Idioms are Oblivious, Arrows are Meticulous, Monads are Promiscuous
Sam Lindley, Philip L. Wadler, Jeremy Yallop · Electronic Notes in Theoretical Computer Science · 2011
We revisit the connection between three notions of computation: Moggiʼs monads, Hughesʼs arrows and McBride and Patersonʼs idioms (also called applicative functors). We show that idioms are equivalent to arrows that satisfy the type isomorphism A↝B≃1↝(A→B) and that monads are equivalent to arrows that satisfy the type isomorphism A↝B≃A→(1↝B). Further, idioms embed into arrows and arrows embed into monads.