### dodo

parent 2e48a891
 ... ... @@ -610,7 +610,7 @@ Furthermore, this function satisfies the condition (\E^*)_{n+1} \ar[r,"(G^*)_{n+1}"]&(\E'^*)_{n+1} \end{tikzcd} \] is commutative. Notice also that $j$ is compatible with source and targets in the sense that for every $x \in \Sigma$, we have is commutative. Notice also that $j$ is compatible with source and target in the sense that for every $x \in \Sigma$, we have $\src(j(x))=\sigma(x) \text{ and } \trgt(j(x))=\tau(x).$ ... ... @@ -629,9 +629,14 @@ Furthermore, this function satisfies the condition \] 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 consequence of the previous lemma, we will often identify $\Sigma$ to a subset of $(\E^*)_{n+1}$. When we do so, it will \emph{always} be via tha map $j$. Note that this identification is compatible with source and target in the sense that the source (resp. target) of $x \in \Sigma$, seen as $(n+1)$-cell of $\E^*$, is exactly $\sigma(x)$ (resp. $\tau(x)$). \end{paragr} \iffalse \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 $(\E^{*})_{n+1}$ and $j$ as the canonical inclusion. \end{paragr} \fi 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 $\E^*$. ... ... @@ -809,9 +814,11 @@ Recall that an $n$-category is a particular case of $n$-magma. \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 $0\leq k1$, 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 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 \begin{example}\label{example:categoricalcongruence} 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 $x\; \R \;y \text{ if } F(x)=F(y)$ ... ... @@ -902,8 +909,8 @@ In the following definition, we consider that a binary relation $\R$ on a set $E \end{example} Similarly to the stupid'' truncation of$\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)$. The following lemma is trivial but nonetheless important. Its immediate proof is omitted. \begin{lemma} The following lemma is trivial but nonetheless important. Its immediate proof is ommited. \begin{lemma}\label{lemma:quotientmagma} Let$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$-category. \end{lemma} We wish now to see how to prove the existence of a congruence defined with a condition such as the smallest congruence that contains a given binary relation on the$(n+1)$-cells''. ... ... @@ -940,6 +947,110 @@ We wish now to see how to prove the existence of a congruence defined with a con \begin{proof} Each four axioms of Definition \ref{def:categoricalcongruence} says that some pairs of parallel$n$-cells must be equivalent under a congruence$\R$for it to be categorical. The result follows then from Lemma \ref{lemma:congruencegenerated}. \end{proof} \iffalse \begin{paragr} Let$\E=(C,\Sigma,\sigma,\tau)$be an$n$-cellular extension,$D$an$(n+1)$-category and $G=(\varphi,F) : \E \to U_n(D)$ a morphism of$n$-cellular extensions. \end{paragr} \fi \begin{paragr} Let$\E=(C,\Sigma,\sigma,\tau)$be an$n$-cellular extension and consider the injective map$j : \Sigma \to (\E^*)_{n+1}$constructed in Paragraph \ref{paragr:freecext}. By induction, we define a map In particular, for$\E$an$n$-cellular extension, there exists a smallest categorical congruence on$\E^+$, which we denote here by$\equiv$. Applying Lemma \ref{lemma:quotientmagma} gives us that$\E^+/{\equiv}$is an$(n+1)$-category. The construction $\E \mapsto \E^+/{\equiv}$ clearly defines a functor$\Cat_n^+ \to (n+1)\Cat$. \end{paragr} We can now prove the expected result. \begin{proposition} Let$\E$be an$n$-cellular extension and let$\equiv$be the smallest categorical congruence on$\E^+$. The$(n+1)$-category$\E^+/{\equiv}$is naturally isomorphic to$\E^*$. \end{proposition} \begin{proof} The strategy of the proof is to show that the functor$\E \mapsto \E^+/{\equiv}$is left adjoint to$U_n : (n+1)\Cat \to \Cat_n^+$. The result will follow then from the uniqueness (up to natural isomorphism) of left adjoints. Let$\E = (C,\Sigma,\sigma,\tau)$be an$n$-cellular extension,$D$an$(n+1)$-category and let $G=(\varphi,F) : \E \to U_n(D)$ be a morphism of$n$-cellular extensions. We recursively define a map $\overline{\varphi} : \T[\E] \to D_{n+1}$ as: \begin{itemize}[label=-] \item$\overline{\varphi}(\cc_{\alpha}) = \varphi(\alpha)$for$\alpha \in \Sigma$, \item$\overline{\varphi}(\ii_x)=1_{F(x)}$for$x \in C_n$, \item$\overline{\varphi}((w \fcomp_k w'))=\overline{\varphi}(w)\comp_k\overline{\varphi}(w')$. \end{itemize} A straightforward induction shows that$\overline{\varphi}$is compatible with source, target and units. Hence, we can define a morphism of$(n+1)$-magmas $\overline{G} : \E^+ \to D$ as $\overline{G}_{n+1}=\overline{\varphi}$ and $\overline{G}_k=F_k \text{ for }0 \leq k \leq n.$ Since$D$is an$(n+1)$-category, the binary relation$\R$on$\T[\E]$defined as $x\; \R \; y \text{ if } \overline{\varphi}(x)=\overline{\varphi}(y)$ is a \emph{categorical} congruence on$\E^+$(Example \ref{example:categoricalcongruence}). If we denote by$\equiv$the smallest categorical congruence on$\E^+$then, by definition,$\equiv$is included in$\R$. In particular, the map$\overline{\varphi}$induces a map $\widehat{\varphi} : \T[\E]/{\equiv} \to D_{n+1}.$ Since$\equiv$is a congruence, this map is compatible with source and target, and it is straightforward to check that it is also compatible with units. Hence, we have an$(n+1)-functor $\widehat{G} : \E^+/{\equiv} \to D$ defined as $\widehat{G}_{n+1} = \widehat{\varphi}$ and $\tau_{\leq n }^s(\widehat{G})=F.$ Altogether, we have constructed a map \begin{align*} \Hom_{\Cat_n^+}(\E,U_n(D)) &\to \Hom_{(n+1)\Cat}(\E^+/{\equiv},D)\\ G &\mapsto \hat{G} \end{align*} which is clearly natural in\E$and$D$. What we want to prove is that this map is a bijection. Let us start with the surjectivity. Let$H : \E^+/{\equiv} \to \D$be a$(n+1)$-functor. We define a map$\varphi : \Sigma \to D_{n+1}as \begin{align*} \varphi : \Sigma &\to D_{n+1}\\ \alpha &\mapsto H_{n+1}([\cc_{\alpha}]), \end{align*} where[w]$is the equivalence class under$\equiv$of an element$w \in \T[\E]$. All we need to show is that $\widehat{\varphi}=H_{n+1}.$ Let$z$be an element of$\T[\E]/{\equiv}$and let us choose$w \in \T[\E]$such that$z=[w]$. We proceed to show that$\widehat{\varphi}(z)=H_{n+1}(z)$by induction on the size of$w$(Definition \ref{def:sizeword}). If$|w|=0$, then either$w=\cc_{\alpha}$for some$\alpha \in \Sigma$or$w = \ii_x$for some$x \in C_n$. In the first case, we have $\widehat{\varphi}([\cc_{\alpha}])=\overline{\varphi}(\cc_{\alpha})=\varphi(\alpha)=H_{n+1}([\cc_{\alpha}]),$ and in the second case we have, $\widehat{\varphi}([\ii_x])=\overline{\varphi}(\ii_x)=1_{H_n(x)}=H_{n+1}([\ii_x])$ where for the last equality, we used the fact that$[\ii_x]$is the unit on$x$in$\E^+/{\equiv}$. Now if$|w| = n+1$with$n \geq 0$, then$w=(w'\fcomp_k w'')$for some$w', w'' \in \T[\E]$that are$k$-composable with$k\leq n. Hence, using the induction hypothesis, we have \begin{align*} \widehat{\varphi}([w])=\widehat{\varphi}([(w' \fcomp_k w'')])=\widehat{\varphi}([w']\comp_k [w''])&=\widehat{\varphi}([w'])\comp_k\widehat{\varphi}([w''])\\&= H_{n+1}([w'])\comp_kH_{n+1}([w''])\\&=H_{n+1}([w']\comp_k [w''])\\&=H_{n+1}([(w' \fcomp_k w'')])\\&=H_{n+1}([w]). \end{align*} We now turn to the injectivity. LetG=(\varphi,F)$and$G'=(\varphi',F')$be two morphism of$n$-cellular extensions$\E \to U_n(D)$such that$\widehat{G}=\widehat{G'}$. Since$F=\tau_{\leq n}^s(\widehat{G})=\tau_{\leq n}^s(\widehat{G'})=F'$, all we have to show is that $\varphi=\varphi'.$ But, by definition, for every$\alpha \in \Sigma$we have $\varphi(\alpha)=\widehat{\varphi}([\cc_{\alpha}])=\widehat{\varphi'}([\cc_{\alpha}])=\varphi'(\alpha).$ \end{proof} \begin{paragr} In particular, the previous proposition tells us that$n$-cells in a free$\oo$-category can be represented as a (well formed) words made up of the generating$n\$-cells and units on lower dimensionan cells. \end{paragr}
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