Modules over monads and initial semantics

Inspired by the classical theory of modules over a monoid, we introduce the natural notion of module over a monad. The associated notion of morphism of left modules (“linear” natural transformations) captures an important property of compatibility with substitution, not only in the so-called homogeneous case but also in the heterogeneous case where “terms” and variables therein could be of different types. In this paper, we present basic constructions of modules and we show how modules allow a new point of view concerning higher-order syntax and semantics.