nLab simplex

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 a simplicial set, 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 topological $n$-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 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 homeomorphic $n$-simplices:

• Barycentric coordinates

• Cartesian coordinates

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:

• Transformation between Barycentric and Cartesian coordinates.

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

$$\Delta^n \;\coloneqq\; \Big\{ \vec x \in (\mathbb{R}_{\geq 0})^{n+1} \,\big|\, \textstyle{\sum}_{i = 0 }^n x^i = 1 \Big\} \;\subset\; \mathbb{R}^{n+1}$$

of the Cartesian space $\mathbb{R}^{n+1}$, and whose topology is the subspace topology inherited from the Euclidean topology on $\mathbb{R}^{n+1}$.

(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. , by the inclusion

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

which omits the $k$th coordinate

$$(x_0, \cdots , x_{n-1}) \mapsto (x_0, \cdots, x_{k-1} , 0 , x_{k}, \cdots, x_{n-1}) \,.$$

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 $0 \leq k \lt n$ 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. 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. and degeneracy projections, def. 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 standard 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. , and

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

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

Write

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

for the continuous function given in the standard coordinates by

$$(x_0, \cdots, x_{n}) \mapsto (x_0, x_0 + x_1, \cdots, \sum_{i = 0}^k x_i, \cdots, \sum_{i = 0}^{n-1} x_i) \,.$$

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 degeneracy 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 relate simplices to globes.

Related concepts

• vertex, edge

• triangle, tetrahedron

• horn, boundary of a simplex, spine

• differential forms on simplices

• simplicial set

  • simplicial complex

  • Kan complex

  • weak Kan complex

• simplicial category

• globe,

• tree, dendrex

• Bloch region

• Delta-generated space

References

• Paul Goerss, J. F. Jardine, Exp. 1.7 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) [doi:10.1007/978-3-0346-0189-4, webpage]