Equivalence of Applicative Functors and Multifunctors

Abstract

McBride and Paterson introduced Applicative functors to Haskell, which are equivalent to the lax monoidal functors (with strength) of category theory. Applicative functors F are presented via idiomatic application $\_\circledast\_ : F (A \to B) \to F A \to F B$ and laws that are a bit hard to remember. Capriotti and Kaposi observed that applicative functors can be conceived as multifunctors, i.e., by a family liftA$_n$ : $(A_1 \to ... \to A_n \to C) \to F A_1 \to ... \to F A_n \to F C$ of zipWith-like functions that generalize pure $(n=0)$, fmap $(n=1)$ and liftA2 $(n=2)$. This reduces the associated laws to just the first functor law and a uniform scheme of second (multi)functor laws, i.e., a composition law for liftA. In this note, we rigorously prove that applicative functors are in fact equivalent to multifunctors, by interderiving their laws.