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Kripke Resource Models of a Dependently-typed, Bunched λ-calculus

The λΛ-calculus is a dependent type theory with both linear and intuitionistic dependent function spaces. It can be seen to arise in two ways. Firstly, in logical frameworks, where it is the language of the RLF logical framework and can uniformly represent linear and other relevant logics. Secondly, it is a presentation of the proof-objects of a structural variation, with Dereliction, of a fragment of BI, the logic of bunched implications. As such, it is also closely related to linear logic. BI is a logic which directly combines linear and intuitionistic implication and, in its predicate version, has both linear and intutionistic quantifiers. The λΛ-calculus is the dependent type theory which generalizes both implications and quantifiers. In this paper, we study the categorical semantics of the λΛ-calculus, gives a theory of 'Kripke resource models', i.e. monoid-indexed sets of functorial Kripke models, in which the monoid gives an account of resource consumption. A class of concrete, set-theoretic models is given by the category of families of sets parametrized over a small monoidal category, in which the intuitionistic dependent function space is described in the established way, but the linear dependent function space is described using Day's tensor product.