Poset-valued sets or how to build models for linear logics

Abstract

We describe a method for constructing models of linear logic based on the category of sets and relations. The resulting categories are non-degenerate in general; in particular they are not compact closed nor do they have biproducts. The construction is simple, lifting the structure of a poset to the new category. The underlying poset thus controls the structure of this category, and different posets give rise to differently-flavoured models. As a result, this technique allows the construction of models for both, intuitionistic or classical linear logic as desired. A number of well-known models, for example coherence spaces and hypercoherences, are instances of this method.