Editing Simplicial complex (section)

== Definitions ==
A '''simplicial complex''' <math>\mathcal{K}</math> is a set of [[Simplex|simplices]] that satisfies the following conditions:
# Every [[Simplex#Elements|face]] of a simplex from <math>\mathcal{K}</math> is also in <math>\mathcal{K}</math>.
# The non-empty [[Set intersection|intersection]] of any two simplices <math>\sigma_1, \sigma_2 \in \mathcal{K}</math> is a face of both <math>\sigma_1</math> and <math>\sigma_2</math>.

See also the definition of an [[abstract simplicial complex]], which loosely speaking is a simplicial complex without an associated geometry.

A '''simplicial ''k''-complex''' <math>\mathcal{K}</math> is a simplicial complex where the largest dimension of any simplex in <math>\mathcal{K}</math> 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 <math>\mathcal{K}</math> is a simplicial complex where every simplex of dimension less than ''k'' is a face of some simplex <math>\sigma \in \mathcal{K}</math> 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. Pure simplicial complexes can be thought of as [[Triangulation (topology)|triangulations]] and provide a definition of [[polytope]]s.

A '''facet''' is a maximal simplex, i.e., any simplex in a complex that is ''not'' a face of any larger simplex.<ref>{{citation |title=Triangulations: Structures for Algorithms and Applications |volume=25 |series=Algorithms and Computation in Mathematics |first1=Jesús A. |last1=De Loera |author1-link=Jesús A. De Loera |first2=Jörg |last2=Rambau |first3=Francisco |last3=Santos |author3-link=Francisco Santos Leal |publisher=Springer |year=2010 |isbn=9783642129711 |page=493 |url=https://books.google.com/books?id=SxY1Xrr12DwC&pg=PA493 }}</ref> (Note the difference from a [[Face of a simplex|"face" of a simplex]]). A pure simplicial complex can be thought of as a complex where all facets have the same dimension. For (boundary complexes of) [[simplicial polytope]]s this coincides with the meaning from polyhedral combinatorics.

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 [[Embedding|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 [[Homeomorphism|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 (set theory)|union]] of its simplices. It is usually denoted by <math>|\mathcal{K}|</math> or <math>\|\mathcal{K}\|</math>.