Modules over Monads and Linearity
نویسندگان
چکیده
Inspired by the classical theory of modules over a monoid, we give a first account of the natural notion of module over a monad, which encompasses the notion of algebra. The associated notion of morphism of modules (”linear” natural transformations) captures important formal properties about substitution. In this paper, we present basic constructions of modules and we show examples concerning in particular abstract syntax and lambda-calculus.
منابع مشابه
Modules Over Monads and Their Algebras
Modules over monads (or: actions of monads on endofunctors) are structures in which a monad interacts with an endofunctor, composed either on the left or on the right. Although usually not explicitly identified as such, modules appear in many contexts in programming and semantics. In this paper, we investigate the elementary theory of modules. In particular, we identify the monad freely generat...
متن کاملAzumaya Monads and Comonads
The definition of Azumaya algebras over commutative rings R requires the tensor product of modules over R and the twist map for the tensor product of any two R-modules. Similar constructions are available in braided monoidal categories, and Azumaya algebras were defined in these settings. Here, we introduce Azumaya monads on any category A by considering a monad (F,m, e) on A endowed with a dis...
متن کاملModules over Monads, Monadic Syntax and the Category of Untyped Lambda-calculi
We define a notion of module over a monad and use it to propose a new definition (or semantics) for abstract syntax (with binding constructions). Using our notion of module, we build a category of exponential monads, which can be understood as the category of lambda-calculi, and prove that it has an initial object (the pure untyped lambda-calculus). Our definitions and results are formalized in...
متن کاملModules 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 diff...
متن کاملInitial Semantics for higher-order typed syntax
We present an initial semantics result for typed higher-order syntax based on monads and modules over monads. The notion of module generalizes the substitution structure of monads. For a simply typed binding signature S we define a representation of S to be a monad equipped with a morphism of modules for each of its arities. The monad of abstract syntax of S then is the initial object in the ca...
متن کامل