homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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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 simplicial sets, 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 toplogical $n$-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 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. 1, 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. 1 under the surjection
which sends
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
from the simplex category $\Delta$ to the category Top of topological spaces. This is, up to isomorphism, the canonical 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. 1, and
for the topological $n$-simplex in Cartesian coordinate presentation, def. 4.
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 degenracy 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 related simplices to globes.