Initiality for Typed Syntax and Semantics

We give an algebraic characterization of the syntax and semantics of a class of simply–typed languages, such as the language PCF: we characterize simply–typed binding syntax equipped with reduction rules via a universal property, namely as the initial object of some category. For this purpose, we employ techniques developed in two previous works: in [2], we model syntactic translations between languages over different sets of types as initial morphisms in a category of models. In [1], we characterize untyped syntax with reduction rules as initial object in a category of models. In the present work, we show that those techniques are modular enough to be combined: we thus characterize simply–typed syntax with reduction rules as initial object in a category. The universal property yields an operator which allows to specify translations — that are semantically faithful by construction — between languages over possibly different sets of types.