Categorical Glueing and Logical Predicates for Models of Linear Logic

We give a series of glueing constructions for categorical models of fragments of linear logic. Specifically, we consider the glueing of (i) symmetric monoidal closed categories (models of Multiplicative Intuitionistic Linear Logic), (ii) symmetric monoidal adjunctions (for interpreting the modality !) and (iii) -autonomous categories (models of Multiplicative Linear Logic); the glueing construction for -autonomous categories is a mild generalization of the double glueing construction due to Hyland and Tan. Each of the glueing techniques can be used for creating interesting models of linear logic. In particular, we use them, together with the free symmetric monoidal cocompletion, for deriving Kripke-like parameterized logical predicates (logical relations) for the fragments of linear logic. As an application, we show full completeness results for translations between linear type theories.