E-CCC: between CCC and topos, — its expressive power from the viewpoint of data type theory

Abstract

The cartesian closed category (ccc) and topos differ from each other in both the descriptive power and the executability. A ccc cannot express the rich structure of data types, in particular, the concept of subtypes, while it has an executable structure as a model of typed λ-calculus. On the othe rhand, topos has the strong expressive power of the concept of subtypes, although it does not in general have a good correspondence to any computation system. This article introduces the structure of e-ccc, as an intermediate of the ccc and topos. The e-ccc has the correspondence to λ-calculus based on an extended abstract data type theory and thus can be considered to be executable. This theory can be considered to be between typed λ-calculus(ccc) and higher order intuitionistic type theory(topos). General discussion on data type theory and category theory is also made. Moreover, relations between e-ccc and ccc or topos are discussed. In particular, topos is proved to be a specially-structured e-ccc.