Weak Omega Categories I

Abstract

We develop a theory of weak omega categories that will be accessible to anyone who is familiar with the language of categories and functors and who has encountered the definition of a strict 2-category. The most remarkable feature of this theory is its simplicity. We build upon an idea due to Jacques Penon by defining a weak omega category to be a span of omega magmas with certain properties. (An omega magma is a reflexive, globular set with a system of partially defined, binary composition operations which respects the globular structure.) Categories, bicategories, strict omega categories and Penon's weak omega categories are all instances of our weak omega categories. We offer a heuristic argument to justify the claim that Batanin's weak omega categories also fit into our framework. We show that the Baez-Dolan stabilization hypothesis is a direct consequence of our definition of weak omega categories. We define a natural notion of a pseudo-functor between weak omega categories and show that it includes the classical notion of a homomorphism between bicategories. In any weak omega category the operation of composition with a fixed 1-cell defines such a pseudo-functor. Finally, we define a notion of weak equivalence between weak omega categories which generalizes the standard definition of an equivalence between ordinary categories.