# Contents

## Idea

The 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 simplicial sets, the topological /cellular one, and categorical contexts, plus others.

## Definitions

### Simplicial simplices

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

### Cellular (simplicial) simplex

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 dimensions. 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 application, but of course there is a simple coordinate transformation that exhibits an explicit homeomorphism between the two:

#### Barycentric coordinates

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

###### Definition

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

$\Delta^n \coloneqq \{ \vec x \in \mathbb{R}^{n+1} | \sum_{i = 0 }^n x_i = 1 \; and \; \forall i . x_i \geq 0 \} \subset \mathbb{R}^{n+1}$

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

###### Definition

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

$\delta_k : \Delta^{n-1} \hookrightarrow \Delta^n$

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

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

which omits the $k+1$st canonical coordinate

$(x_1, \cdots , x_n) \mapsto (x_1, \cdots, x_{k-1} , 0 , x_k, \cdots, x_n) \,.$
###### Example

The inclusion

$\delta_0 : \Delta^0 \to \Delta^1$

is the inclusion

$\{1\} \hookrightarrow [0,1]$

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

$\delta_1 : \Delta^0 \to \Delta^1$

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

###### Definition

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

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

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

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

which sends

$(x_0, \cdots, x_n) \mapsto (x_0, \cdots, x_{k} + x_{k+1}, \cdots, x_n) \,.$
###### 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

$\Delta^\bullet : \Delta \to Top$

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

#### Cartesian coordinates

###### Definition

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

$\Delta^n \coloneqq \{ \vec x \in \mathbb{R}^n | 0 \leq x_1 \leq \cdots \leq x_n \leq 1 \} \hookrightarrow \mathbb{R}^n$

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

###### Remark

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 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

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

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

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

$\left\{ (x_0,x_1) | 0 \leq x_0 \leq x_1 \leq 1 \right\} = \left\{ \array{ && && (1,1) \\ && & \nearrow & \downarrow \\ && (\tfrac{1}{2}, \tfrac{1}{2}) && (\tfrac{1}{2},1) \\ & \nearrow & && \downarrow \\ (0,0) &\stackrel{}{\to}& (0,\tfrac{1}{2}) & \to & (0,1) } \right\}$

#### Transformation between Barycentric and Cartesian coordinates

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

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

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

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

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

Write

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

for the continuous function given in the standard coordinates by

$(x_0, \cdots, x_{n}) \mapsto (x_0, x_0 + x_1, \cdots, \sum_{i = 0}^k x_i, \cdots, \sum_{i = 0}^n x_i) \,.$

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

$\array{ \Delta^n_{bar} &\hookrightarrow& \mathbb{R}^{n+1} \\ \downarrow^{\mathrlap{S_n|_{\Delta^n_{bar}}}} && \downarrow^{p_n} \\ \Delta^n_{cart} &\hookrightarrow& \mathbb{R}^n } \,.$
###### Proposition

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

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

$S_\bullet : \Delta^\bullet_{bar} \stackrel{\simeq}{\to} \Delta^\bullet_{cart} \,.$

### Singular simplex

###### Definition

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

$\sigma : \Delta^n \to X \,.$

Write

$(Sing X)_n \coloneqq Hom_{Top}(\Delta^n , X)$

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 January 26, 2014 00:38:59 by Tim Porter (2.26.18.36)