---- > [!definition] Definition. ([[(abstract) simplicial complex]]) > > A collection $\mathcal{K}$ of nonempty finite subsets of a set $M$ is called an **(abstract) simplicial complex** if it is stable with respect to inclusion.[^1] Elements $F$ of $\mathcal{K}$ are called **faces**, or **simplices**, of **dimension $|F|-1$**. > > The **vertex set** of $V=V(\mathcal{K})$ of $\mathcal{K}$ is the union of all its faces. It follows from the definition that every face $F$ is a union of vertices and for every vertex $v \in V$, $\{ v \} \in \mathcal{K}$. It is common to 'forbid ghost vertices' by enforcing $M=V$. Note that $\mathcal{K}$ is finite (i.e., there are finitely many faces) if and only if the vertex set $V$ is finite. > > > $\mathcal{K}$ is [[poset|partially ordered]] by inclusion; the resulting [[poset]] is denoted by $\text{Cells}(\mathcal{K})$ or just (again) by $\mathcal{K}$. Usually $V$ is assumed to be [[well-ordered set|well-ordered]]. > > One recovers the [[simplicial complex|standard geometrical]] [[simplex|realization]] of an abstract simplicial complex $\mathcal{K}$ as the [[compact]] Euclidean subspace $|\mathcal{K}|= \bigcup_{F \in \mathcal{K}} \Delta_{F} \subset \mathbb{R}^{|V|} $ > where $\Delta_{F}$ is the [[simplex|standard simplex]] $\Delta_{F}=\text{ConvHull}(e_{i}: i \in F)$. It is manifestly a [[simplicial complex]]. Note that there is one dimension per vertex! > > When one refers to [[topological space|topological properties]] ([[homeomorphism]], [[homotopy equivalent]], etc.) of an abstract simplicial complex $\mathcal{K}$, it is understood they are referring to the geometric realization $\mathcal{K}$. > ![[CleanShot 2025-03-18 at [email protected]]] The solid simplex $\Delta(k)$ has ${k+1 \choose i}$ faces of codimension $i \leq k$.[^5] > [!definition] Definition. (Orientation of a simplex) > An **orientation** of a simplex $F$ is a choice of ordering of its vertices, modulo [[parity of a permutation|even permutation]]. Thus, $F$ has exactly two orientations, and switching the order of two vertices reverses the orientation. We fix notation $F=[v_{0},\dots,v_{|F|}]$ to designate $F$ as an oriented simplex. ^definition ---- #### [^1]: That is, if $I \in \mathcal{K}$ and $J \subset I$, then $J \in \mathcal{K}$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` For each pair $i \leq k$ of nonnegative integers, how many faces of codimension $i$ does the solid $k$-simplex $\Delta(k)$ have? [^5]: Indeed, the solid $k$-simplex $\Delta(k)$ has $k+1$ vertices, $\# V=k+1$, $\Delta(k)=2^{V}$. A codimension-$i$ face has dimension $k-i$. Thus number of codimension-$i$ faces equals the number of cardinality-($k-i+1$) subsets of $V$, i.e., equals $\# \{ \sigma \in 2^{V}: \# \sigma=k+1-i \}.$ This equals ${k+1 \choose k+1 - i}={k + 1 \choose i}$.