A Morse function is a generic map to the real line. A Morse 2-function is generic map to the plane, or other 2-manifold. These functions featured prominently in my dissertation where I used them to give a classification of 2D topological field theories in terms of generators and relations. Now David Gay and Robion Kirby are using them to study 4-manifolds.
Given a Morse 2-function $f: X^4 \to S^2$, we give minimal conditions on the fold curves and fibers so that $X^4$ and $f$ can be reconstructed from a certain combinatorial diagram attached to $S^2$. Additional remarks are made in other dimensions.