# Is it possible for three simplices to share the same face (for the standard definition of simplical complex)?

Is it possible for three simplices to share the same face (for the standard definition of simplical complex)?

In this website (http://www.cs.cmu.edu/afs/cs/project/pscico/pscico/src/simpcomp/README.html), it is written, "A simplicial set is similar to a simplicial complex but more general. A simplicial set allows three or more simplices share the same face.", implying that for regular simplicial complexes this is not allowed.

However, consider the simplicial complex made up of 5 vertices and 4 lines in the shape of an X. Then, surely, the 4 1-simplices share the same face, i.e. the vertex (0 simplex) in the middle?

Thanks for any help.

• Or just take the simplicial complex for a closed $d$-dimensional simplex with $d \geqslant 3$, at each vertex, $d$ one-simplices have a common "face". $d-1$ two-simplices share each edge, … – Daniel Fischer Sep 13 '15 at 9:11
• I don't know what they were thinking. The main differences (IMO) between a simplicial complex and a simplicial set is that in a simplicial set, a set of $n+1$ vertices doesn't necessarily uniquely specify a unique $n$-simplex, and the same vertex can appear multiple times in a single simplex. – Najib Idrissi Sep 13 '15 at 9:18

## 1 Answer

For the standard definition of a simplicial complex, it is allowed that an arbitrary number of simplices share a face. Just consider the simplicial complex consisting of a $d$-dimensional simplex and its faces.

• Each vertex is shared by $d$ edges.
• Each edge is shared by $d-1$ triangles.
• Each triangle is shared by $d-2$ three-simplices.
• And so on.