  We study the topology of toric maps. We show that if $f\colon X\to Y$ is a
proper toric morphism, with $X$ simplicial, then the cohomology of every fiber
of $f$ is pure and of Hodge-Tate type. When the map is a fibration, we give an
explicit formula for the Betti numbers of the fibers in terms of a relative
version of the $f$-vector, extending the usual formula for the Betti numbers of
a simplicial complete toric variety. We then describe the Decomposition Theorem
for a toric fibration, giving in particular a nonnegative combinatorial
invariant attached to each cone in the fan of $Y$, which is positive precisely
when the corresponding closed subset of $Y$ appears as a support in the
Decomposition Theorem. The description of this invariant involves the stalks of
the intersection cohomology complexes on $X$ and $Y$, but in the case when both
$X$ and $Y$ are simplicial, there is a simple formula in terms of the relative
$f$-vector.
