Lifting Operators and Laws

Mathematicians routinely lift operators to structures. For instance, almost every textbook on calculus lifts addition pointwise to functions: ( f + g)(x) = f (x)+ g(x). In this particular example, the lifted operator inherits the properties of the base-level operator. Does this hold in general? In order to approach this problem, one has to make the concept of lifting precise. I argue that lifting can be defined generically using the notion of an applicative functor or idiom. In this setting, the paper answers two questions: “Which lifted base-level identities hold in every idiom?” and “Which idioms satisfy every lifted base-level identity?”