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.