Pasting diagrams beyond acyclicity

Many of the available algebro-combinatorial frameworks for presenting pasting diagrams, or more general diagrams in n-categories, rely on a global acyclicity
condition to ensure that a "combinatorial diagram" is equivalent to the n-functor that it presents.
This is somewhat inconvenient, as global properties tend to be unstable, and many diagrams that arise in practice are not acyclic.
In my talk, I would like to give an overview of the combinatorics of higher-dimensional diagrams when (global) acyclicity is relaxed to (local) "regularity",
a condition topological in nature: what works equally well, what works better, and what still requires some milder form of acyclicity.