A monad gives rise to an applicative functor, but not all applicative functors result from monads. Unlike monads, applicative functors are closed under composition.

Conor Mcbride, Ross Paterson, Applicative programming with effects, Journal of Functional Programming. 18 (01): 1–13. (paper)

Ross Paterson, Constructing Applicative Functors, in Mathematics of Program Construction, Madrid, 2012, Lecture Notes in Computer Science vol. 7342, pp. 300-323, Springer, 2012. (paper)

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