end of the working day

2cat.tex

@@ -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}

mystyle.sty

 \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}}