# Abstract simplicial complex

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## Definition

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

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

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

## Terminology

• We make the following definitions regarding dimension of an abstract simplicial complex:
1. For any simplex, [ilmath]A\in\mathcal{S} [/ilmath], we define: [ilmath]\text{Dim}(A):\eq\vert A\vert-1[/ilmath] - the dimension of [ilmath]A[/ilmath] 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 [ilmath]\mathcal{S} [/ilmath] be a abstract simplicial complex, we define the vertex set of [ilmath]\mathcal{S} [/ilmath], denoted as just [ilmath]V[/ilmath] or [ilmath]V_\mathcal{S} [/ilmath], 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 [ilmath]\mathcal{S} [/ilmath]

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

### Vertex Scheme

The vertex scheme of a simplicial complex, [ilmath]K[/ilmath], is an abstract simplicial complex.

#### Definition

Let [ilmath]K[/ilmath] be a simplicial complex and let [ilmath]V_K[/ilmath] be the vertex set of [ilmath]K[/ilmath] (not to be confused with the vertex set of an abstract simplicial complex), then we may define [ilmath]\mathcal{K} [/ilmath] - 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 [ilmath]\mathcal{K} [/ilmath] is the set containing all collections of vertices such that the vertices span a simplex in [ilmath]K[/ilmath]