Categorical models for intuitionistic and linear type theory

This paper describes the categorical semantics of a system of mixed intuitionistic and linear type theory (ILT). ILT was proposed by G. Plotkin and also independently by P. Wadler. The logic associated with ILT is obtained as a combination of intuitionistic logic with intuitionistic linear logic, and can be embedded in Barber and Plotkin’s Dual Intuitionistic Linear Logic (DILL). However, unlike DILL, the logic for ILT lacks an explicit modality ! that translates intuitionistic proofs into linear ones. So while the semantics of DILL can be given in terms of monoidal adjunctions between symmetric monoidal closed categories and cartesian closed categories, the semantics of ILT is better presented via fibrations. These interpret double contexts, which cannot be reduced to linear ones. In order to interpret the intuitionistic and linear identity axioms acting on the same type we need fibrations satisfying the comprehension axiom. Research supported by EPSRC-grant GR/L28296 under the title “The eXplicit Substitution Linear Abstract Machine”.