In
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a
family of sets
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set fami ...
that is closed under taking
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s, 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 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 ...
.
[ Lee, John M., Introduction to Topological Manifolds, Springer 2011, , p153] For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1).
In the context of
matroids and
greedoid
In combinatorics, a greedoid is a type of set system. It arises from the notion of the matroid, which was originally introduced by Whitney in 1935 to study planar graphs and was later used by Edmonds to characterize a class of optimization pro ...
s, abstract simplicial complexes are also called
independence systems.
An abstract simplex can be studied algebraically by forming its
Stanley–Reisner ring; this sets up a powerful relation between
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
and
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
.
Definitions
A collection of non-empty finite subsets of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''S'' is called a set-family.
A set-family is called an abstract simplicial complex if, for every set in , and every non-empty subset , the set also belongs to .
The finite sets that belong to are called faces of the complex, and a face is said to belong to another face if , so the definition of an abstract simplicial complex can be restated as saying that every face of a face of a complex is itself a face of . The vertex set of is defined as , the union of all faces of . The elements of the vertex set are called the vertices of the complex. For every vertex ''v'' of , the set is a face of the complex, and every face of the complex is a finite subset of the vertex set.
The maximal faces of (i.e., faces that are not subsets of any other faces) are called facets of the complex. The dimension of a face in is defined as : faces consisting of a single element are zero-dimensional, faces consisting of two elements are one-dimensional, etc. The dimension of the complex is defined as the largest dimension of any of its faces, or infinity if there is no finite bound on the dimension of the faces.
The complex is said to be finite if it has finitely many faces, or equivalently if its vertex set is finite. Also, is said to be pure if it is finite-dimensional (but not necessarily finite) and every facet has the same dimension. In other words, is pure if is finite and every face is contained in a facet of dimension .
One-dimensional abstract simplicial complexes are mathematically equivalent to
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
undirected graph
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' ve ...
s: 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.
A subcomplex of is an abstract simplicial complex ''L'' such that every face of ''L'' belongs to ; that is, and ''L'' is an abstract simplicial complex. A subcomplex that consists of all of the subsets of a single face of is often called a simplex of . (However, some authors use the term "simplex" for a face or, rather ambiguously, for both a face and the subcomplex associated with a face, by analogy with the non-abstract (geometric)
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 ...
terminology. To avoid ambiguity, we do not use in this article the term "simplex" for a face in the context of abstract complexes).
The
''d''-skeleton of is the subcomplex of consisting of all of the faces of that have dimension at most ''d''. In particular, the
1-skeleton is called the underlying graph of . The 0-skeleton of can be identified with its vertex set, although formally it is not quite the same thing (the vertex set is a single set of all of the vertices, while the 0-skeleton is a family of single-element sets).
The link of a face in , often denoted or , is the subcomplex of defined by
:
Note that the link of the empty set is itself.
Simplicial maps
Given two abstract simplicial complexes, and , a
simplicial map is a
function that maps the vertices of to the vertices of and that has the property that for any face of , the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
is a face of . There is a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
SCpx with abstract simplicial complexes as objects and simplicial maps as
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s. This is equivalent to a suitable category defined using non-abstract
simplicial complexes.
Moreover, the categorical point of view allows us to tighten the relation between the underlying set ''S'' of an abstract simplicial complex and the vertex set of : for the purposes of defining a category of abstract simplicial complexes, the elements of ''S'' not lying in are irrelevant. More precisely, SCpx is equivalent to the category where:
* an object is a set ''S'' equipped with a collection of non-empty finite subsets that contains all singletons and such that if is in and is non-empty, then also belongs to .
* a morphism from to is a function such that the image of any element of is an element of .
Geometric realization
We can associate to any abstract simplicial complex (ASC) ''K'' a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, called its geometric realization. There are several ways to define
.
Geometric definition
Every
geometric simplicial complex (GSC) determines an ASC:''
[, Section 4.3]'' the vertices of the ASC are the vertices of the GSC, and the faces of the ASC are the vertex-sets of the faces of the GSC. For example, consider a GSC with 4 vertices , where the maximal faces are the triangle between and the lines between and . Then, the corresponding ASC contains the sets , , , and all their subsets. We say that the GSC is the geometric realization of the ASC.
Every ASC has a geometric realization. This is easy to see for a finite ASC.''
'' Let
. Identify the vertices in
with the vertices of an (''N-1'')-dimensional simplex in
. Construct the GSC . Clearly, the ASC associated with this GSC is identical to ''K'', so we have indeed constructed a geometric realization of ''K.'' In fact, an ASC can be realized using much fewer dimensions. 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
, but might not have a geometric realization in
''
'' 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
in this way.
If ''K'' is the standard combinatorial ''n''-simplex, then
can be naturally identified with .
Every two geometric realizations of the same ASC, even in Euclidean spaces of different dimensions, are
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
.''
'' Therefore, given an ASC ''K,'' one can speak of ''the'' geometric realization of ''K''.
Topological definition
The construction goes as follows. First, define
as a subset of
consisting of functions