  A combinatorial category Disks was introduced by Andr\'e Joyal to play a role
in his definition of weak omega-category. He defined the category Theta to be
dual to Disks. 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.
  In this second version the paper's contents have been reorganized with the
goal of a more readable presentation. We define augmented categories and their
reduced counterparts (which lack a single trivial object of the augmented
category). These augmented categories are more suitable for inductive arguments
and their reduced counterparts are equivalent to Disk and Theta. The
equivalence between Disk and Theta is demonstrated in Sections 4 and 6 using
categories inductively defined (in Section 3) from intervals and ordinals. The
last two sections take a more categorical perspective, constructing categories
of so-called labeled trees and showing that they are equivalent to their
inductively defined counterparts, and so to Disk and Theta. The distinction
between augmented and reduced categories corrects an error in the first version
where the terminal tree was included in the category Disk.
