Abstract

There are many notions of type in computing. The most classical notion is "types as sets", which has been extended to cover many features of modern programming languages. This paper shows that such features are handled perhaps even more naturally by an extension of the "types as algebras" notion to a "types as theories" notion. This notion naturally supports object oriented concepts, including inheritance and local state, as well as generic modules and dependent types. Moreover, it explains why polymorphic operations are natural transformations and provides a systematic foundation for a powerful form of programming in the large. This paper also presents a new semantics for dependent types based on strict fibrations. NOTE: This version of the paper contains some material that was inadvertently omitted from the version printed in Topology and Category Theory in Computer Science (Oxford, 1991). 1

Citations