How a PDE degeneration inspired new combinatorics and the notion of (A_\infty,2)-category

This is the story of a geometric analyst with category number 0 (i.e. hard to inspire with questions of functoriality) pursuing the singularity formation in a natural PDE degeneration -- finding a mysterious new bubbling effect -- giving up -- finding a friendly combinatorialist -- and graduating to category number (\infty,2), i.e. developing new notions of "2-categories up to homotopy".