Thin fillers in the cubical nerves of omega-categories

Richard Steiner

It is shown that the cubical nerve of a strict omega-category is a sequence of sets with cubical face operations and distinguished subclasses of thin elements satisfying certain thin filler conditions. It is also shown that a sequence of this type is the cubical nerve of a strict omega-category unique up to isomorphism; the cubical nerve functor is therefore an equivalence of categories. The sequences of sets involved are the analogues of cubical T-complexes appropriate for strict omega-categories. Degeneracies are not required in the definition of these sequences, but can in fact be constructed as thin fillers. The proof of the thin filler conditions uses chain complexes and chain homotopies.

Keywords: omega-category, cubical nerve, stratified precubical set, cubical T-complex, thin filler

2000 MSC: 18D05

Theory and Applications of Categories, Vol. 16, 2006, No. 8, pp 144-173.

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