \to A \text{ and } A \to C$ ... ... @@ -308,10 +308,10 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp Then, this above square is homotopy cocartesian in $\Cat$ (equipped with Thomason equivalences). Indeed, it obviously is the image of a square in $\Rgrph$ by the functor $L$ and since the source and target of $f$ are different, the top map comes from a monomorphism of $\Rgrph$. \end{example} \begin{remark} Note that in the previous example, we see that it was useful to consider the category of reflexive graphs and not only the category of graphs because the map $\sD_1 \to \sD_0$ does not come from a morphism in the category of graphs. \todo{À mieux dire ?} \end{remark} Note that in the previous example, we see that it was useful to consider the category of reflexive graphs and not only the category of graphs because the map $\sD_1 \to \sD_0$ does not come from a morphism in the category of graphs. \todo{Préciser que si A=B dans l'exemple précédent ça ne marche pas ?} Note also that the hypothesis that $A\neq B$ was fundamental in the previous example as for $A=B$ the square is \emph{not} homotopy cocartesian. \end{remark} \section{Preliminaries : Bisimplicial sets} \begin{paragr} ... ...
 ... ... @@ -78,7 +78,7 @@ We end this section with an important result on slices $\oo$\nbd{}category (Para For every $n \in \mathbb{N}$, the $\oo$\nbd{}category $\sD_n$ is oplax contractible. \end{lemma} \begin{proof} Recall that we write $e_n$ for the unique non-trivial $n$\nbd{}cell of $\sD_n$ and that by definition $\sD_n$ has exactly two non-trivial $k$\nbd{}-cells for every $k$ such that $0\leq kn$:] Since $\sS_n$ is an $n$\nbd{}category, the image of $\alpha_k$ is necessarily of the form $\varphi(\alpha_k)=\1^k_{x}$ with $x$ a cell of $\sS_n$ of dimension non-greater than $n$. If the dimension of $x$ is strictly lower than $n$, then everything works like in the case $k  ... ... @@ -628,7 +628,7 @@ We now turn to the most important way of obtaining op-prederivators. \] The result follows then from \cite[Proposition 3.12(2)]{groth2013derivators}. \end{proof} \begin{lemma}[Pasting lemma for homotopy cocartesian squares] \begin{lemma}[Pasting lemma for homotopy cocartesian squares]\label{lemma:pastinghmtpycocartesian} Let$(\C,\W)$be a homotopy cocomplete localizer and let \[ \begin{tikzcd} ... ... No preview for this file type  ... ... @@ -383,7 +383,7 @@ We can now state the promised result, whose proof can be found in \cite[Section We often say refer to the elements of the$n$-basis of a free$\oo$-category as the \emph{generating$n$-cells}. This sometimes leads to use the alternative terminology \emph{set of generating$n$-cells} instead of \emph{$n$-basis}. \end{paragr} \begin{definition} \begin{definition}\label{def:rigidmorphism} Let$X$and$Y$be two free$\oo$-categories. An$\oo$-functor$f : X \to Y$is \emph{rigid} if for every$n\geq 0$and every generating$n$-cell$x$of$X$,$f(x)$is a generating$n$-cell of$Y$. \end{definition} So far, we have not yet seen examples of free$\oo$-categories. In order to do that, we will explain in a further section a recursive way of constructing free$\oo\$-categories; but first we take a little detour. ... ...
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