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Omega-categories and chain complexes

Richard Steiner
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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.

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Richard Steiner. 2004-05-17. Omega-categories and chain complexes. https://arxiv.org/abs/math/0403237

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