- Abstract simplicial complex
In

mathematics , an**abstract simplicial complex**is a purely combinatorial description of the geometric notion of asimplicial complex , consisting of a family offinite set s closed under the operation of takingsubset s. In the context ofmatroid s andgreedoid s, abstract simplicial complexes are also called**independence systems**. [*cite book*]

author = Korte, Bernard; Lovász, László; Schrader, Rainer

year = 1991

title = Greedoids

publisher = Springer-Verlag

id = ISBN 3-540-18190-3

pages = 19–43**Definition and example**Given a

universal set "S", and a family "K" of finite nonempty subsets of "S" (that is, "K" is a subset of thepower set of "S"; ahypergraph ), "K" is an**abstract simplicial complex**if the following is true::∀ "X" ∈ "K", ∀ "Y" ⊂ "X", "Y" ∈ "K"

The elements of "K" are called

**faces**or**simplices**of the complex, so the line above states that any subset ("Y") of a simplex ("X") is itself a simplex of the complex ("K"). The**dimension**of a simplex "X" ∈ "K" is defined as dim("X") = |"X"| − 1. Consequently one can define dim("K") to be max{dim("X"), "X" ∈ "K"}. Note that the dimension of "K" might be infinite.A

**subcomplex**of "K" is a simplicial complex "L" such that every face of "L" is a face of "K". A subcomplex that consists of all nonempty subsets of a face is often called a**simplex**of "K" (and many authors use the term "simplex" for both a face and a subcomplex).An abstract simplicial complex is

**pure**if every maximal face has the same dimension. In other words, dim("K") is finite and every face is contained in a face whose dimension equals dim('K").One-dimensional simplicial complexes are (simple) graphs.

The subcomplex of "K" defined by

:"K"

^{("d")}= {"X" ∈ "K", dim("X") ≤ "d"}is the

of "K". In particular, the 1-skeleton is called thed-skeleton **underlying graph**. The union of all 0-dimensional simplices, ∪"K"^{0}, is called the**vertex set**of "K"."K" is said to be a

**finite simplicial complex**if "K" is a finite family of sets. This is easily seen to be equivalent to requiring that the underlying vertex set be finite.A

**simplicial map**"f": "K" → "L" is a function between the underlying sets, ∪"K"^{0}→ ∪"L"^{0}, such that for any "X" ∈ "K" the image set "f"("X") is a face of "L".**Geometric realization**We can associate to an abstract simplicial complex "K" a topological space |"K"|, called its

**geometric realization**, which is asimplicial complex . The construction goes as follows.First let $mathcal\{K\}$ denote the category whose objects are the simplices in "K" and whose morphisms are inclusions. Next choose a

total order on the underlying vertex set of "K" and define a functor "F" from $mathcal\{K\}$ to the category of topological spaces as follows. For any simplex "X" ∈ "K" of dimension "n", let "F"("X") = Δ^{"n"}be the standard "n"-simplex. The partial order on the underlying vertex set of "K" then specifies a unique bijection between the elements of "X" and vertices of Δ^{"n"}, ordered in the usual way "e"_{0}< "e"_{1}< ... < "e"_{"n"}. If "Y" ⊂ "X" is a subsimplex of dimension "m" < "n", then this bijection specifies a unique "m"-dimensional face of Δ^{"n"}. Define "F"("Y") → "F"("X") to be the unique affine linear embedding of Δ^{"m"}as that distinguished face of Δ^{"n"}, such that the map on vertices is order preserving.We can then define the geometric realization |"K"| as the

colimit of the functor "F". More specifically |"K"| is thequotient space of thedisjoint union :$coprod\_\{X\; in\; K\}\{F(X)\}$

by the equivalence relation which identifies a point "y" ∈ "F"("Y") with its image under the map "F"("Y") → "F"("X"), for every inclusion "Y" ⊂ "X".

If "K" is a finite simplicial complex, then we can describe |"K"| more simply. Choose an embedding of the underlying vertex set of "K" as an

affinely independent subset of some Euclidean space**R**^{"N"}of sufficiently high dimension "N". Then any abstract simplex "X" ∈ "K" can be identified with the geometric simplex in**R**^{"N"}spanned by the corresponding embedded vertices. Take |"K"| to be the union of all such simplices.If "K" is the standard combinatorial "n"-simplex, then clearly |"K"| can be naturally identified with Δ

^{"n"}.**Examples***As an example, let "V" be a finite subset of "S" of cardinality "n"+1 and let "K" be the

power set of "V". Then "K" is called a**combinatorial**"n"-**simplex**with vertex set "V". If "V" = "S" = {0, 1, 2, ..., "n"}, "K" is called the**standard**combinatorial "n"-simplex.* In the theory of

partially ordered set s ("posets"), the**order complex**of a poset is the set of all finite chains. Its homology groups and other topological invariants contain important information about the poset. The order complex of a poset is pureif and only if the poset is graded.* The

Vietoris–Rips complex is defined from any metric space "M" and distance δ by forming a simplex for every finite subset of "M" with diameter at most δ. It has applications inhomology theory,hyperbolic group s,image processing , andmobile ad-hoc network ing.**ee also***

Kruskal-Katona theorem **References**

*Wikimedia Foundation.
2010.*

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