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Keywords:
pasting diagrams; strict omega-categories; polygraphs; computads; parity complexes; pasting schemes; augmented
pasting diagrams; strict omega-categories; polygraphs; computads; parity complexes; pasting schemes; augmented
Summary:
In this work, we relate the three main formalisms for the notion of pasting diagram in strict $\omega$: Street’s {\it parity complexes}, Johnson’s {\it pasting schemes} and Steiner’s {\it 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 {\it 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.
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