The semantic marriage of monads and effects

Wadler and Thiemann unified type-and-effect systems with monadic semantics via a syntactic correspondence and soundness results with respect to an operational semantics. They conjecture that a general, “coherent” denotational semantics can be given to unify effect systems with a monadic-style semantics. We provide such a semantics based on the novel structure of an indexed monad, which we introduce. We redefine the semantics of Moggi’s computational λ-calculus in terms of (strong) indexed monads which gives a oneto-one correspondence between indices of the denotations and the effect annotations of traditional effect systems. Dually, this approach yields indexed comonads which gives a unified semantics and effect system to contextual notions of effect (called coeffects), which we have previously described [9]. Previously, Wadler and Thiemann established a syntactic correspondence between type-and-effect systems and the monadic semantics approach by annotating monadic type constructors with the effect sets of the type-and-effect system [10]. They established soundness results between the effect system and an operational semantics, and conjectured a “coherent semantics” of effects and monads in a denotational style. One suggestion was to associate to each effect set σ a different monad T . We take a different approach to a coherent semantics, unifying effect systems with a monadic-style semantics in terms of the novel notion of indexed monads, which generalises monads. Indexed monads Indexed monads comprise a functor