Categorical Models of Explicit Substitutions

This paper concerns itself with the categorical semantics of -calculi extended with explicit substitutions. For the simply-typed -calculus, indexed categories seem to provide the right categorical framework but because these structures are inherently non-linear, alternate models are needed for linear -calculi extended with explicit substitutions. We propose to replace indexed categories by presheaves and obtain a semantics which can be specialised to both the linear and the intuitionistic case. The basic models of a calculi of linear (or intuitionistic) explicit substitutions are called linear (or cartesian) context-handling categories. Then we add extra categorical structure to model the connectives of the logic, obtaining L-categories as models of the ( ; I; )fragment of intuitionistic linear logic and E-categories as models of the simply typed -calculus. The ! type constructor is then modelled by a monoidal adjunction between an E and L-category. Finally, soundness and completeness of our categorical model is proven.