2004/03/31 by Richard Steiner · 1 citation
Mathematics · #math.CT #msc:18D05
published as Homology, Homotopy and Applications, vol 6(1), 2004, pp. 175-200 · 18 pages; as published, with minor changes from version 1
arxiv created 2004/05/17 · arxiv updated 2009/12/01
There are several ways to construct omega-categories from combinatorial objects such as pasting schemes or parity complexes. We make these constructions into a functor on a category of chain complexes with additional structure, which we call augmented directed complexes. This functor from augmented directed complexes to omega-categories has a left adjoint, and the adjunction restricts to an equivalence on a category of augmented directed complexes with good bases. The omega-categories equivalent to augmented directed complexes with good bases include the omega-categories associated to globes, simplexes and cubes; thus the morphisms between these omega-categories are determined by morphisms between chain complexes. It follows that the entire theory of omega-categories can be expressed in terms of chain complexes; in particular we describe the biclosed monoidal structure on omega-categories and calculate some internal homomorphism objects.