I am reading "An elementary illustrated introduction to simplicial sets" by Greg Friedman, available online here. He defines Delta sets (or semi-simplicial sets) as a generalisation of a simplicial complex, and then simplicial sets as generalising Delta sets. I am struggling to see how simplicial sets generalise Delta sets.
After defining simplicial sets, Friedman says "every Delta set can be “completed” to a simplicial set by an analogous process, though some additional care is necessary as we know that an element of a Delta set is not necessarily determined by its vertices". This is left as an exercise to the reader, but I don't see how to proceed with it.
One example of a Delta set is a graph with 1 vertex $v$, and 2 loop edges $e_1, e_2$ on $v$. In particular, the face maps applied to either edge give us $v$, and so they are not uniquely determined by the face maps.
What would the simplicial set arising from the above example look like? If $s_0$ is the degeneracy map on the 0-simplices, then $s_0(v)$ should be a 1-simplex different from either edge (we have no reason to pick one over the other!). So I think we should have 3 1-simplices, viz. $e_1, e_2$, and $[v,v]$.
I think we should also have 6 2-simplices, as (using the relations between face and degeneracy maps) $s_i$ for $i=0,1$ applied to any 1-simplex gives a unique 2-simplex. Is this right?
In general, given a simplicial set $S$ arising from a Delta set and an n-simplex $c\in S$ from the simplicial set, I'm interested in being able to say when $d_i(c)$ is an $(n-1)$ simplex that is also an $(n-1)$ simplex of the Delta set.