Abstract

A category theorist models Girard's linear logic by comonads (with some additional structure), and Moggi's computational metalanguage by monads (with different additional structure). Both of these systems are are of interest, in part because of the existence of various translations of the simply-typed lambda calculus (or, equivalently, propositional intuitionistic logic) into each of them. Are the similarities between the two systems a mirage, or the reflection of a deep connection? Recently, Benton described an adjoint model that subsumes both models of intuitionistic linear logic and a restricted class of models of the computational metalanguage. This shows that Girard's and Moggi's systems are indeed connected. Here we strengthen that link by showing that the various translations of the lambda calculus proposed by Girard and Moggi are intimately related. In particular, the well-known `Girard translation' corresponds to the trivial embedding of the lambda calculus into Moggi's metala...