# Contents

## Idea

The cellular simplex is one of the basic geometric shapes for higher structures.

## Definition

### Cellular (simplicial) simplex

For $n\in ℕ$, the standard simplicial $n$-simplex is the simplicial set which is represented (as a presheaf) by the object $\left[n\right]$ in the simplex category.

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

### Topological simplex

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

Each of these has its advantages and disadvantages, depending on aplication. But of course there is a simple coordinate transformation that exhibits an explicit homeomorphism between the two:

#### Barycentric coordinates

In the following, for $n\in ℕ$ we regard the Cartesian space ${ℝ}^{n}$ as equipped with the canonical coordinates labeled ${x}_{0},{x}_{1},\cdots ,{x}_{n-1}$.

###### Definition

For $n\in ℕ$, the topological $n$-simplex is, up to homeomorphism, the topological space whose underlying set is the subset

${\Delta }^{n}≔\left\{\stackrel{⇀}{x}\in {ℝ}^{n+1}\mid \sum _{i=0}^{n}{x}_{i}=1\phantom{\rule{thickmathspace}{0ex}}\mathrm{and}\phantom{\rule{thickmathspace}{0ex}}\forall i.{x}_{i}\ge 0\right\}\subset {ℝ}^{n+1}$\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}

of the Cartesian space ${ℝ}^{n+1}$, and whose topology is the subspace topology induces from the canonical topology in ${ℝ}^{n+1}$.

###### Definition

For $n\in ℕ$, $n\ge 1$ and $0\le k\le n$, the $k$th $\left(n-1\right)$-face (inclusion) of the topological $n$-simplex is the subspace inclusion

${\delta }_{k}:{\Delta }^{n-1}↪{\Delta }^{n}$\delta_k : \Delta^{n-1} \hookrightarrow \Delta^n

induced under the barycentric coordinates of def. 1, by the inclusion

${ℝ}^{n}↪{ℝ}^{n+1}$\mathbb{R}^n \hookrightarrow \mathbb{R}^{n+1}

which omits the $k+1$st canonical coordinate

$\left({x}_{1},\cdots ,{x}_{n}\right)↦\left({x}_{1},\cdots ,{x}_{k-1},0,{x}_{k},\cdots ,{x}_{n}\right)\phantom{\rule{thinmathspace}{0ex}}.$(x_1, \cdots , x_n) \mapsto (x_1, \cdots, x_{k-1} , 0 , x_k, \cdots, x_n) \,.
###### Example

The inclusion

${\delta }_{0}:{\Delta }^{0}\to {\Delta }^{1}$\delta_0 : \Delta^0 \to \Delta^1

is the inclusion

$\left\{1\right\}↪\left[0,1\right]$\{1\} \hookrightarrow [0,1]

of the “right” end of the standard interval. The other inclusion

${\delta }_{1}:{\Delta }^{0}\to {\Delta }^{1}$\delta_1 : \Delta^0 \to \Delta^1

is that of the “left” end $\left\{0\right\}↪\left[0,1\right]$.

###### Definition

For $n\in ℕ$ and $0\le k\le n$ the $k$th degenerate $n$-simplex (projection) is the surjective map

${\sigma }_{k}:{\Delta }^{n}\to {\Delta }^{n-1}$\sigma_k : \Delta^{n} \to \Delta^{n-1}

induced under the barycentric coordinates of def. 1 under the surjection

${ℝ}^{n+1}\to {ℝ}^{n}$\mathbb{R}^{n+1} \to \mathbb{R}^n

which sends

$\left({x}_{0},\cdots ,{x}_{n}\right)↦\left({x}_{0},\cdots ,{x}_{k}+{x}_{k+1},\cdots ,{x}_{n}\right)\phantom{\rule{thinmathspace}{0ex}}.$(x_0, \cdots, x_n) \mapsto (x_0, \cdots, x_{k} + x_{k+1}, \cdots, x_n) \,.
###### Proposition

The collection of face inclusions, def. 2 and degenracy projections, def. 3 satisfy the (dual) simplicial identities. Equivalently, they constitute the components of a functor

${\Delta }^{•}:\Delta \to \mathrm{Top}$\Delta^\bullet : \Delta \to Top

from the simplex category $\Delta$ to the category Top of topological spaces. This is, up to isomorphism, the canonical cosimplicial object in $\mathrm{Top}$.

#### Cartesian coordinates

###### Definition

The standard topological $n$-simplex is, up to homeomorphism, the subset

${\Delta }^{n}≔\left\{\stackrel{⇀}{x}\in {ℝ}^{n}\mid 0\le {x}_{1}\le \cdots \le {x}_{n}\le 1\right\}↪{ℝ}^{n}$\Delta^n \coloneqq \{ \vec x \in \mathbb{R}^n | 0 \leq x_1 \leq \cdots \leq x_n \leq 1 \} \hookrightarrow \mathbb{R}^n

equipped with the subspace topology of the standard topology on the Cartesian space ${ℝ}^{n}$.

###### Remark

This definition identifies the topological $n$-simplex with the space of interval maps (preserving top and bottom) $\left\{0<1<\dots into the topological interval. This point of view takes advantage of the 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

$\Delta \simeq {\nabla }^{\mathrm{op}}\stackrel{\mathrm{Int}\left(-,I\right)}{\to }\mathrm{Top}.$\Delta \simeq \nabla^{op} \stackrel{Int(-, I)}{\to} Top.
###### Example
• For $n=0$ this is the point, ${\Delta }^{0}=*$.

• For $n=1$ this is the standard interval object ${\Delta }^{1}=\left[0,1\right]$.

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

$\left\{\left({x}_{0},{x}_{1}\right)\mid 0\le {x}_{0}\le {x}_{1}\le 1\right\}=\left\{\begin{array}{ccccc}& & & & \left(1,1\right)\\ & & & ↗& ↓\\ & & \left(\frac{1}{2},\frac{1}{2}\right)& & \left(\frac{1}{2},1\right)\\ & ↗& & & ↓\\ \left(0,0\right)& \stackrel{}{\to }& \left(0,\frac{1}{2}\right)& \to & \left(0,1\right)\end{array}\right\}$\left\{ (x_0,x_1) | 0 \leq x_0 \leq x_1 \leq 1 \right\} = \left\{ \array{ && && (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\}

#### Transformation between Barycentric and Cartesian coordinates

For $n\in ℕ$, write now explicitly

${\Delta }_{\mathrm{bar}}^{n}↪{ℝ}^{n+1}$\Delta^n_{bar} \hookrightarrow \mathbb{R}^{n+1}

for the topological $n$-simplex in barycentric coordinate presentation, def. 1, and

${\Delta }_{\mathrm{cart}}^{n}↪{ℝ}^{n}$\Delta^n_{cart} \hookrightarrow \mathbb{R}^{n}

for the topological $n$-simplex in Cartesian coordinate presentation, def. 4.

Write

${S}_{n}:{ℝ}^{n+1}\to {ℝ}^{n}$S_n : \mathbb{R}^{n+1} \to \mathbb{R}^n

for the continuous function given in the standard coordinates by

$\left({x}_{0},\cdots ,{x}_{n}\right)↦\left({x}_{0},{x}_{0}+{x}_{1},\cdots ,\sum _{i=0}^{k}{x}_{i},\cdots ,\sum _{i=0}^{n}{x}_{i}\right)\phantom{\rule{thinmathspace}{0ex}}.$(x_0, \cdots, x_{n}) \mapsto (x_0, x_0 + x_1, \cdots, \sum_{i = 0}^k x_i, \cdots, \sum_{i = 0}^n x_i) \,.

By restriction, this induces a continuous function on the topological $n$-simplices

$\begin{array}{ccc}{\Delta }_{\mathrm{bar}}^{n}& ↪& {ℝ}^{n+1}\\ {↓}^{{S}_{n}{\mid }_{{\Delta }_{\mathrm{bar}}^{n}}}& & {↓}^{{p}_{n}}\\ {\Delta }_{\mathrm{cart}}^{n}& ↪& {ℝ}^{n}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \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 } \,.
###### Proposition

For every $n\in ℕ$ the function ${S}_{n}$ is a homeomorphism and respects the face and degenracy maps.

Equivalently, ${S}_{•}$ is a natural isomorphism of functors ${\Delta }^{n}\to \mathrm{Top}$, hence an isomorphism of cosimplicial objects

${S}_{•}:{\Delta }_{\mathrm{bar}}^{•}\stackrel{\simeq }{\to }{\Delta }_{\mathrm{cart}}^{•}\phantom{\rule{thinmathspace}{0ex}}.$S_\bullet : \Delta^\bullet_{bar} \stackrel{\simeq}{\to} \Delta^\bullet_{cart} \,.

### Singular simplex

###### Definition

For $X\in$ Top and $n\in ℕ$, a singular $n$-simplex in $X$ is a continuous map

$\sigma :{\Delta }^{n}\to X\phantom{\rule{thinmathspace}{0ex}}.$\sigma : \Delta^n \to X \,.

Write

$\left(\mathrm{Sing}X{\right)}_{n}≔{\mathrm{Hom}}_{\mathrm{Top}}\left({\Delta }^{n},X\right)$(Sing X)_n \coloneqq Hom_{Top}(\Delta^n , X)

for the set of singular $n$-simplices of $X$.

As $n$ varies, this forms the singular simplicial complex of $X$.

## Properties

### Relation to globes

The orientals related simplices to globes.

Revised on April 29, 2013 20:24:54 by Urs Schreiber (89.204.138.79)