  Shapes of four dimensional spaces can be studied effectively via maps to
standard surfaces. We explain, and illustrate by quintessential examples, how
to simplify such generic maps on 4-manifolds topologically, in order to derive
simple decompositions into much better understood manifold pieces. Our methods
not only allow us to produce various interesting families of examples, but also
to establish a correspondence between simplified broken Lefschetz fibrations
and simplified trisections of closed, oriented 4-manifolds.
