Paths and cylinders
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 , the standard simplicial -simplex is the simplicial set which is represented (as a presheaf) by the object in the simplex category, so .
Cellular (simplicial) simplex
Likewise, there is a standard toplogical -simplex, which is (more or less by definition) the geometric realization of the standard simplicial -simplex.
The topological -simplex is a generalization of the standard filled triangle in the plane, from dimension 2 to arbitrary dimensions. Each is homeomorphic to the closed -ball , 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 -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 -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 we regard the Cartesian space as equipped with the canonical coordinates labeled .
For , the topological -simplex is, up to homeomorphism, the topological space whose underlying set is the subset
of the Cartesian space , and whose topology is the subspace topology induces from the canonical topology in .
For , and , the th -face (inclusion) of the topological -simplex is the subspace inclusion
induced under the barycentric coordinates of def. 1, by the inclusion
which omits the st canonical coordinate
is the inclusion
of the “right” end of the standard interval. The other inclusion
is that of the “left” end .
For and the th degenerate -simplex (projection) is the surjective map
induced under the barycentric coordinates of def. 1 under the surjection
The standard topological -simplex is, up to homeomorphism, the subset
equipped with the subspace topology of the standard topology on the Cartesian space .
For this is the point, .
For this is the standard interval object .
For this is a triangle sitting in the plane like this:
Transformation between Barycentric and Cartesian coordinates
For , write now explicitly
for the topological -simplex in barycentric coordinate presentation, def. 1, and
for the topological -simplex in Cartesian coordinate presentation, def. 4.
for the continuous function given in the standard coordinates by
By restriction, this induces a continuous function on the topological -simplices
For every the function is a homeomorphism and respects the face and degenracy maps.
Equivalently, is a natural isomorphism of functors , hence an isomorphism of cosimplicial objects
For Top and , a singular -simplex in is a continuous map
for the set of singular -simplices of .
As varies, this forms the singular simplicial complex of .
Relation to globes
The orientals related simplices to globes.