TITLE: n-categorical pasting diagram overview
QUESTION [12 upvotes]: There has been a lots of approaches to the notion of n-categorical diagram and n-categorical pasting diagram:
Street : "Parity complexes"
Power : " An n-categorical pasting theorem"
Johnson : "The combinatorics of n-categorical pasting"
Steiner : "Omega-category and chain complexes"
I'm wondering about the relations between these approaches.
It seem very clear for each of these how to translate from one approach to the other, but it is not obvious at all, and probably not true in general that these notion are exactly equivalent. In particular the way the "loop free" condition is formulated for each of these is rather subtle and they do not look equivalent.
Are there some known equivalences between these notions ? or maybe inclusion of some of them into others ? I would also be interested in examples that can be covered by one notion and not by the others...
Another way to put it: each notion of n-categorical diagram in the papers above define a class of polygraph, what are there known inclusion/equality between these classes.

REPLY [5 votes]: The answer above by Simon does address Steiner's much newer and immensely more computable structure, the Augmented Directed Complexes, but he doesn't really cover its benefits over all of the previous methods.  ADCs have been used to extend the lax/oplax tensor product to strict $\omega$-categories as well as the lax/oplax join and its various slice adjoints by Dimitri Ara and Georges Maltsiniotis because working with chain complexes of free modules is, all told, pretty easy.
It reduces a lot of the combinatorics of pasting to doing linear algebra in a chain complex of free abelian groups with a particular basis.  The morphisms are just morphisms that preserve the free commutative monoids of 'positive elements' generated by the basis.  Steiner does dip back into some of his earlier directed complex work in order to prove that Augmented Directed Complexes with a unital loop-free basis embed fully and faithfully into strict $\omega$-categories and are dense in it.
This answer is mostly supplementary to Simon's but if you're looking into working with pasting diagrams, I would highly highly recommend you look into the work of Steiner.
Also important is the fact that in Steiner's case, because the embedding is full and faithful, you can find 'relevant' elements as the images of maps from the ADC associated with an $n$-globe.
Edit: Avoid Steiner's 1993 version and read the 2004 version. They are (almost) totally unrelated. Basically, the 1993 version sort of describes the image of the embedding directly.  It is not really useful for computation.