I am looking for the definition of the subdivision of a simplicial complex. When the complex is defined in a geometric way, then the definition is pretty simple : the complex σ(C) is a subdivision of C if each simplex of σ(C) is contained in a simplex in C, and if each simplex of C is the union of finitely many simplexes in σ(C) (definition found in Herlihy-Shavit 1999).

When we deal with the abstract definition of simplicial complexes (i.e. a subset-closed family of subsets), I fail to find a similar definition. I found definitions of specific subdivisions (e.g. standard chromatic subdivision of the n-simplex) but not a general formulation like the one I gave. A difficulty is the fact that an abstract simplex cannot be the union of its subdivision, since the subdivision contains new edges that do not appear in the first simplex (they are not in the same "space" because there is no convenient notion of "space" in the abstract definition).

Thanks a lot for your help !