# On the duality between trees and disks

## David Oury

A combinatorial category Disk was introduced by André Joyal to play a role in his definition of weak $\omega$-category. He defined the category $\Theta$ to be dual to Disk. In the ensuing literature, a more concrete description of $\Theta$ was provided. In this paper we provide another proof of the dual equivalence and introduce various categories equivalent to Disk or $\Theta$, each providing a helpful viewpoint.

Keywords: delta, disk, duality, globular set, omega-category, theta-category, tree

2000 MSC: 18D05, 18D20, 18D35

Theory and Applications of Categories, Vol. 24, 2010, No. 16, pp 418-450.

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