Structural Limits and Approximations of Mappings

We extend the general framework of structural limits from graphs and relational structures to finite structures (including function symbols). For perhaps the simplest model of this type — sets with single unary function — we determine limit objects with respect to the three main fragments of first order. In each of these cases we solve an analog of Aldous-Lyons conjecture. This builds on the experience gained when studying limits of sequences of trees.