Modules over Monads and Linearity

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Inspired by the classical theory of modules over a monoid, we give a first account of 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, in the heterogeneous case where ”terms” and variables therein could be of different types as well as in the homogeneous case. In this paper, we present basic constructions of modules and we show examples concerning in particular abstract syntax and lambda-calculus.