### security commit

parent 498aacaf
 ... ... @@ -576,71 +576,84 @@ Furthermore, this function satisfies the condition $\sigma'(\varphi(x))=F(\sigma(x)) \text{ and } \tau'(\varphi(x))=F(\tau(x)).$ \end{definition} \end{definition} \begin{paragr} We denote by $n\Cat^{+}$ the category of $n$-cellular extensions and morphisms of $n$-cellular extensions. Every $(n+1)$-category $C$ canonically defines an $n$-cellular extension $(\tau^s_{\leq n }(C),C_{n+1},\src,\trgt)$ where $\src,\trgt : C_{n+1} \to C_n$ are the source and target maps of $C$. This defines a functor \begin{align*} U_n : (n+1)\Cat &\to n\Cat^+\\ C &\mapsto (\tau^s_{\leq n }(C),C_{n+1},\src,\trgt). \end{align*} \end{paragr} The following proposition is the key result of this section. It is slightly less trivial than it appears. \begin{proposition}\label{prop:fromcexttocat} Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension and let $\E^*$ be the $(n+1)$-category defined as the following amalgamated sum: $On the other hand, any \E=(D,\Sigma,\sigma,\tau), cellular extension of an n-category D, yields an (n+1)-category \E^* defined as the following amalgamated sum: \begin{equation}\label{squarefreecext} \begin{tikzcd}[column sep=huge, row sep=huge] \displaystyle\coprod_{x \in \Sigma}\sS^n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \sigma(x),\tau(x)\rangle_{x \in \Sigma}}"] & C \ar[d] \\ \displaystyle\coprod_{x \in \Sigma}\sS^n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \sigma(x),\tau(x)\rangle_{x \in \Sigma}}"] & D \ar[d] \\ \displaystyle\coprod_{x \in \Sigma}\sD_{n+1}\ar[r]&\E^* \ar[from=1-1,to=2-2,very near end,phantom,"\ulcorner"] \end{tikzcd}.$ Then, $\E^*$ has an $(n+1)$-basis which is isomorphic to $\Sigma$. Moreover, the identification of $\Sigma$ as a subset of $(\E^*)_{n+1}$ is natural in that, for any morphism of $n$-cellular extensions $G=(F,\varphi) : \E \to \E'$, the induced $(n+1)$-functor $G^* : \E^* \to \E^*$ is such that the square \end{equation} This actually defines a functor \begin{align*} n\Cat^+ &\to (n+1)\Cat \\ \E &\mapsto \E^*, \end{align*} which is easily checked to be left adjoint to $U_n$. Now let $\phi : \coprod_{x \in \Sigma} \sD_{n} \to \E^*$ the bottom map of square \eqref{squarefreecext}. It induces a canonical map \begin{align*} j: \Sigma &\to (\E^*)_{n+1}\\ x &\mapsto \phi_x(e_{n+1}), \end{align*} where $e_{n+1}$ is the principal $(n+1)$-cell of $\sD_{n+1}$ (\ref{paragr:defglobe}). Notice that this map is natural in that, for any morphism of $n$-cellular extensions $G=(F,\varphi) : \E \to \E'$, the square $\begin{tikzcd} \Sigma \ar[d,hook] \ar[r,"\varphi"] & \Sigma' \ar[d,hook] \\ \Sigma \ar[d,"j"] \ar[r,"\varphi"] & \Sigma' \ar[d,"j'"] \\ (\E^*)_{n+1} \ar[r,"(G^*)_{n+1}"]&(\E'^*)_{n+1} \end{tikzcd}$ is commutative. \end{proposition} is commutative. Notice also that $j$ is compatible with source and targets in the sense that for every $x \in \Sigma$, we have $\src(j(x))=\sigma(x) \text{ and } \trgt(j(x))=\tau(x).$ \end{paragr} \begin{lemma}\label{lemma:basisfreecext} Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension. The canonical map $j: \Sigma \to (\E^*)_{n+1}$ is injective. \end{lemma} \begin{proof} Notice first that since the map $i_{n+1} : \sS^n \to \sD_{n+1}$ is nothing but the canonical inclusion $\sk_{n}(\sD_{n+1}) \to \sk_{n+1}(\sD_{n+1})=\sD_{n+1}$, it follows easily from the fact the the skeleton functors preserve colimits that $C$ is canonically isomorphic to $\sk_n(\E^*)$ and that the map $C \to \E^*$ can be identified with the canonical inclusion $\sk_n(\E^*) \to \sk_{n+1}(\E^*)=\E^*$. A thorough reading of the techniques used in the proofs of Lemma \ref{lemma:nfunctortomonoid}, Lemma \ref{lemma:freencattomonoid} and Proposition \ref{prop:countingfunction} shows that the universal property defining $\E^*$ as the amalgamated sum \eqref{squarefreecext} is sufficient enough to prove the existence, for each $x \in \Sigma$, of a function $w_{x} : (\E^*)_{n+1} \to \mathbb{N}$ such that $w_{x}(j(x))=1$ and $w_{x}(j(y))=0$ for any $y \in \Sigma$ with $y\neq x$. In particular, this implies that $j$ is injective. \end{proof} \begin{paragr} In particular, the previous lemma tells us that for any $n$-cellular extension $\E=(C,\Sigma,\sigma,\tau)$, the set of indeterminates $\Sigma$ canonically defines a subset of the $(n+1)$-cells of $\E^*$. As of now, we will consider $\Sigma$ as a subset of $(\E^{*})_{n+1}$. \end{paragr} We can now prove the following proposition, which is the key result of this section. It is slightly less trivial than it appears. \begin{proposition}\label{prop:fromcexttocat} For any $n$-cellular extension $\E=(C,\Sigma,\sigma,\tau)$, the subset $\Sigma \subseteq (\E^{*})_{n+1}$ is an $(n+1)$-basis of $\E^*$. \end{proposition} \begin{proof} Notice first that since the map $i_{n+1} : \sS^n \to \sD_{n+1}$ is nothing but the canonical inclusion $\sk_{n}(\sD_{n+1}) \to \sk_{n+1}(\sD_{n+1})=\sD_{n+1}$, it follows easily from square \eqref{squarefreecext} and the fact the the skeleton functors preserve colimits that $C$ is canonically isomorphic to $\sk_n(\E^*)$ and that the map $C \to \E^*$ can be identified with the canonical inclusion $\sk_n(\E^*) \to \sk_{n+1}(\E^*)=\E^*$. Now let $\phi : \coprod_{x \in \Sigma} \sD_{n} \to \E^*$ the bottom map of the cocartesian square of the proposition and consider the map \begin{align*} j: \Sigma &\to (\E^*)_{n+1}\\ x &\mapsto \phi_x(e_{n+1}), \end{align*} where $e_{n+1}$ is the principal $(n+1)$-cell of $\sD_{n+1}$ (\ref{paragr:defglobe}). The only subtlety of the proposition is to show that $j$ is injective and hence, that $\Sigma$ can be identified to a subset of $(n+1)$-cells of $E^*$. Now a thorough reading of the techniques used in the proofs of Lemma \ref{lemma:nfunctortomonoid}, Lemma \ref{lemma:freencattomonoid} and Proposition \ref{prop:countingfunction} shows that the universal property defining $\E^*$ is sufficient enough to prove the existence, for each $x \in \Sigma$, of a function $w_{x} : (\E^*)_{n+1} \to \mathbb{N}$ such that $w_{x}(j(x))=1$ and $w_{x}(j(y))=0$ for any $y \in \Sigma$ with $y\neq x$. In particular, this implies that $j$ is injective. Altogether, we have shown that there exists a subset of $(\E^*)_{n+1}$ (which we identify with $\Sigma$ via the map $j$) such that the square where $e_{n+1}$ is the principal $(n+1)$-cell of $\sD_{n+1}$ (\ref{paragr:defglobe}). Hence, cocartesian square \eqref{squarefreecext} can be identified with $\begin{tikzcd}[column sep=huge, row sep=huge] \displaystyle\coprod_{x \in \Sigma}\sS^n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \src(x),\trgt(x)\rangle_{x \in \Sigma}}"] & \sk_{n}(\E^*) \ar[d,hook] \\ \displaystyle\coprod_{x \in \Sigma}\sD_{n+1}\ar[r]&\sk_{n+1}(\E^*) \displaystyle\coprod_{x \in \Sigma}\sD_{n+1}\ar[r]&\sk_{n+1}(\E^*). \ar[from=1-1,to=2-2,very near end,phantom,"\ulcorner"] \end{tikzcd}$ is cocartesian, which proves the first part of the proposition. The second part, concerning the naturality of the identification of $\Sigma$ as a subset of $(\E^*)_{n+1}$ is clear from the definition of $j$. \end{proof} \begin{corollary} Let $n \geq 0$. The functor \begin{align*} n\Cat^+ &\to (n+1)\Cat \\ \E &\mapsto \E^* \end{align*} is left adjoint to the functor $U_n : (n+1)\Cat \to n\Cat^+$. \end{corollary} \begin{proof} This is simply a reformulation of the universal property defining $\E^*$. Since from $j : \Sigma \to (\E^{*})_{n+1}$ is injective, we can consider $\Sigma$ as a subset of $(\E^{*})_{n+1})$, and then, by definition, $\E^*$ has $\Sigma$ as an $(n+1)$-basis. \end{proof} \begin{paragr} Let $C$ be an $(n+1)$-category and $E$ be a subset $E \subseteq C_{n+1}$. This defines a cellular extension ... ... @@ -733,7 +746,7 @@ The following proposition is the key result of this section. It is slightly less The previous proposition admits an obvious truncated version for free $n$-categories with $n$ finite. In that case, we only need a finite sequence $(\E^{(k)}))_{ -1 \leq k \leq n-1}$ of cellular extensions. \end{remark} \begin{remark} The data of a sequence $(\E^{(n)})_{n \geq -1}$ as in Proposition \ref{prop:freeonpolygraph} is commonly referred to in the litterature of the filed as a \emph{computad} \cite{street1976limits} or \emph{polygraph} \cite{burroni1993higher}; and consequently a $\oo$-category which is free in the sense of definition \ref{def:freeoocat} is sometimes referred to as \emph{free on a computad}-or-\emph{polygraph} in the litterature. However, the underlying polygraph of a free $\oo$-category is uniquely determined by the free $\oo$-category itself (a straightforward consequence of Proposition \ref{prop:uniquebasis}), and this is why we chose the shorter terminology \emph{free $\oo$-category}. The data of a sequence $(\E^{(n)})_{n \geq -1}$ as in Proposition \ref{prop:freeonpolygraph} is commonly referred to in the litterature of the field as a \emph{computad} \cite{street1976limits} or \emph{polygraph} \cite{burroni1993higher}; and consequently a $\oo$-category which is free in the sense of definition \ref{def:freeoocat} is sometimes referred to as \emph{free on a computad}-or-\emph{polygraph} in the litterature. However, the underlying polygraph of a free $\oo$-category is uniquely determined by the free $\oo$-category itself (a straightforward consequence of Proposition \ref{prop:uniquebasis}), and this is why we chose the shorter terminology \emph{free $\oo$-category}. \end{remark} \begin{paragr} 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: ... ... @@ -859,7 +872,7 @@ In the following definition, we consider that a binary relation $\R$ on a set $E \] 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} \begin{definition}\label{def:categoricalcongruence} 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. \begin{example} Let$C$be an$n$-category with$n>1$, 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 Another important example 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 $x \R y \text{ if } F(x)=F(y) x\; \R \;y \text{ if } F(x)=F(y)$ 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 Similarly to the stupid'' truncation of$\oo$-categories (\ref{paragr:defncat}), for an$(n+1)$-magma$X$, we write$\tau_{\leq n}^s(X)$for the$n$-magma obtained by simply forgetting the cells of dimension$(n+1)$. The following lemma is trivial but nonetheless important. Its immediate proof is omitted. \begin{lemma} Let$X$be an$n$-magma with$n>1$and c Let$X$be an$n$-magma with$n>1$and$\R$a congruence on$X$. If$\tau_{\leq n}^s(X)$is an$(n-1)$-category and$\R$is categorical, then$X/{\R}$is an$n$-category. \end{lemma} We wish now to see on which conditions there exists a smallest'' congruence on an$n$-magma that contains a given binary relation$\R$on the set of$n$-cells. \begin{lemma}\label{lemma:intersectioncongruence} Let$X$be an$n$-magma with$n \geq 1$and$(\R_i)_{i \in I}$a \emph{non-empty} family of congruences on$X$(i.e.\$I$is not empty). Then, the binary relation $\R:=\bigcap_{i \in I}\R_i$ is a congruence. \end{lemma} \begin{proof} The fact that$\R$satisfies that first and third axiom of Definition \ref{def:congruence} is immediate and do not even require that$I$be non empty. The only left thing to prove is that if$x \; \R \; x'$, then$x$and$x'$are parallel. To see that, notice that since$I$is not empty, we can choose$i \in I$. Then, by definition, we have$x \; \R_i \; x'$and thus,$x$and$x'$are parallel. \end{proof} \begin{remark} The hypothesis that$I$be non empty in the previous lemma cannot be ommited because, in this case, we would have $\R=\bigcap_{\emptyset}\R_i = X_n\times X_n,$ which means that \emph{all}$n$-cells would be equivalent under$\R$. This certainly does not guarantee that the second axiom of Definition \ref{def:congruence}, i.e.\ that equivalent cells are parallel, is satisfied. \end{remark} \begin{lemma}\label{lemma:congruencegenerated} Let$X$be an$n$-magma with$n\geq 1$and$E$a set of pairs of parallel$n$-cells of$X$. There exists a smallest congruence$\R$on$X$such that for every$(x,y)\in E$, we have$x \; \R \;y$. \end{lemma} \begin{proof} Let$I$be the set of congruence$\mathcal{S}$on$X$such that for every$(x,y) \in E$, we have$x\; \mathcal{S} \; y$. All we have to prove is that$I$is not empty, since in that case, we can apply Lemma \ref{lemma:intersectioncongruence} to the binary relation $\R:=\bigcup_{\mathcal{S} \in I}\S,$ which will obviously be the smallest congruence satisfying the desired condition. To see that$I$is not empty, it suffices to notice that the binary relation being parallel$n$-cells'' is a congruence, which obviously is in$I$. \end{proof} \begin{proposition} Let$X$be an$n$-magma with$n\geq 1$. There exists a smallest categorical congruence on$X$. \end{proposition} \begin{proof} Each four axioms of Definition \ref{def:categoricalcongruence} says that some pairs of parallel$n$-cells must be equivalent under a congruence$\R$for it to be categorical. The result follows then from Lemma \ref{lemma:congruencegenerated}. \end{proof} \begin{paragr} Let$\E=(C,\Sigma,\sigma,\tau)$be an$n\$-cellular extension \end{paragr}
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