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# Homology, Homotopy and Applications

## Volume 6 (2004)

### Number 1

### Omega-categories and chain complexes

Pages: 175 – 200

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

#### Author

#### Abstract

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.

#### Keywords

Omega-category, augmented directed complex

#### 2010 Mathematics Subject Classification

18D05