Commit 4fe5f13d authored by Leonard Guetta's avatar Leonard Guetta
Browse files

edited a lot of typos

parent 1466d455
......@@ -10,8 +10,7 @@
\sigma^i : [n+1] \to [n]
\]
the only surjective non-decreasing map such that the pre-image of $i \in [n]$ contains exactly two elements.
dualized
The category $\Psh{\Delta}$ of simplicial sets is the category of presheaves on $\Delta$. For a simplicial set $X$, we use the notations
The category $\Psh{\Delta}$ of \emph{simplicial sets} is the category of presheaves on $\Delta$. For a simplicial set $X$, we use the notations
\[
\begin{aligned}
X_n &:= X([n]) \\
......@@ -22,7 +21,7 @@
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}.
\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 needs 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}, \cite{street1991parity,street1994parity}, \cite{steiner2004omega}, \cite{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{$n$-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}
......@@ -43,9 +42,9 @@
for $i \in \{0,\cdots,n\}$.
\end{itemize}
\begin{description}
\item[(OR2)] For $n>0$, the source (resp.\ target) of the principal cell of $\Or_n$ can be expressed as a composite of all the generating $(n-1)$\nbd{}cells corresponding to $\delta^i$ with $i$ odd (resp.\ even); each of these generating $(n-1)$\nbd{}cell appearing exactly once in the composite.
\item[(OR2)] For $n>0$, the source (resp.\ target) of the principal cell of $\Or_n$ can be expressed as a composition of all the generating $(n-1)$\nbd{}cells corresponding to $\delta^i$ with $i$ odd (resp.\ even); each of these generating $(n-1)$\nbd{}cell appearing exactly once in the composite.
\end{description}
Another way of formulating \textbf{(OR2)} is: for $n>0$ the weight of the $(n-1)$-cell corresponding to $\delta_i$ in the \emph{source} of the principal cell of $\Or_n$ (see \ref{paragr:weight}) is $1$ if $i$ is odd and $0$ if $i$ is even and the other way around for the \emph{target} of the principal cell of $\Or_n$.
Another way of formulating \textbf{(OR2)} is: for $n>0$ the weight (see \ref{paragr:weight}) of the $(n-1)$\nbd{}cells corresponding to $\delta_i$ in the \emph{source} of the principal cell of $\Or_n$ is $1$ if $i$ is odd and $0$ if $i$ is even and the other way around for the \emph{target} of the principal cell of $\Or_n$.
Here are some pictures in low dimension:
\[
\Or_0 = \langle 0 \rangle,
......@@ -75,7 +74,7 @@
\overset{\langle 0123 \rangle}{\Rrightarrow}
\begin{tikzcd}
& \langle 1 \rangle \ar[rd,"\langle 12 \rangle"] \ar[dd,"\langle 13 \rangle"' description,""{name=B,right}] & \\
\langle 0 \rangle \ar[ru,"\langle 01 \rangle"] \ar[rd,"\langle 03 \rangle"',""{name=A,above}] & & \langle 2 \rangle \ar[ld,"\langle 23 \rangle"]\\
\langle 0 \rangle \ar[ru,"\langle 01 \rangle"] \ar[rd,"\langle 03 \rangle"',""{name=A,above}] & & \langle 2 \rangle. \ar[ld,"\langle 23 \rangle"]\\
& \langle 3 \rangle &
\ar[from=A,to=1-2,Rightarrow,"\langle 013 \rangle", near start, shorten <= 1em, shorten >= 1.5em]
\ar[from=B,to=2-3,Rightarrow,"\langle 123 \rangle", shorten <= 0.75em, shorten >=0.75em]
......@@ -101,7 +100,7 @@
\[
N_n := N_{\oo}{\big |}_{n\Cat} : n\Cat \to \Psh{\Delta}.
\]
By the usual Kan extension technique, we obtain for each $n \in \nbar$ a functor \[c_n : \Psh{\Delta} \to n\Cat,\] left adjoint of $N_n$.
By the usual Kan extension technique, we obtain for every $n \in \nbar$ a functor \[c_n : \Psh{\Delta} \to n\Cat,\] left adjoint to $N_n$.
\end{paragr}
\iffalse
\begin{lemma}
......@@ -115,7 +114,7 @@
\]
For $m > 0$ and $0 \leq i \leq m$, the $(m-1)$-simplex $\partial_i(X)$ is obtained by composing arrows at $X_i$ (or simply deleting it for $i=0$ or $m$). For $m \geq 0$ and $0 \leq i \leq m$, the $(m+1)$-simplex $s_i(X)$ is obtained by inserting a unit map at $X_i$.
For $n=2$, the functor $N_2$ is what is sometimes known as the \emph{Duskin nerve} \cite{duskin2002simplicial} (restricted from bicategories to $2$-categories). For a $2$-category $C$, a $m$-simplex $X$ of $N_2(C)$ consists of:
For $n=2$, the functor $N_2$ is what is sometimes known as the \emph{Duskin nerve} \cite{duskin2002simplicial} (restricted from bicategories to $2$-categories). For a $2$-category $C$, an $m$-simplex $X$ of $N_2(C)$ consists of:
\begin{itemize}[label=-]
\item for every $0\leq i \leq m$, an object $X_i$ of $C$,
\item for all $0\leq i \leq j \leq m$, an arrow $X_{i,j} : X_i \to X_j$ of $C$,
......@@ -128,7 +127,7 @@
\end{tikzcd}
\]
\end{itemize}
subject to the following axiom:
subject to the following axioms:
\begin{itemize}[label=-]
\item for all $0 \leq i \leq m$, we have
\[
......
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