The integration of logical and algebraic types

Abstract

Three versions of the lambda calculus are discussed: typed lambda calculus, untyped lambda calculus, and polymorphically typed lambda calculus. It is shown that the syntax for these calculi can be described as initial algebras for suitable endofunctors. Models of the first two calculi are described as in [Lambek and Scott 1985] and the Per model of the third is recalled. Algebraic types are described in terms of sketches. A main contribution of the paper is the internalization of the theory of sketches in the category Per. Sketches themselves, models of sketches, categories of models of sketches and the category of all models of all sketches can all be realized in this setting. Next, dinatural transformations are reviewed in the context of the category Per and the relation between certain end formulas and initial algebras for finite signatures is discussed. The point of the theory of algebraic semantics in Per is to provide a setting for thinking about algebraic types and algebraic operations in conjunction with logical types and polymorphic operations. Two specific examples are discussed and a general principle for combining algebraic and polymorphic operations is proposed.KeywordsNatural TransformationCategory TheoryLogical TypeDenotational SemanticGraph ObjectThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.