Commit 498aacaf by Leonard Guetta

### security commit

parent 56004c82
 ... ... @@ -123,6 +123,8 @@ % useful stuff \newcommand{\ii}{\mathbf{i}} % a boldfont i \newcommand{\cc}{\mathbf{c}} % a boldfont c \newcommand{\bs}[1]{\ensuremath{\boldsymbol{#1}}} % a shortcut for \boldsymbol \newcommand{\nbd}{\nobreakdash} % A shortcut for \nobreakdash ... ...
 ... ... @@ -147,7 +147,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo \end{itemize} we have $((x \comp_k x')\comp_l (y \comp_k y'))=((x \comp_l y)\comp_k (x' \comp_l y')). (x \comp_k x')\comp_l (y \comp_k y')=(x \comp_l y)\comp_k (x' \comp_l y').$ \end{description} We will use the same letter to denote an$\oo$-category and its underlying$\oo$\nbd-magma. A \emph{morphism of$\oo$-categories} (or \emph{$\oo$-functor}),$f : X \to Y$, is simply a morphism of the underlying$\oo$\nbd-magmas. We denote by$\oo\Cat$the category of$\oo$-categories and morphisms of$\oo$-categories. This category is clearly locally presentable. ... ... @@ -739,11 +739,11 @@ The following proposition is the key result of this section. It is slightly less Concretely, Proposition \ref{prop:freeonpolygraph} gives us a recipe to construct free$\oo$-categories. It suffices to give a formal list of generating cells of the form: \begin{itemize}[label=-] \item generating$0$-cells :$x^0, y^0, \dots$\item generating$1$-cells :$x^1 : \sigma(x^1) \to \tau(x^1): y^1 : \sigma(y^1) \to \tau(y^1), \dots$\item generating$2$-cells :$x^2 : \sigma(x^2) \to \tau(x^2): y^2 : \sigma(y^2) \to \tau(y^2), \dots$\item generating$1$-cells :$x^1 : \sigma(x^1) \to \tau(x^1), y^1 : \sigma(y^1) \to \tau(y^1), \dots$\item generating$2$-cells :$x^2 : \sigma(x^2) \to \tau(x^2), y^2 : \sigma(y^2) \to \tau(y^2), \dots$\item$\dots$, \end{itemize} where for a generating$k$-cell$x$with$k>0$,$\sigma(x)$and$\tau(x)$are$(k-1)$-cells of the free$(k-1)$-category recursively generated by the generating cells of dimension strictly lower than$k$. where for a generating$k$-cell$x$with$k>0$,$\sigma(x)$and$\tau(x)$are parallel$(k-1)$-cells of the free$(k-1)$-category recursively generated by the generating cells of dimension strictly lower than$k$. \end{paragr} \begin{example} The data of$1$-cellular extension$\E$is nothing but the data of a graph$G$(or$1$-graph in the terminology of of \ref{paragr:defncat}), and it is not hard to see that, in that case,$\E^*$is nothing but the free category on$G$. That is to say, the category whose objects are those of$G$and whose arrows are strings of composable arrows of$G$; the composition being given by concatenation of strings. Hence, from Proposition \ref{prop:freeonpolygraph}, a ($1$-)category is free in the sense of Definition \ref{def:freeoocat} if and only if it is (isomorphic to) a free category on a graph. ... ... @@ -771,3 +771,130 @@ The following proposition is the key result of this section. It is slightly less \end{example} \section{Cells of free$\oo$-categories as words} In this section, we undertake to give a more explicit construction of the$(n+1)$\nbd-category$\E^*$generated by an$n$-cellular cellular extension$\E=(C,\Sigma,\sigma,\tau)$. By definition of$\E^*$, this amounts to give an explicit description of a particular type of colimit in$\oo\Cat$. Note also that since$\tau_{\leq n}(\E^*)=C$, all we need to do is to describe the$(n+1)$-cells of$\E^*$. This will take place in two steps: first we construct what ought to be called the \emph{free$(n+1)$-magma generated by$\E$}, for which the$(n+1)$-cells are really easy to describe, and then we quotient out these cells as to obtain an$(n+1)$-category, which will be$\E^*$. Recall that an$n$-category is a particular case of$n$-magma. \begin{paragr}\label{paragr:defwords} Let$\E=(C,\Sigma,\sigma,\tau)$be an$n$-cellular extension. We denote by$\W[\E]$the set of finite words of the alphabet that has: \begin{itemize}[label=-] \item a symbol$\cc_{\alpha}$for each$\alpha \in \Sigma$, \item a symbol$\ii_{x}$for each$x \in C_n$, \item a symbol$\fcomp_k$for each$0 \leq k \leq n$, \item a symbol of opening parenthesis$($, \item a symbol of closing parenthesis$)$. \end{itemize} If$w$and$w'$are two elements of$\W[\E]$, we write$ww'$for their concatenation. We now define the subset$\T[\E] \subseteq \W[\E]$of \emph{well formed words} (or \emph{terms}) on the previous alphabet together with two maps$\src, \trgt : \T[E] \to C_n$in the following recursive way: \begin{itemize}[label=-] \item for every$\alpha \in \Sigma$, the word$(\cc_{\alpha})$is well formed and we have $\src((\cc_{\alpha}))=\sigma(\alpha) \text{ and } \trgt((\cc_{\alpha}))=\tau(\alpha),$ \item for every$x \in C_n$, the word$(\ii_{x})$is well formed and we have $\src((\ii_{x}))=\trgt((\ii_{x}))=x,$ \item if$w$and$w'$are well formed words such that$\src(w)=\trgt(w')$, then the word$(w\fcomp_n w')$is well formed and we have $\src((w\fcomp_n w'))=s(w') \text{ and } \trgt((w\fcomp_n w'))=t(w),$ \item if$w$and$w'$are well formed words such that$\src_k(\src(w))=\trgt_k(\trgt(w))$for some$0 \leq k < n$, then the word$(w\fcomp_k w')$is well formed and we have $\src((w \fcomp_k w'))=\src(w)\comp_k s(w') \text{ and } \trgt((w \fcomp_k w'))=\trgt(w) \comp_k \trgt(w').$ \end{itemize} As usual, for$0\leq k1$. The binary relation$\R$on$X_n$defined as $x\R y \text{ if } F(x)=F(y)$ is a congruence. \end{example} \begin{paragr} Let$X$be an$n$-magma with$n \geq 1$and$\R$a congruence on$X$. By the first axiom of Definition \ref{def:congruence},$\R$is an equivalence relation and we can consider the quotient set$X_n/\R$. We write$[x]$for the equivalence class of an$n$-cell$x$of$X$. From the second axiom of Definition \ref{def:congruence}, we can define unambiguously $\src([x]):=\src(x) \text{ and } \trgt([x]):=\trgt(x),$ for$x \in X_n$and from the third axiom, we can define unambiguously $[x]\comp_k[y] := [x \comp_k y]$ for$x$and$yk$-composable$n$-cells of$X$. Altoghether, this defines an$n$-magma, which we denote by$X/{\R}$, whose set$k$-cells is$X_k$for$0 \leq k < n$, and$X_n/{\R}$for$k=n$. The composition, source, target and units of cells of dimension strictly lower than$n$being those of$X$and the composition, source and target of$n$-cells being given by the above formulas. \end{paragr} \begin{definition} Let$\R$be a congruence on an$n$-magma$X$with$n \geq 1$. We say that$\R$is \emph{categorical} if it satisfies all four following axioms: \begin{enumerate} \item for every$k1$, which we consider as an$n$-magma. The equality on the set of$n$-cells of$C$, is, by definition, categorical. \end{example} \begin{example} Another important example of categorical congruence is the following. Let$F : X \to Y$be a morphism of$n$-magmas with$n>1$and suppose that$Y$is an$n$-category. Then the binary relation$\R$on$X_n$defined as $x \R y \text{ if } F(x)=F(y)$ is obviously a \emph{categorical} congruence. \end{example} In the following lemma, we use the notation$\tau_{\leq n}^s(X)$for an$n$-mag \begin{lemma} Let$X$be an$n$-magma with$n>1\$ and c \end{lemma}
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