Idioms are Oblivious, Arrows are Meticulous, Monads are Promiscuous

Authors:
Sam Lindley;Philip Wadler;Jeremy Yallop
Affiliations:
Laboratory for Foundations of Computer Science, School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, United Kingdom;Laboratory for Foundations of Computer Science, School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, United Kingdom;Laboratory for Foundations of Computer Science, School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, United Kingdom
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2011

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Abstract

We revisit the connection between three notions of computation: Moggi@?s monads, Hughes@?s arrows and McBride and Paterson@?s 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.