Abstract simplicial complex
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family.
It is a purely combinatorial description of the geometric notion of a simplicial complex.
In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems.
A collection Δ of non-empty finite subsets of a set S is called a set-family.
The elements of the vertex set are called the vertices of the complex.
Also, Δ is said to be pure if it is finite-dimensional (but not necessarily finite) and every facet has the same dimension.
One-dimensional abstract simplicial complexes are mathematically equivalent to simple undirected graphs: the vertex set of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspond to undirected edges of a graph.
In this view, one-element facets of a complex correspond to isolated vertices that do not have any incident edges.
To avoid ambiguity, we do not use in this article the term "simplex" for a face in the context of abstract complexes).
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.
This is equivalent to a suitable category defined using non-abstract simplicial complexes.
More precisely, SCpx is equivalent to the category where: We can associate to any abstract simplicial complex (ASC) K a topological space
For example, consider a GSC with 4 vertices {1,2,3,4}, where the maximal faces are the triangle between {1,2,3} and the lines between {2,4} and {3,4}.
If an ASC is d-dimensional (that is, the maximum cardinality of a simplex in it is d+1), then it has a geometric realization in
[3]: 16 The special case d=1 corresponds to the well-known fact, that any graph can be plotted in
where the edges are straight lines that do not intersect each other except in common vertices, but not any graph can be plotted in
Every two geometric realizations of the same ASC, even in Euclidean spaces of different dimensions, are homeomorphic.
where A ranges over finite subsets of S, and give that direct limit the induced topology.
denote the category whose objects are the faces of K and whose morphisms are inclusions.
Next choose a total order on the vertex set of K and define a functor F from
The order on the vertex set then specifies a unique bijection between the elements of X and vertices of Δn, ordered in the usual way e0 < e1 < ... < en.
Define F(Y) → F(X) to be the unique affine linear embedding of Δm as that distinguished face of Δn, such that the map on vertices is order-preserving.
is the quotient space of the disjoint union by the equivalence relation that identifies a point y ∈ F(Y) with its image under the map F(Y) → F(X), for every inclusion Y ⊆ X.
The combinatorial n-simplex with vertex-set V is an ASC whose faces are all nonempty subsets of V (i.e., it is the power set of V).
The order complex of P is an ASC whose faces are all finite chains in P. Its homology groups and other topological invariants contain important information about the poset P. 5.
The Vietoris–Rips complex is an ASC whose faces are the finite subsets of M with diameter at most δ.
It has applications in homology theory, hyperbolic groups, image processing, and mobile ad hoc networking.
In fact, there is a bijection between (non-empty) abstract simplicial complexes on n vertices and square-free monomial ideals in S. If
For any open covering C of a topological space, the nerve complex of C is an abstract simplicial complex containing the sub-families of C with a non-empty intersection.
This corresponds to the number of antichain covers of a labeled n-set; there is a clear bijection between antichain covers of an n-set and simplicial complexes on n elements described in terms of their maximal faces.
