Categorical Proof Theory of Co-Intuitionistic Linear Logic

To provide a categorical semantics for co-intuitionistic logic one has to face the fact, noted by Tristan Crolard, that the definition of co-exponents as adjuncts of coproducts does not work in the category Set, where coproducts are disjoint unions. Following the familiar construction of models of intuitionistic linear logic with exponential"!", we build models of co-intuitionistic logic in symmetric monoidal left-closed categories with additional structure, using a variant of Crolard's term assignment to co-intuitionistic logic in the construction of a free category.