Homology, Homotopy and Applications

Volume 6 (2004)

Number 1

Omega-categories and chain complexes

Pages: 175 – 200

DOI: https://dx.doi.org/10.4310/HHA.2004.v6.n1.a12


Richard Steiner (Department of Mathematics, University of Glasgow, Scotland, United Kingdom)


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.


Omega-category, augmented directed complex

2010 Mathematics Subject Classification


Published 1 January 2004