  We use the author's combinatorial theory of full heaps (defined in
math.QA/0605768) to categorify the action of a large class of Weyl groups on
their root systems, and thus to give an elementary and uniform construction of
a family of faithful permutation representations of Weyl groups. Examples
include the standard representations of affine Weyl groups as permutations of
${\Bbb Z}$ and geometrical examples such as the realization of the Weyl group
of type $E_6$ as permutations of 27 lines on a cubic surface; in the latter
case, we also show how to recover the incidence relations between the lines
from the structure of the heap. Another class of examples involves the action
of certain Weyl groups on sets of pairs $(t, f)$, where $t \in {\Bbb Z}$ and
$f$ is a function from a suitably chosen set to the two-element set $\{+, -\}$.
Each of the permutation representations corresponds to a module for a
Kac--Moody algebra, and gives an explicit basis for it.
