An oriented simplicial complex is a simplicial complex K equipped with a partial ordering on its vertices that restricts to a linear ordering on each simplex. I am wondering if there is a standard name for simplicial maps $f\colon K\to K$ which preserve the partial order on vertices. I would have liked to call them orientation preserving but I think this is usually used to just mean that the linear ordering on each simplex is preserved up to an even permutation. I had no luck with Google.
Example: If P is a poset, then the order complex (=nerve) of the poset is naturally oriented by the original ordering on P and any order preserving map on P induces a simplicial map preserving orientation in the strong sense I described above. In particular, after a barycentric subdivision one always can get my property.
Also an oriented simplicial complex can be viewed as a simplicial set in a natural way but not all simplicial morphisms translate into morphisms of simplicial sets. The ones I am considering do.
Question: Is there a standard terminology for a simplicial map on an oriented simplicial complex that preserves the ordering on the vertices?
Best Answer
A simplicial complex with partially ordered vertices such that the vertex set of each simplex is a chain of the poset is called an ordered simplicial complex. This avoids the confusion with orientability, etc. The terminology is not new, you can find it in this paper from 1969. So, your maps are just maps of ordered simplicial complexes. Regarding the paragraph just before your question, if a map $f:K\to K$ of ordered simplicial complexes is injective on simplices, then it can be realized as a map of $\Delta$-sets. I think this is covered Friedman's wonderful notes, but it's been a while since I went through those.