Commit 241c7c49 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

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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