Syntax and Semantics of Linear Dependent Types
For researchers in type theory and logic, this provides a foundational framework that bridges linear and dependent types, though the work is largely theoretical and incremental in nature.
The paper presents a type theory that unifies intuitionistic linear types with type dependency, generalizing both intuitionistic dependent type theory and linear logic. It develops syntax and categorical semantics, and introduces multiplicative quantifiers and identity types derived from categorical principles.
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are developed, the latter in terms of (strict) indexed symmetric monoidal categories with comprehension. Various optional type formers are treated in a modular way. In particular, we will see that the historically much-debated multiplicative quantifiers and identity types arise naturally from categorical considerations. These new multiplicative connectives are further characterised by several identities relating them to the usual connectives from dependent type theory and linear logic. Finally, one important class of models, given by families with values in some symmetric monoidal category, is investigated in detail.