Commit 533c6bbb by Leonard Guetta

### edited a lot of typos

parent efb96c27
 ... ... @@ -15,13 +15,13 @@ \begin{aligned} X_n &:= X([n]) \\ \partial_i &:= X(\delta^i): X_n \to X_{n\shortminus 1}\\ s_i &:= X(\sigma^i): X_{n+1} \to X_n. s_i &:= X(\sigma^i): X_{n} \to X_{n+1}. \end{aligned} \] Elements of $X_n$ are referred to as \emph{$n$-simplices of $X$}, the maps $\partial_i$ are the \emph{face maps} and the maps $s_i$ are the \emph{degeneracy maps}. Elements of $X_n$ are referred to as \emph{$n$\nbd{}simplices of $X$}, the maps $\partial_i$ are the \emph{face maps} and the maps $s_i$ are the \emph{degeneracy maps}. \end{paragr} \begin{paragr}\label{paragr:orientals} We denote by $\Or : \Delta \to \omega\Cat$ the cosimplicial object introduced by Street in \cite{street1987algebra}. The $\omega$-category $\Or_n$ is called the \emph{$n$-oriental}. There are various ways to give a precise definition of the orientals, but each of them requires some machinery that we don't want to introduce here. Instead, we only recall some important facts on orientals that we shall need in the sequel and refer to the literature on the subject (such as \cite{street1987algebra,street1991parity,street1994parity,steiner2004omega,buckley2016orientals} or \cite[chapitre 7]{ara2016joint}) for details. We denote by $\Or : \Delta \to \omega\Cat$ the cosimplicial object introduced by Street in \cite{street1987algebra}. The $\omega$-category $\Or_n$ is called the \emph{$n$\nbd{}oriental}. There are various ways to give a precise definition of the orientals, but each of them requires some machinery that we don't want to introduce here. Instead, we only recall some important facts on orientals that we shall need in the sequel and refer to the literature on the subject (such as \cite{street1987algebra,street1991parity,street1994parity,steiner2004omega,buckley2016orientals} or \cite[chapitre 7]{ara2016joint}) for details. The two main points to retain are: \begin{description} ... ... @@ -33,8 +33,8 @@ \end{description} We use the notation $\langle i_1\, i_2\cdots i_k\rangle$ for such a cell. In particular, we have that: \begin{itemize}[label=-] \item There is no generating $k$-cells for $k>n$. Hence, $\Or_n$ is an $n$-category. \item There is exactly one generating $n$-cell of $\Or_n$, which is $\langle 0 \,1 \cdots n\rangle$. We refer to this cell as the \emph{principal cell of $\Or_n$}. \item There is no generating $k$-cells for $k>n$. Hence, $\Or_n$ is an $n$\nbd{}category. \item There is exactly one generating $n$\nbd{}cell of $\Or_n$, which is $\langle 0 \,1 \cdots n\rangle$. We refer to this cell as the \emph{principal cell of $\Or_n$}. \item There are exactly $n+1$ generating $(n-1)$-cells of $\Or_n$. They correspond to the maps \delta^i : [n-1] \to [n] ... ... @@ -96,7 +96,7 @@ C &\mapsto N_{\omega}(C), \end{aligned} which we refer to as the \emph{nerve functor for $\oo$\nbd{}categories}. Furthermore, for every $n \in \mathbb{N}$, we also define a nerve functor for $n$-categories as the restriction of $N_{\oo}$ to $n\Cat$ (seen as a full subcategory of $\oo\Cat$) which we refer to as the \emph{nerve functor for $\oo$\nbd{}categories}. Furthermore, for every $n \in \mathbb{N}$, we also define a nerve functor for $n$\nbd{}categories as the restriction of $N_{\oo}$ to $n\Cat$ (seen as a full subcategory of $\oo\Cat$) $N_n := N_{\oo}{\big |}_{n\Cat} : n\Cat \to \Psh{\Delta}.$ ... ... @@ -172,7 +172,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with Let $n \in \nbar$. A morphism $f : X \to Y$ of $n\Cat$ is a \emph{Thomason equivalence} when ${N_n(f) : N_n(X) \to N_n(Y)}$ is a weak equivalence of simplicial sets. We denote by $\W_n^{\mathrm{Th}}$ the class of Thomason equivalences. \end{definition} \begin{paragr}\label{paragr:notationthom} We usually make reference to Thomason equivalences in the notations of homotopic constructions induced by these equivalences. For example, we write $\Ho(n\Cat^{\Th})$ for the homotopy op\nbd{}prederivator of $(n\Cat,\W_n^{\Th})$ and We usually make reference to the name Thomason'' in the notations of homotopic constructions induced by Thomason equivalences. For example, we write $\Ho(n\Cat^{\Th})$ for the homotopy op\nbd{}prederivator of $(n\Cat,\W_n^{\Th})$ and $\gamma^{\Th} : n\Cat \to \Ho(n\Cat^{\Th})$ ... ... @@ -296,15 +296,15 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with $\alpha_x : u(x) \to v(x),$ \item for every $n$-cell of $x$ of $X$ with $n>0$, an $(n+1)$-cell of $Y$ \item for every $n$\nbd{}cell of $x$ of $X$ with $n>0$, an $(n+1)$-cell of $Y$ $\alpha_x : \alpha_{\trgt_{n-1}(x)}\comp_{n-1}\cdots\comp_1\alpha_{\trgt_0(x)}\comp_0u(x) \to v(x)\comp_0\alpha_{\src_0(x)}\comp_1\cdots\comp_{n-1}\alpha_{\src_{n-1}(x)}$ subject to the following axioms: \begin{enumerate} \item for every $n$-cell $x$ of $X$, \item for every $n$\nbd{}cell $x$ of $X$, $\alpha_{1_x}=1_{\alpha_x},$ \item for all $0\leq k < n$, for all $n$-cells $x$ and $y$ of $X$ that are $k$-composable, \item for all $0\leq k < n$, for all $n$\nbd{}cells $x$ and $y$ of $X$ that are $k$-composable, $\begin{multlined} \alpha_{x \comp_k y}={\left(v(\trgt_{k+1}(x))\comp_0\alpha_{\src_0(x)}\comp_1\cdots\comp_{n-1}\alpha_{\src_{n-1}(x)}\comp_k\alpha_y\right)}\\ ... ... @@ -327,7 +327,7 @@ From now on, we will consider that the category \Psh{\Delta} is equipped with \[ (1_u)_{x}:=1_{u(x)}$ for every $n$-cell $x$ of $C$. More abstractly, this oplax transformation corresponds to the $\oo$\nbd{}functor for every cell $x$ of $C$. More abstractly, this oplax transformation corresponds to the $\oo$\nbd{}functor $\sD_1 \otimes C \overset{p\otimes i}{\longrightarrow} \sD_0 \otimes D \simeq D,$ ... ... @@ -382,10 +382,10 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with Every oplax homotopy equivalence is a Thomason equivalence. \end{proposition} \begin{paragr}\label{paragr:defrtract} An $\oo$\nbd{}functor $i : C \to D$ is an \emph{oplax deformation retract} if there exists an $\oo$\nbd{}functor $r : C \to D$ such that: An $\oo$\nbd{}functor $i : C \to D$ is an \emph{oplax deformation retract} if there exists an $\oo$\nbd{}functor $r : D \to C$ such that: \begin{enumerate}[label=(\alph*)] \item $r\circ i=\mathrm{id}_C$, \item there exists an oplax transformation $\alpha : \mathrm{id}_B \Rightarrow i\circ r$. \item there exists an oplax transformation $\alpha : \mathrm{id}_D \Rightarrow i\circ r$. \end{enumerate} Furthermore, $i$ is a \emph{strong oplax deformation retract} if $\alpha$ can be chosen such that: \begin{enumerate}[label=(\alph*),resume] ... ... @@ -423,13 +423,13 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with From the commutativity of the following solid arrow diagram $\begin{tikzcd} A \ar[r,"u"] \ar[d,"i"] & A' \ar[d,"i'"] \ar[dd,bend left=75,"\mathrm{id}_{B'}"] \\ A \ar[r,"u"] \ar[d,"i"] & A' \ar[d,"i'"] \ar[dd,bend left=75,"\mathrm{id}_{A'}"] \\ B \ar[d,"r"] \ar[r,"v"] & B' \ar[d,"r'",dashed ] \\ A \ar[r,"u"] & A', \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] \end{tikzcd}$ we deduce the existence of $r' : B' \to A'$ that makes the whole diagram commutes. In particular, we have $r' \circ i' = \mathrm{id}_{B'}$. we deduce the existence of $r' : B' \to A'$ that makes the whole diagram commutes. In particular, we have $r' \circ i' = \mathrm{id}_{A'}$. From the commutativity of (\ref{diagramstrong}), we easily deduce the commutativity of the following solid arrow diagram $... ... @@ -503,7 +503,7 @@ From now on, we will consider that the category \Psh{\Delta} is equipped with Let C be an \omega-category. We define the equivalence relation \sim_{\omega} on the set C_n by co-induction on n \in \mathbb{N}. For x, y \in C_n, we have x \sim_{\omega} y  when: \begin{itemize} \item[-] x and y are parallel, \item[-] there exists r, s \in C_{n+1} such that r : x \to y, s : y \to x, \item[-] there exist r, s \in C_{n+1} such that r : x \to y, s : y \to x, \[ r\ast_{n}s \sim_{\omega} 1_y$ ... ... @@ -516,15 +516,15 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with For details on this definition and the proof that it is an equivalence relation, see \cite[section 4.2]{lafont2010folk}. \end{paragr} \begin{example} Let $x$ and $y$ be two $0$-cells of an $n$-category $C$. Let $x$ and $y$ be two $0$-cells of an $n$\nbd{}category $C$. \begin{itemize}[label=-] \item When $n=1$, $x \sim_{\omega} y$ means that $x$ and $y$ are isomorphic. \item When $n=2$, $x \sim_{\omega} y$ means that $x$ and $y$ are equivalent, i.e.\ there exists $f : x \to y$ and $g : y \to x$ such that $fg$ is isomorphic to $1_y$ and $gf$ is isomorphic to $1_x$. \item When $n=2$, $x \sim_{\omega} y$ means that $x$ and $y$ are equivalent, i.e.\ there exist $f : x \to y$ and $g : y \to x$ such that $fg$ is isomorphic to $1_y$ and $gf$ is isomorphic to $1_x$. \end{itemize} \end{example} For later reference, we put here the following trivial but important lemma, whose proof is omitted. \begin{lemma}\label{lemma:ooequivalenceisfunctorial} Let $F : C \to D$ be an $\oo$\nbd{}functor and $x$,$y$ be $n$-cells of $C$ for some $n \geq 0$. If $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$. Let $F : C \to D$ be an $\oo$\nbd{}functor, $n \geq 0$ and $x,y$ be $n$\nbd{}cells of $C$. If $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$. \end{lemma} \begin{definition}\label{def:eqomegacat} An $\omega$-functor $F : C \to D$ is an \emph{equivalence of $\oo$\nbd{}categories} when: ... ... @@ -663,7 +663,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences \] Let us now give an alternative definition of the $\oo$\nbd{}category $A/a_0$ using explicit formulas. The equivalence with the previous definition follows from the dual of \cite[Proposition B.5.2]{ara2016joint} \begin{itemize}[label=-] \item An $n$-cell of $A/a_0$ is a table \item An $n$\nbd{}cell of $A/a_0$ is a table $(x,a)=\begin{pmatrix} \begin{matrix} ... ... @@ -676,10 +676,10 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends equivalences where x_0 and x_0' are 0-cells of A, and: \begin{tabular}{ll} x_i : x_{i-1} \longrightarrow x'_{i-1} &for every 1 \leq i \leq n,\\[0.75em] x_i': x_{i-1} \longrightarrow x'_{i-1} &for every 1 \leq i < n,\\[0.75em] a_i : a'_{i-1}\comp_{i-2} a'_{i-2} \comp_{i-3} \cdots \comp_0 x_{i-1} \longrightarrow a_{i-1}, &for every 1 \leq i \leq n+1,\\[0.75em] a'_i : a'_{i-1}\comp_{i-2} a'_{i-2} \comp_{i-3} \cdots \comp_0 x'_{i-1} \longrightarrow a_{i-1}, &for every 1 \leq i \leq n\\ x_i : x_{i-1} \longrightarrow x'_{i-1}, &for every 1 \leq i \leq n,\\[0.75em] x_i': x_{i-1} \longrightarrow x'_{i-1}, &for every 1 \leq i \leq n-1,\\[0.75em] a_i : a'_{i-1}\comp_{i-2} a'_{i-2} \comp_{i-3} \cdots \comp_1 a'_1\comp_0 x_{i-1} \longrightarrow a_{i-1}, &for every 1 \leq i \leq n+1,\\[0.75em] a'_i : a'_{i-1}\comp_{i-2} a'_{i-2} \comp_{i-3} \cdots \comp_1 a'_1 \comp_0 x'_{i-1} \longrightarrow a_{i-1}, &for every 1 \leq i \leq n\\ \end{tabular} are i-cells of A. In low dimension, this gives: ... ... @@ -703,7 +703,7 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends equivalences & (x_2,a_{3}) \end{pmatrix}}:&{\begin{tikzcd}[column sep=small] x_0 \ar[rr,"x_1"] \ar[rd,"a_1"',""{name=A,left}] && x_0' \ar[ld,"a_1'"] \\ &a_0 & \ar[from=1-3,to=A,Rightarrow,"a_2", shorten <=1em, shorten >=1em]\end{tikzcd}\; \overset{a_3}{\Lleftarrow} \; \begin{tikzcd}[column sep=small] x_0\ar[rr,bend left=50,"x_1",pos=11/20,""{name=toto,below}] \ar[rr,"x_1'"description,""{name=titi,above}] \ar[rd,"a_1"',""{name=A,left}] && x_0' \ar[ld,"a_1'"] \\ &a_0 & \ar[from=1-3,to=A,Rightarrow,"a_2'", shorten <=1em, shorten >=1em] \ar[from=toto,to=titi,Rightarrow,"x_2",pos=1/5]\end{tikzcd}} \end{tabular} \item The source, target of the n-cell (a,x) are given by the matrices: \item The source, target of the n\nbd{}cell (a,x) are given by the matrices: \[ s(x,a)=\begin{pmatrix} \begin{matrix} ... ... @@ -723,7 +723,7 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends equivalences \end{pmatrix}$ % It is understood that when $n=1$, the source is simply $(x_0,a_1)$ and the target $(x_0,a_1')$ \item The unit of the $n$-cell $(a,x)$ is given by the table: \item The unit of the $n$\nbd{}cell $(a,x)$ is given by the table: $1_{(x,a)}=\begin{pmatrix} \begin{matrix} ... ... @@ -733,7 +733,7 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends equivalences & (1_{x_n},1_{a_{n+1}}) \end{pmatrix}$ \item The composition of $n$-cells $(x,a)$ and $(y,b)$ such that $\src_k(y,b)=\trgt_k(a,x)$, is given by the table: \item The composition of $n$\nbd{}cells $(x,a)$ and $(y,b)$ such that $\src_k(y,b)=\trgt_k(a,x)$, is given by the table: (y,b)\comp_k (x,a)=\begin{pmatrix} \begin{matrix} ... ... @@ -746,10 +746,10 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends equivalences where: \begin{tabular}{ll} z_{i}=y_i\comp_k x_i & for every k+1 \leq i \leq n, \\[0.75em] z'_i=y'_i \comp_k x'_i & for every k+1 \leq i \leq n-1, \\[0.75em] c_i=a_i\comp_k b_i \comp_{k-1} a'_{k-1} \comp_{k-2} a'_{k-2} \comp_{k-3} \cdots \comp_{1} a'_1\comp_0 x_k&for every k+1 \leq i \leq n+1, \\[0.75em] c'_i=a'_i\comp_k b'_i \comp_{k-1} a'_{k-1} \comp_{k-2} a'_{k-2} \comp_{k-3} \cdots \comp_{1} a'_1\comp_0 x'_k&for every k+1 \leq i \leq n.\\ z_{i}=y_i\comp_k x_i, & for every k+1 \leq i \leq n, \\[0.75em] z'_i=y'_i \comp_k x'_i, & for every k+1 \leq i \leq n-1, \\[0.75em] c_i=a_i\comp_k b_i \comp_{k-1} a'_{k-1} \comp_{k-2} a'_{k-2} \comp_{k-3} \cdots \comp_{1} a'_1\comp_0 x_k,&for every k+1 \leq i \leq n+1, \\[0.75em] c'_i=a'_i\comp_k b'_i \comp_{k-1} a'_{k-1} \comp_{k-2} a'_{k-2} \comp_{k-3} \cdots \comp_{1} a'_1\comp_0 x'_k,&for every k+1 \leq i \leq n.\\ \end{tabular} \end{itemize} We leave it to the reader to check that the formulas are well defined and that the axioms of \oo\nbd{}category are satisfied. The canonical forgetful \oo\nbd{}functor \pi : A/a_0 \to A is simply expressed as: ... ... @@ -757,7 +757,7 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends equivalences A/a_0 &\to A \\ (x,a) &\mapsto x_n. \end{align*} Notice that if A is an n-category, then so is A/a_0. In this case, for an n-cell (x,a), a_{n+1} is a unit, hence Notice that if A is an n\nbd{}category, then so is A/a_0. In this case, for an n\nbd{}cell (x,a), a_{n+1} is a unit, hence \[ a'_n \comp_{n-1} a'_{n-1} \comp_{n-2} \cdots \comp_1 a'_1 \comp_0 x_n = a_n. ... ... @@ -774,7 +774,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences \ar[from=1-1,to=2-2,phantom,description,very near start,"\lrcorner"] \end{tikzcd} \] More explicitly, an $n$-cell $(x,b)$ of $A/b_0$ is a table More explicitly, an $n$\nbd{}cell $(x,b)$ of $A/b_0$ is a table $(x,b)=\begin{pmatrix} \begin{matrix} ... ... @@ -794,7 +794,7 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends equivalences & (u(x_n),b_{n+1}) \end{pmatrix}$ is an $n$-cell of $B/b_0$. is an $n$\nbd{}cell of $B/b_0$. The canonical $\oo$\nbd{}functor $A/b_0 \to A$ is simply expressed as \begin{align*} ... ... @@ -834,7 +834,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences is an equivalence of $\oo$\nbd{}categories, then so is $u$. \end{theorem} \begin{proof} Before anything else, recall from Lemma \ref{lemma:ooequivalenceisfunctorial} that given an $\oo$\nbd{}functor $F : X \to Y$ and $n$-cells $x$ and $y$ of $X$, if $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$. Before anything else, recall from Lemma \ref{lemma:ooequivalenceisfunctorial} that given an $\oo$\nbd{}functor $F : X \to Y$ and $n$\nbd{}cells $x$ and $y$ of $X$, if $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$. \begin{enumerate}[label=(\roman*)] \item Let $b_0$ be $0$\nbd{}cell of $B$ and set $c_0:=v(b_0)$. By definition, the pair $(b_0,1_{c_0})$ is a $0$-cell of $B/c_0$. By hypothesis, we know that there exists a $0$\nbd{}cell $(a_0,c_1)$ of $A/c_0$ such that $(u(a_0),c_1)\sim_{\oo} (b_0,1_{c_0})$. Hence, by applying the canonical functor $B/c_0 \to B$, we obtain that $u(a_0) \sim_{\oo} b_0$. ... ... @@ -886,7 +886,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences & (\alpha,\Lambda) \end{pmatrix} \] whose image by $u/c_0$ is equivalent for the relation $\sim_{\oo}$ to the above $(n+1)$\nbd{}cell of $B/c_0$ . In particular, the source and target of $\alpha$ are respectively $f$ and $f'$. Finally, we obtain that $\alpha \sim_{\oo} \beta$ by applying the canonical $\oo$\nbd{}functor $A/b_0 \to A$, . whose image by $u/c_0$ is equivalent for the relation $\sim_{\oo}$ to the above $(n+1)$\nbd{}cell of $B/c_0$. In particular, the source and target of $\alpha$ are respectively $f$ and $f'$. Finally, we obtain that $\alpha \sim_{\oo} \beta$ by applying the canonical $\oo$\nbd{}functor $A/b_0 \to A$, . \end{enumerate} \end{proof} %\todo{Il faudrait vérifier que je n'ai pas écrit de bêtises dans la preuve précédente.} ... ...
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