Difference between revisions of "Abstract simplicial complex"

Latest revision as of 11:54, 19 February 2017

Definition

Let $\mathcal{S}$ be a collection of sets, it is an abstract simplicial complex, or ASC for short, if^[1]:

1. $\forall A\in\mathcal{S}[A\neq\emptyset]$
2. $\forall A\in\mathcal{S}[\vert A\vert\in\mathbb{N}_0]$ - where $\vert\cdot\vert$ denotes cardinality of a set
3. $\forall A\in\mathcal{S}\forall B\in\mathcal{P}(A)[B\neq\emptyset\implies B\in\mathcal{S}]$^{[Note 1]}

Any $A\in\mathcal{S}$ is called a simplex of $\mathcal{S}$, and any $B\in(\mathcal{P}(A)-\{\emptyset\})$ is called a face of $A$

• Caveat:We may allow both the empty set to be an asc and we may also allow the empty set to be a simplex - as per Books:Combinatorial Algebraic Topology - Dmitry Kozlov

Terminology

• We make the following definitions regarding dimension of an abstract simplicial complex:
  1. For any simplex, $A\in\mathcal{S}$, we define: $\text{Dim}(A):\eq\vert A\vert-1$ - the dimension of $A$ is one less than the number of items in the simplex considered as a set
  2. We define the dimension of the abstract simplicial complex itself as follows: $$\text{Dim}(\mathcal{S}):\eq\mathop{\text{Sup} }_{A\in\mathcal{S} }\Big(\text{Dim}(A)\Big)$$

Related terminology

Warning: do not confuse vertex scheme with the vertex set!

Vertex Set

Let $\mathcal{S}$ be a abstract simplicial complex, we define the vertex set of $\mathcal{S}$, denoted as just $V$ or $V_\mathcal{S}$, as follows^[1]:

• $$V_\mathcal{S}:\eq\bigcup_{A\in\{B\in\mathcal{S}\ \vert\ \vert B\vert\eq 1 \} } A$$ - the union of all one-point sets in $\mathcal{S}$

Note: we do not usually distinguish between $v\in V_\mathcal{S}$ and $\{v\}\in\mathcal{S}$^[1], they are notionally identified.

Vertex Scheme

The vertex scheme of a simplicial complex, $K$, is an abstract simplicial complex.

Definition

Let $K$ be a simplicial complex and let $V_K$ be the vertex set of $K$ (not to be confused with the vertex set of an abstract simplicial complex), then we may define $\mathcal{K}$ - an abstract simplicial complex - as follows^[1]:

• $$\mathcal{K}:\eq\left\{\{a_0,\ldots,a_n\}\in \mathcal{P}(V_K)\ \big\vert\ \text{Span}(a_0,\ldots,a_n)\in K\right\}$$ ^{Warning:[Note 2]} - that is to say $\mathcal{K}$ is the set containing all collections of vertices such that the vertices span a simplex in $K$

See next

• isomorphism

See also

• Simplicial complex

Notes

1. Jump up ↑ Perhaps better written as:
   • $\forall A\in\mathcal{S}\forall B\in(\mathcal{P}(A)-\{\emptyset\})[B\in\mathcal{S}]$
   This is an exercise in equivalent ways of expressing a sentence in FOL, and should be easy to see
2. Jump up ↑ $n\in\mathbb{N}_0$ here so $n$ may be zero, we are expressing our interest in only those finite members of $\mathcal{P}(V_K)$ here, and that are non-empty.
   • TODO: This needs to be rewritten!

References

1. ↑ ^{Jump up to: 1.0 1.1 1.2 1.3} Elements of Algebraic Topology - James R. Munkres