I have a graph-like object which distributes across multiple "levels":
Each "level" of the graph in and of itself represents geometric relationships between $k$- and $k+1$-simplices, e.g. a 1-simplex (a line) would have its two endpoints (0-simplices) as "geometric children" and zero or more 2-simplices (triangles) as "geometric parents". Valid $k$-simplices exist for $0 \le k \le N$. As a consequence, the set of all graph nodes, i.e. the set of all simplices, at a single level $l$ would form a non-homogeneous simplicial $N$-complex.
This is where it gets more complicated. Each $k$-simplex $T$ on level $l$ can be subdivided into a set of $k$-simplices on level $l+1$ which are nested within $T$ (imagine a red-refinement of $T$). So every $k$-simplex in the aforementioned graph has zero or more "hierarchical child" $k$-simplices on level $l+1$, as well as zero or one "hierarchical parent" $k$-simplices on level $l-1$. It must be mentioned, however, that there exist $k$-simplices on level $l$ which are not "hierarchical children" of $k$-simplices on level $l-1$. These are, however, "geometric descendants" of $K>k$-simplices on level $l$ which are "hierarchical children" of $K$-simplices on level $l-1$.
Is there a canonical classification of such an object? For each level separately, I assume (correct me if I'm wrong) that this is simply an undirected graph in which the edges represent geometric parent-child relationships and the nodes represent instances of simplices. However, I'm unsure as to what the construction as a whole (across levels) could be classified as?