Arrows are strong monads

Abstract

Hughes' arrows were shown, by Jacobs et al., to be roughly monads in the bicategory Prof of profunctors (distributors, modules). However in their work as well as others', the categorical nature of the first operator was not pursued and its formulation remained rather ad hoc. In this paper, we identify first with strength for a monad, therefore: arrows are strong monads in Prof. Strong monads have been widely used in the semantics of functional programming after Moggi's seminal work, therefore our observation establishes categorical canonicity of the notion of arrow.