# What are the vertices of an abstract simplicial complex?

So a simplex is the $$k$$-dimensional convex hull of $$k+1$$ vertices. The convex hulls of a subset of it's vertices are it's faces. A simplicial complex is a collection of simplices such that each face of the simplices is in the complex, and the intersections of the simplices are faces.

Wikipedia tells me that an abstract simplicial complex is a family $$\Delta$$ of subsets of some set $$S$$. The finite sets are the faces of the abstract simplicial complex, and faces are within other faces if they are subsets of the sets. That all seems fine.

However, it goes on the define the vertices of the abstract simplicial complex as $$\bigcup \Delta$$, the union of all the faces. But isn't the union of the faces simply $$\Delta$$ itself? It seems to me the "vertices" of some kind of abstraction of the usual notion of simplicial complex should be something like all the pairwise intersections of all the faces.

Obviously I am missing something - can someone clarify?

Let the elements of $$S$$ be called points. $$\Delta$$ is a collection of sets of points. $$\bigcup \Delta$$ is a set of points. A set of points is not a collection of sets of points.
The idea is that in an abstract simplicial complex, a set of points represents an entire geometric simplex with those points as vertices. So for instance, the abstract simplicial complex $$\Delta=\{\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}\}$$ represents (the boundary of) a triangle. Each of $$\{a\},\{b\},\{c\}$$ just represents a single vertex. But then $$\{a,b\}$$ represents an edge between the vertices $$a$$ and $$b$$, and similarly for the other two-element sets. So we have three vertices, and an edge between each pair of them--that's a triangle. If we then ask what the vertices are, those will just be all the points that are in any of our finite sets, so that's $$\bigcup \Delta=\{a,b,c\}$$.
• if $\Delta$ contained $\{a,b,c\}$ would the abstract simplicial complex just be the entire triangle? Mar 3, 2020 at 23:02