Thin fillers in the cubical nerves of omega-categories

TL;DR

This paper characterizes the cubical nerve of strict omega-categories as sequences of sets with specific thin filler properties, establishing an equivalence of categories and connecting to cubical T-complexes.

Contribution

It proves that the cubical nerve functor provides an equivalence between strict omega-categories and certain sequences of sets with thin filler conditions, extending the understanding of omega-categories.

Findings

Cubical nerve of strict omega-categories characterized as sequences with thin fillers

Establishment of an equivalence of categories via the cubical nerve functor

Connection to cubical T-complexes and chain complex techniques

Abstract

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 in effect 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.