  We study a diagrammatic categorification (the "anti-spherical category") of
the anti-spherical module for any Coxeter group. We deduce that Deodhar's
(sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients,
and that a monotonicity conjecture of Brenti's holds. The main technical
observation is a localisation procedure for the anti-spherical category, from
which we construct a "light leaves" basis of morphisms. Our techniques may be
used to calculate many new elements of the $p$-canonical basis in the
anti-spherical module.
