Simplicial 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 theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.Two simplices and their closure.A vertex and its star.A vertex and its link. 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 theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A simplicial complex K {displaystyle {mathcal {K}}} is a set of simplices that satisfies the following conditions: See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial k-complex K {displaystyle {mathcal {K}}} is a simplicial complex where the largest dimension of any simplex in K {displaystyle {mathcal {K}}} equals k. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimensional simplices. A pure or homogeneous simplicial k-complex K {displaystyle {mathcal {K}}} is a simplicial complex where every simplex of dimension less than k is a face of some simplex σ ∈ K {displaystyle sigma in {mathcal {K}}} of dimension exactly k. Informally, a pure 1-complex 'looks' like it's made of a bunch of lines, a 2-complex 'looks' like it's made of a bunch of triangles, etc. An example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices. A facet is any simplex in a complex that is not a face of any larger simplex. (Note the difference from a 'face' of a simplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension. Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex. For a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. The term cell is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the definition of cell complex. The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices.