  In this work, we relate the three main formalisms for the notion of pasting
diagram in strict $\omega$-categories: Street's parity complexes, Johnson's
pasting schemes and Steiner's augmented directed complexes. In the process, we
show that the axioms of parity complexes and pasting schemes are not strong
enough for them to correctly represent pasting diagrams, and we do so by
providing a counter-example. Then, we introduce a new formalism, called
torsion-free complexes, which aims at encompassing the three other ones. We
prove its correctness by providing a detailed proof that an instance induces a
free $\omega$-category. Next, we prove that the three other formalisms can be
embedded in some sense in the new one. Finally, we show that there are no other
embedding between these four formalisms.
