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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.

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    $\begingroup$ 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, … $\endgroup$ – Daniel Fischer Sep 13 '15 at 9:11
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    $\begingroup$ 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. $\endgroup$ – Najib Idrissi Sep 13 '15 at 9:18
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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.
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