  We define a 2-category that categorifies the covering Kac-Moody algebra for
sl(2) introduced by Clark and Wang. This categorification forms the structure
of a super-2-category as formulated by Kang, Kashiwara, and Oh. The
super-2-category structure introduces a (Z x Z_2)-grading giving its
Grothendieck group the structure of a free module over the group algebra of Z x
Z_2. By specializing the Z_2-action to +1 or to -1, the construction
specializes to an "odd" categorification of sl(2) and to a
supercategorification of osp(1|2), respectively.
