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Duality for Simple $\omega$-Categories and Disks

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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.

Keywords: omega-category, globular set, omega-graph, disk, schizophrenic object, duality, theta-category.

2000 MSC: 18D05, 18D10, 18D35.

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

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