📂Topological Data Analysis

Abstract Simplicial Complexes: Definitions

Definition ¹

Let’s say an arbitrary set $X$ is given.

1. A (Abstract Simplicial) Complex $A \subset 2^{X}$ that satisfies the following among the finite subsets of the power set $2^{X}$ of $X$ is defined as: $$ \alpha \in A \land \beta \subset \alpha \implies \beta \in A $$
2. The elements $\alpha \in A$ of the complex $A$ are called Simplices.
3. The Dimension of a Simplex $\alpha$ $\dim$ is defined as the value obtained by subtracting $1$ from the cardinality of $\alpha$. $$ \dim \alpha := | \alpha | - 1 $$ The Dimension of the Complex $A$ is defined as the maximum value among the dimensions of all simplices in $A$. $$ \dim A := \max_{\alpha \in A} \left( \dim \alpha \right) $$
4. A proper subset $\beta \subsetneq \alpha$ of the simplex $\alpha$, which is not the empty set, is called a Face of $\alpha$.
5. The union $V(A)$ of all simplices in $A$, calculated as follows, is called the Vertex Set of $A$. $$ V(A) := \bigcup_{\alpha \in A} \alpha $$
6. If a subset $B \subset A$ of a complex is a complex, it is called a Subcomplex.
7. If there exists a bijective $b : V(A) \to V(B)$ satisfying the following, two complexes $A, B$ are said to be Isomorphic. $$ \alpha \in A \iff b (\alpha) \in B $$
8. For a (Geometric) Simplicial Complex $K$, the (Abstract) Simplicial Complex $A$ obtained by ignoring all its constructions but maintaining the relationships between vertices is called the Vertex Scheme of $K$, and in this case, $K$ is referred to as the Geometric Realization of $A$.

Explanation

The Abstract Simplicial Complex is, as its name suggests, an abstraction of the simplicial complex stripped of its geometric meaning. From a mathematician’s point of view, conditions like convex hulls are just annoying restrictions.

For example, when considering $X = \mathbb{N}$ $$ \begin{align*} T :=& \left\{ \left\{ 1 \right\}, \left\{ 2 \right\} , \left\{ 3 \right\}, \left\{ 4 \right\} , \right. \\ & \left\{ 1,2 \right\}, \left\{ 2,3 \right\}, \left\{ 3,4 \right\}, \left\{ 4,1 \right\}, \left\{ 2,4 \right\} \\ & \left. \left\{ 1,2,4 \right\} , \left\{ 2,3,4 \right\} \right\} \end{align*} $$ it perfectly satisfies all the conditions of an abstract simplicial complex, and in this case, it’s completely fine to disregard any geometric meaning like the Euclidean space $\mathbb{R}$. $T$ has $0$-dimensional simplices, $5$ $1$-dimensional simplices, and $2$ $2$-dimensional simplices, making the complex itself $2$-dimensional with the vertex set $V(T) = \left\{ 1,2,3,4 \right\}$. Meanwhile, if a Geometric Simplicial Complex $G$ is given, it’s entirely appropriate to consider $T$ as the vertex scheme of $G$.