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A simplicial 3-complex. In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). ). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy th
Simplicial sets can be viewed as a higher-dimensional generalization of directed multigraphs. A simplicial set contains vertices (known as "0-simplices" in this context) and arrows ("1-simplices") between some of these vertices. Two vertices may be connected by several arrows, and directed loops that connect a vertex to itself are also allowed.
A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of a simplex always span a simplex. [1] Simplicial maps can be used to approximate continuous functions between topological spaces that can be triangulated ; this is formalized by the simplicial ...
Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. [2]: sec.8.6 As a result, it gives a computable way to distinguish one space from another.
Given two abstract simplicial complexes, Δ and Γ, a simplicial map is a function f that maps the vertices of Δ to the vertices of Γ and that has the property that for any face X of Δ, the image f (X) is a face of Γ. There is a category SCpx with abstract simplicial complexes as objects and simplicial maps as morphisms.
The simplicial complex recognition problem is a computational problem in algebraic topology. Given a simplicial complex, the problem is to decide whether it is homeomorphic to another fixed simplicial complex. The problem is undecidable for complexes of dimension 5 or more. [1] [2]: 9–11
A simplicial set is called a Kan complex if the map from {}, the one-point simplicial set, is a Kan fibration. In the model category for simplicial sets, { ∗ } {\displaystyle \{*\}} is the terminal object and so a Kan complex is exactly the same as a fibrant object .
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they ...