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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.
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 $\Delta[n]= \Delta(-,[n])$.
Likewise, there is a standard topological $n$-simplex, which is (more or less by definition) the geometric realization of the standard simplicial $n$-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:
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:
In the following, for $n \in \mathbb{N}$ we regard the Cartesian space $\mathbb{R}^n$ as equipped with the canonical coordinates labeled $x_0, x_1, \cdots, x_{n-1}$.
For $n \in \mathbb{N}$, the topological $n$-simplex is, up to homeomorphism, the topological space whose underlying set is the subset
of the Cartesian space $\mathbb{R}^{n+1}$, and whose topology is the subspace topology induces from the canonical topology in $\mathbb{R}^{n+1}$.
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
induced under the barycentric coordinates of def. , by the inclusion
which omits the $k$th coordinate
The inclusion
is the inclusion
of the “right” end of the standard interval. The other inclusion
is that of the “left” end $\{0\} \hookrightarrow [0,1]$.
For $n \in \mathbb{N}$ and $0 \leq k \lt n$ the $k$th degenerate $n$-simplex (projection) is the surjective map
induced under the barycentric coordinates of def. under the surjection
which sends
The collection of face inclusions, def. and degeneracy projections, def. satisfy the (dual) simplicial identities. Equivalently, they constitute the components of a functor
from the simplex category $\Delta$ to the category Top of topological spaces. This is, up to isomorphism, the standard cosimplicial object in $Top$.
The standard topological $n$-simplex is, up to homeomorphism, the subset
equipped with the subspace topology of the standard topology on the Cartesian space $\mathbb{R}^n$.
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
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:
For $n \in \mathbb{N}$, write now explicitly
for the topological $n$-simplex in barycentric coordinate presentation, def. , and
for the topological $n$-simplex in Cartesian coordinate presentation, def. .
Write
for the continuous function given in the standard coordinates by
By restriction, this induces a continuous function on the topological $n$-simplices
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
For $X \in$ Top and $n \in \mathbb{N}$, a singular $n$-simplex in $X$ is a continuous map
Write
for the set of singular $n$-simplices of $X$.
As $n$ varies, this forms the singular simplicial complex of $X$.
The orientals relate simplices to globes.
Last revised on July 23, 2021 at 04:03:59. See the history of this page for a list of all contributions to it.