Terminology Concerning Oriented Simplicial Complexes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:29:30Z http://mathoverflow.net/feeds/question/100610 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100610/terminology-concerning-oriented-simplicial-complexes Terminology Concerning Oriented Simplicial Complexes Benjamin Steinberg 2012-06-25T17:41:14Z 2012-06-25T19:03:32Z <p>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.</p> <p>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.</p> <p>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.</p> <blockquote> <blockquote> <p>Question: Is there a standard terminology for a simplicial map on an oriented simplicial complex that preserves the ordering on the vertices?</p> </blockquote> </blockquote> http://mathoverflow.net/questions/100610/terminology-concerning-oriented-simplicial-complexes/100624#100624 Answer by Vidit Nanda for Terminology Concerning Oriented Simplicial Complexes Vidit Nanda 2012-06-25T19:03:32Z 2012-06-25T19:03:32Z <p>A simplicial complex with partially ordered vertices such that the vertex set of each simplex is a chain of the poset is called an <strong>ordered</strong> simplicial complex. This avoids the confusion with orientability, etc. The terminology is not new, you can find it in <a href="http://www.maths.ed.ac.uk/~aar/papers/deltars.pdf" rel="nofollow">this paper</a> from 1969. So, your maps are just <em>maps of ordered simplicial complexes</em>. Regarding the paragraph just before your question, if a map $f:K\to K$ of ordered simplicial complexes is <em>injective</em> on simplices, then it can be realized as a map of $\Delta$-sets. I think this is covered Friedman's wonderful <a href="http://faculty.tcu.edu/gfriedman/papers/simp.pdf" rel="nofollow">notes</a>, but it's been a while since I went through those. </p>