### end of the working day

parent 271765db
 ... ... @@ -317,36 +317,38 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp \begin{paragr} A \emph{bisimplicial set} is a presheaf over the category $\Delta \times \Delta$, $X : \Delta^{op} \times \Delta^{op} \to \Set. X : \Delta^{\op} \times \Delta^{\op} \to \Set.$ In a similar fashion as for simplicial sets (\ref{paragr:simpset}), for $n,m \geq 0$, we use the following notations for the face and degeneracy operators: \begin{align*} In a similar fashion as for simplicial sets (\ref{paragr:simpset}), for $n,m \geq 0$, we use the notation \begin{align*} X_{n,m} &:= X([n],[m]) \\ \partial_i^h &:=X(\delta^i,\mathrm{id}) : X_{n+1,m} \to X_{n,m}\\ \partial_j^v &:=X(\mathrm{id},\delta^j) : X_{n,m+1} \to X_{n,m}\\ s_i^h &:=X(\sigma^i,\mathrm{id}): X_{n,m} \to X_{n+1,m}\\ s_j^v&:=X(\mathrm{id},\sigma^j) : X_{n,m} \to X_{n,m+1}. \end{align*} The maps $\partial_i^h$ and $s_i^h$ will be referred to as the \emph{horizontal} face and degeneracy operators; and $\partial_i^v$ and $s_i^v$ as the \emph{vertical} face and degeneracy operators. Note that for every $n\geq 0$, we have simplicial sets \begin{align*} X_{\bullet,n} : \Delta^{op} &\to \Set \\ X_{\bullet,n} : \Delta^{\op} &\to \Set \\ [k] &\mapsto X_{k,n} \end{align*} and \begin{align*} X_{n,\bullet} : \Delta^{op} &\to \Set \\ X_{n,\bullet} : \Delta^{\op} &\to \Set \\ [k] &\mapsto X_{n,k}. \end{align*} The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$. \iffalse Moreover for every $n \geq 0$, if we fix the first variable to $n$, we obtain a simplical set \begin{align*} X_{n,\bullet} : \Delta^{op} &\to \Set \\ X_{n,\bullet} : \Delta^{\op} &\to \Set \\ [m] &\mapsto X_{n,m}. \end{align*} Similarly, if we fix the second variable to $n$, we obtain a simplicial \begin{align*} X_{\bullet,n} : \Delta^{op} &\to \Set \\ X_{\bullet,n} : \Delta^{\op} &\to \Set \\ [m] &\mapsto X_{m,n}. \end{align*} The correspondances ... ... @@ -355,9 +357,9 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp \] actually define functors $\Delta \to \Psh{\Delta}$. They correspond to the two currying'' operations $\Psh{\Delta\times\Delta} \to \underline{\Hom}(\Delta^{op},\Psh{\Delta}), \Psh{\Delta\times\Delta} \to \underline{\Hom}(\Delta^{\op},\Psh{\Delta}),$ which are isomorphisms of categories. In other words, the category of bisimplicial sets can be identified with the category of functors $\underline{\Hom}(\Delta^{op},\Psh{\Delta})$ in two canonical ways. which are isomorphisms of categories. In other words, the category of bisimplicial sets can be identified with the category of functors $\underline{\Hom}(\Delta^{\op},\Psh{\Delta})$ in two canonical ways. \fi \end{paragr} \begin{paragr} ... ... @@ -483,11 +485,45 @@ We shall now describe a nerve'' for $2$-categories with value in bisimplicial \end{itemize} \end{notation} \begin{paragr} Let $C$ be a $2$-category. The \emph{bisimplicial nerve} of $C$ is the bisimplicial set $\nu(C)$ \todo{notation ?} defined in the following fashion: \begin{itemize} \item[-] For $n,m \geq 0$, we have $\nu(C)_{n,m} = \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times n+1}}N(C(x_0,x_1))_m\times\cdots\times N(C(x_{n-1},x_n))_m.$ \end{itemize} Each $2$-category $C$ defines a simplicial object in $\Cat$, $T(C): \Delta^{\op} \to \Cat,$ where, for each $n \geq 0$, $T(C)_n:= \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}C(x_0,x_1) \times \cdots \times C(x_{n-1},x_n),$ and where, similar to the nerve of categories, the face operators are induced by horizontal composition (of $2$-cells) and the degeneracy operators are induced by the units (for the horizontal composition). Post-composing $T(C)$ with the nerve functor $N : \Cat \to \Psh{\Delta}$, we obtain a functor $NT(C) : \Delta^{\op} \to \Psh{\Delta}.$ \end{paragr} \begin{remark} When $C$ is a $1$-category, the simplicial object $T(C)$ is nothing but the usual nerve of $C$ where, for each $n\geq 0$, $T(C)_n$ is seen as a discrete category. \end{remark} \begin{definition} The \emph{bisimplicial nerve} of a $2$-category $C$ is the bisimplicial set $\binerve(C)$ defined as $\binerve(C)_{n,m}:=N(T(C)_n)_m,$ for all $n,m \geq 0$. \end{definition} %(Or, more conceptually, by uncurryfying'' $NT(C)$). \begin{paragr} Since the nerve $N$ commutes with products and sums, we obtain the formula $\binerve(C)_{n,m} = \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}N(C(x_0,x_1))_m \times \cdots \times N(C(x_{n-1},x_n))_m.$ More intuitively, an element of $\binerve(C)_{n,m}$ consists of a pasting scheme'' in $C$ that look like $m \text{ times }\underbrace{\left\{\begin{tikzcd}[column sep=huge,ampersand replacement=\&] \bullet \ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=A,below}] \ar[r,bend left=35,""{name=B,above}] \ar[r,bend right=35,"\vdots",""{name=G,below}]\ar[r,bend right=90,looseness=1.4,""{name=H,above}] \& \bullet\ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=C,below}] \ar[r,bend left=35,""{name=D,above}] \ar[r,bend right=35,"\vdots",""{name=I,below}]\ar[r,bend right=90,looseness=1.4,""{name=J,above}] \&\bullet\ar[r,phantom,description,"\cdots"]\&\bullet\ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=E,below}] \ar[r,bend left=35,""{name=F,above}] \ar[r,bend right=35,"\vdots",""{name=K,below}]\ar[r,bend right=90,looseness=1.4,""{name=L,above}] \&\bullet \ar[from=A,to=B,Rightarrow] \ar[from=C,to=D,Rightarrow] \ar[from=E,to=F,Rightarrow] \ar[from=G,to=H,Rightarrow] \ar[from=I,to=J,Rightarrow] \ar[from=K,to=L,Rightarrow] \end{tikzcd}\right.}_{ n \text{ times }}$ \todo{parenthèses moches dans le diagramme} \end{paragr}
 \ProvidesPackage{mystyle} % Layout %\usepackage{times} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} ... ... @@ -135,6 +135,11 @@ \newcommand{\good}{homologically coherent} %This is provisional. I need to find a good terminology \newcommand{\op}{\mathrm{op}} %For opposite categories % Nerve \newcommand{\binerve}{N_{\Delta\times\Delta}} % Bisimplicial nerve % squelette \newcommand{\sk}{\mathrm{sk}} ... ...
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