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\begin{document}

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\section*{simplex}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definitions}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{SimplicialSimplex}{Simplicial simplices}\dotfill \pageref*{SimplicialSimplex} \linebreak
\noindent\hyperlink{CellularSimplex}{Cellular (simplicial) simplex}\dotfill \pageref*{CellularSimplex} \linebreak
\noindent\hyperlink{TopologicalSimplex}{Topological simplex}\dotfill \pageref*{TopologicalSimplex} \linebreak
\noindent\hyperlink{BarycentricCoordinates}{Barycentric coordinates}\dotfill \pageref*{BarycentricCoordinates} \linebreak
\noindent\hyperlink{CartesianCoordinates}{Cartesian coordinates}\dotfill \pageref*{CartesianCoordinates} \linebreak
\noindent\hyperlink{CoordinateTransformation}{Transformation between Barycentric and Cartesian coordinates}\dotfill \pageref*{CoordinateTransformation} \linebreak
\noindent\hyperlink{SingularSimplex}{Singular simplex}\dotfill \pageref*{SingularSimplex} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_globes}{Relation to globes}\dotfill \pageref*{relation_to_globes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{cellular simplex} is one of the basic [[geometric shapes for higher structures]]. Variants of the same ``shape archetype'' exist in several settings, e.g., that of a [[simplicial set]], the topological /cellular one, and categorical contexts, plus others.

\hypertarget{Definition}{}\subsection*{{Definitions}}\label{Definition}

\hypertarget{SimplicialSimplex}{}\subsubsection*{{Simplicial simplices}}\label{SimplicialSimplex}

For $n \in \mathbb{N}$, the standard \emph{simplicial $n$-simplex} $\Delta[n]$ is the [[simplicial set]] which is represented (as a [[presheaf]]) by the object $[n]$ in the [[simplex category]], so $\Delta[n]= \Delta(-,[n])$.

\hypertarget{CellularSimplex}{}\subsubsection*{{Cellular (simplicial) simplex}}\label{CellularSimplex}

Likewise, there is a standard topological $n$-simplex, which is (more or less by definition) the [[geometric realization]] of the standard simplicial $n$-simplex.

\hypertarget{TopologicalSimplex}{}\subsubsection*{{Topological simplex}}\label{TopologicalSimplex}

The \emph{topological $n$-simplex} $\Delta^n$ is a generalization of the standard filled \emph{[[triangle]]} in the plane, from [[dimension]] 2 to arbitrary dimensions. Each $\Delta^n$ is [[homeomorphism|homeomorphic]] to the closed $n$-[[ball]] $D^n$, but its defining [[embedding]] into a [[Cartesian space]] equips its [[boundary]] with its cellular decomposition into \emph{[[faces]]}, generalizing the way that the triangle has three [[edges]] (which are 1-simplices) as [[faces]], and three points (which are [[vertices]] or 0-simplices) as corners.

The topological $n$-simplex is naturally defined as a [[subspace]] of a [[Cartesian space]] given by some relation on its canonical [[coordinates]]. There are two standard choices for such coordinate presentation, which of course define [[homeomorphism|homeomorphic]] $n$-simplices:

\begin{itemize}%
\item \hyperlink{BarycentricCoordinates}{Barycentric coordinates}


\item \hyperlink{CartesianCoordinates}{Cartesian coordinates}



\end{itemize}
Each of these has its advantages and disadvantages, depending on application, but of course there is a simple coordinate transformation that exhibits an explicit [[homeomorphism]] between the two:

\begin{itemize}%
\item \hyperlink{CoordinateTransformation}{Transformation between Barycentric and Cartesian coordinates}.

\end{itemize}
\hypertarget{BarycentricCoordinates}{}\paragraph*{{Barycentric coordinates}}\label{BarycentricCoordinates}

In the following, for $n \in \mathbb{N}$ we regard the [[Cartesian space]] $\mathbb{R}^n$ as equipped with the canonical [[coordinates]] labeled $x_0, x_1, \cdots, x_{n-1}$.

\begin{defn}
\label{TopologicalInBarycentricCoords}\hypertarget{TopologicalInBarycentricCoords}{}
For $n \in \mathbb{N}$, the \textbf{topological $n$-simplex} is, up to [[homeomorphism]], the [[topological space]] whose underlying set is the subset

\begin{displaymath}
\Delta^n \coloneqq 
  \{
    \vec x \in \mathbb{R}^{n+1}
    |
    \sum_{i = 0 }^n x_i = 1 \; and \;
    \forall i . x_i \geq 0 
  \}
  \subset \mathbb{R}^{n+1}
\end{displaymath}
of the [[Cartesian space]] $\mathbb{R}^{n+1}$, and whose topology is the [[subspace topology]] induces from the canonical topology in $\mathbb{R}^{n+1}$.

\end{defn}
\begin{defn}
\label{FaceInclusionInBarycentricCoords}\hypertarget{FaceInclusionInBarycentricCoords}{}
For $n \in \mathbb{N}$, $\n \geq 1$ and $0 \leq k \leq n$, the \textbf{$k$th $(n-1)$-face (inclusion)} of the topological $n$-simplex is the subspace inclusion

\begin{displaymath}
\delta_k : \Delta^{n-1} \hookrightarrow \Delta^n
\end{displaymath}
induced under the barycentric coordinates of def. \ref{TopologicalInBarycentricCoords}, by the inclusion

\begin{displaymath}
\mathbb{R}^n \hookrightarrow \mathbb{R}^{n+1}
\end{displaymath}
which omits the $k$th coordinate

\begin{displaymath}
(x_0, \cdots , x_{n-1}) \mapsto (x_0, \cdots, x_{k-1} , 0 , x_{k}, \cdots, x_{n-1})
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
The inclusion

\begin{displaymath}
\delta_0 : \Delta^0 \to \Delta^1
\end{displaymath}
is the inclusion

\begin{displaymath}
\{1\} \hookrightarrow [0,1]
\end{displaymath}
of the ``right'' end of the standard interval. The other inclusion

\begin{displaymath}
\delta_1 : \Delta^0 \to \Delta^1
\end{displaymath}
is that of the ``left'' end $\{0\} \hookrightarrow [0,1]$.

\end{example}
\begin{defn}
\label{DegeneracyProjectionsInBarycentricCoords}\hypertarget{DegeneracyProjectionsInBarycentricCoords}{}
For $n \in \mathbb{N}$ and $0 \leq k \lt n$ the \textbf{$k$th degenerate $n$-simplex (projection)} is the surjective map

\begin{displaymath}
\sigma_k : \Delta^{n} \to \Delta^{n-1}
\end{displaymath}
induced under the barycentric coordinates of def. \ref{TopologicalInBarycentricCoords} under the surjection

\begin{displaymath}
\mathbb{R}^{n+1} \to \mathbb{R}^n
\end{displaymath}
which sends

\begin{displaymath}
(x_0, \cdots, x_n) \mapsto (x_0, \cdots, x_{k} + x_{k+1}, \cdots, x_n)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The collection of face inclusions, def. \ref{FaceInclusionInBarycentricCoords} and degeneracy projections, def. \ref{DegeneracyProjectionsInBarycentricCoords} satisfy the (dual) [[simplicial identities]]. Equivalently, they constitute the components of a [[functor]]

\begin{displaymath}
\Delta^\bullet : \Delta \to Top
\end{displaymath}
from the [[simplex category]] $\Delta$ to the category [[Top]] of [[topological spaces]]. This is, up to [[isomorphism]], the standard [[cosimplicial object]] in $Top$.

\end{prop}
\hypertarget{CartesianCoordinates}{}\paragraph*{{Cartesian coordinates}}\label{CartesianCoordinates}

\begin{defn}
\label{TopologicalInCartesianCoordinates}\hypertarget{TopologicalInCartesianCoordinates}{}
The standard \textbf{topological $n$-simplex} is, up to [[homeomorphism]], the [[subset]]

\begin{displaymath}
\Delta^n 
  \coloneqq
  \{
     \vec x \in \mathbb{R}^n | 0 \leq x_1 \leq \cdots \leq x_n \leq 1
  \}
  \hookrightarrow \mathbb{R}^n
\end{displaymath}
equipped with the [[subspace topology]] of the standard topology on the [[Cartesian space]] $\mathbb{R}^n$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This definition identifies the topological $n$-simplex with the space of [[interval]] maps (preserving top and bottom) $\{0 \lt 1 \lt \ldots \lt n+1\} \to I$ into the topological interval. This point of view takes advantage of the \href{http://ncatlab.org/nlab/show/simplex+category#duality_with_intervals_23}{duality} between the [[simplex category]] $\Delta$ and the category $\nabla$ of finite [[intervals]] with distinct top and bottom. Indeed, it follows from the duality that we obtain a functor

\begin{displaymath}
\Delta \simeq \nabla^{op} \stackrel{Int(-, I)}{\to} Top.
\end{displaymath}
\end{remark}
\begin{example}
\label{}\hypertarget{}{}
\begin{itemize}%
\item For $n = 0$ this is the [[point]], $\Delta^0 = *$.


\item For $n = 1$ this is the standard [[interval object]] $\Delta^1 = [0,1]$.


\item For $n = 2$ this is a triangle sitting in the plane like this:

\begin{displaymath}
\left\{
    (x_0,x_1) | 0 \leq x_0 \leq x_1 \leq 1
  \right\}
  = 
  \left\{
  \itexarray{
     && && (1,1)
     \\
     && & \nearrow & \downarrow
     \\
     && (\tfrac{1}{2}, \tfrac{1}{2}) && (\tfrac{1}{2},1)
     \\
     & \nearrow & && \downarrow
     \\
     (0,0) &\stackrel{}{\to}& (0,\tfrac{1}{2}) & \to & (0,1)
  }
  \right\}
\end{displaymath}


\end{itemize}
\end{example}
\hypertarget{CoordinateTransformation}{}\paragraph*{{Transformation between Barycentric and Cartesian coordinates}}\label{CoordinateTransformation}

For $n \in \mathbb{N}$, write now explicitly

\begin{displaymath}
\Delta^n_{bar} \hookrightarrow \mathbb{R}^{n+1}
\end{displaymath}
for the topological $n$-simplex in barycentric coordinate presentation, def. \ref{TopologicalInBarycentricCoords}, and

\begin{displaymath}
\Delta^n_{cart} \hookrightarrow \mathbb{R}^{n}
\end{displaymath}
for the topological $n$-simplex in Cartesian coordinate presentation, def. \ref{TopologicalInCartesianCoordinates}.

Write

\begin{displaymath}
S_n : \mathbb{R}^{n+1} \to \mathbb{R}^n
\end{displaymath}
for the [[continuous function]] given in the standard coordinates by

\begin{displaymath}
(x_0, \cdots, x_{n})
  \mapsto
  (x_0, x_0 + x_1, \cdots, \sum_{i = 0}^k x_i, \cdots, \sum_{i = 0}^{n-1} x_i)
  \,.
\end{displaymath}
By restriction, this induces a continuous function on the topological $n$-simplices

\begin{displaymath}
\itexarray{
     \Delta^n_{bar} &\hookrightarrow& \mathbb{R}^{n+1}
     \\
     \downarrow^{\mathrlap{S_n|_{\Delta^n_{bar}}}} && \downarrow^{p_n}
     \\
     \Delta^n_{cart} &\hookrightarrow& \mathbb{R}^n
  }
  \,.
\end{displaymath}
\begin{prop}
\label{}\hypertarget{}{}
For every $n \in \mathbb{N}$ the function $S_n$ is a [[homeomorphism]] and respects the face and degeneracy maps.

Equivalently, $S_\bullet$ is a [[natural isomorphism]] of [[functors]] $\Delta^n \to Top$, hence an [[isomorphism]] of [[cosimplicial objects]]

\begin{displaymath}
S_\bullet : \Delta^\bullet_{bar} \stackrel{\simeq}{\to} \Delta^\bullet_{cart}
  \,.
\end{displaymath}
\end{prop}
\hypertarget{SingularSimplex}{}\subsubsection*{{Singular simplex}}\label{SingularSimplex}

\begin{defn}
\label{}\hypertarget{}{}
For $X \in$ [[nLab:Top]] and $n \in \mathbb{N}$, a \textbf{singular $n$-simplex} in $X$ is a [[nLab:continuous map]]

\begin{displaymath}
\sigma : \Delta^n \to X
  \,.
\end{displaymath}
Write

\begin{displaymath}
(Sing X)_n \coloneqq Hom_{Top}(\Delta^n , X)
\end{displaymath}
for the set of singular $n$-simplices of $X$.

\end{defn}
As $n$ varies, this forms the [[singular simplicial complex]] of $X$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_globes}{}\subsubsection*{{Relation to globes}}\label{relation_to_globes}

The [[orientals]] relate simplices to [[globes]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[vertex]], [[edge]]


\item [[triangle]], [[tetrahedron]]


\item [[horn]], [[boundary of a simplex]], [[spine]]


\item [[differential forms on simplices]]


\item [[simplicial set]]

\begin{itemize}%
\item [[simplicial complex]]


\item [[Kan complex]]


\item [[weak Kan complex]]



\end{itemize}

\item [[simplicial category]]


\item [[globe]],


\item [[tree]], [[dendrex]]


\item [[Bloch region]]



\end{itemize}
[[!redirects simplices]]

[[!redirects simplicial simplex]] [[!redirects cellular simplex]] [[!redirects topological simplex]]

[[!redirects simplicial simplices]] [[!redirects cellular simplices]] [[!redirects topological simplices]]

[[!redirects singular simplex]] [[!redirects singular simplices]]

[[!redirects k-simplex]] [[!redirects k-simplices]]

[[!redirects 1-simplex]] [[!redirects 1-simplices]]

[[!redirects 2-simplex]] [[!redirects 2-simplices]]

[[!redirects 3-simplex]] [[!redirects 3-simplices]]

[[!redirects n-simplex]] [[!redirects n-simplices]]



\end{document}