Duality for Simple $\omega$-Categories and Disks

Mihaly Makkai and Marek Zawadowski

A. Joyal has introduced the category $\cal D$ of the so-called finite disks, and used it to define the concept of $\theta$-category, a notion of weak $\omega$-category. We introduce the notion of an $\omega$-graph being composable (meaning roughly that 'it has a unique composite'), and call an $\omega$-category simple if it is freely generated by a composable $\omega$-graph. The category $\cal S$ of simple $\omega$-categories is a full subcategory of the category, with strict $\omega$-functors as morphisms, of all $\omega$-categories. The category $\cal S$ is a key ingredient in another concept of weak $\omega$-category, called protocategory. We prove that $\cal D$ and $\cal S$ are contravariantly equivalent, by a duality induced by a suitable schizophrenic object living in both categories. In [MZ], this result is one of the tools used to show that the concept of $\theta$-category and that of protocategory are equivalent in a suitable sense. We also prove that composable $\omega$-graphs coincide with the $\omega$-graphs of the form $T^*$ considered by M.Batanin, which were characterized by R. Street and called `globular cardinals'. Batanin's construction, using globular cardinals, of the free $\omega$-category on a globular set plays an important role in our paper. We give a self-contained presentation of Batanin's construction that suits our purposes.

Theory and Applications of Categories, Vol. 8, 2001, No. 7, pp 114-243
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