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.

http://www.tac.mta.ca/tac/volumes/24/16/24-16.dvi

http://www.tac.mta.ca/tac/volumes/24/16/24-16.ps

http://www.tac.mta.ca/tac/volumes/24/16/24-16.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/24/16/24-16.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/24/16/24-16.ps