Building Models of Linear Logic

Abstract

A generic method for constructing categorical models of Linear Logic is provided and instantiated to yield traditional models such as coherence spaces, hypercoherences, phase spaces, relations, etc. The generic construction is modular, as expected. Hence we discuss multiplicative connectives, modalities and additive connectives in turn. Modelling the multiplicative connectives of Linear Logic is a generalisation of previous work, requiring a few non-standard concepts. More challenging is the modelling of the modalities ‘!’ (and, respectively ‘?’), which is achieved in the surprisingly general setting of this construction by considering !-candidates and showing that they exist and constitute a modality, under appropriate conditions.