Linear logic, coherence and dinaturality

Abstract

A general coherence theorem for monoidal closed structures is obtained by modifying the logical approach to coherence questions, due to Lambek [1969, 1990] by making use of linear logic. Linear logic, introduced by Girard, has many advantages which are of use in studying coherence. Most notably, its resource-sensitive nature makes it ideal for studying monoidal closed structures. The logical approach is also modified by using natural deduction rather than sequent calculus. The natural deduction system in question is proof nets, also introduced by Girard. Proof nets have several important properties which are exploited to prove the coherence theorem. In particular, the cut elimination procedure is confluent and strongly normalizing. The approach to coherence is to define a general structure, the autonomous deductive system, for defining many theories of monoidal closed categories. An autonomous deductive system is a deductive system with several added features, which are suggested by the properties of proof nets. It is then possible to give a straightforward criterion for whether a given theory of monoidal closed categories, specified by an autonomous deductive system, is coherent. Finally, a relationship is established between coherence and the composition problem for dinatural transformations. Thus, the dinatural approach to modelling polymorphic types, due to Bainbridge et al. [1990], can be extended to linear polymorphism.