  We prove odd analogs of results of Chuang and Rouquier on
sl(2)-categorification. Combined also with recent work of the second author
with Livesey, this allows us to complete the proof of Brou\'e's Abelian Defect
Conjecture for the double covers of symmetric groups. The article also develops
the theory of odd symmetric functions initiated a decade ago by Ellis, Khovanov
and Lauda. A key role in our approach is played by a 2-category consisting of
odd Grassmannian bimodules over superalgebras which are odd analogs of
equivariant cohomology algebras of Grassmannians. This is the odd analog of the
category of Grassmannian bimodules which was at the heart of Lauda's
independent approach to categorification of sl(2). We also construct an action
of the odd Kac-Moody 2-category of sl(2) on the 2-category of odd Grassmannian
bimodules, and use this to give a new proof of its non-degeneracy.

Representation Theory;Group Theory;Quantum Algebra