Proof-Net Categories

Star-autonomous categories are a brand of symmetric monoidal closed categories of particular interest for classical linear logic. This work formulates equationally a precise notion of star-autonomous category without unit objects, which is called proof-net category. A coherence theorem analogous to the coherence theorem for symmetric monoidal closed categories with respect to graphs is proved for proof-net categories. It is also proved that the free proof-net category generated by a set of objects is isomorphic to a full subcategory of the free star-autonomous category generated by the same set of objects. This yields a very useful coherence theorem for star-autonomous categories involving the unit objects, exactly analogous to the coherence theorem for symmetric monoidal closed categories. An analogous coherence result is proved also for proof-net categories with the mix principle of linear logic. The graphs involved in these coherence theorems are the relevant portions of proof nets that one needs to solve the question whether a diagram of arrows commutes. Proofs are inspired by methods of proof theory. The results of this work are of interest for general proof theory. They show how generality of proofs provides a criterion for identity of proofs in a fragment of linear logic. They also make a contribution to the study of coherence in symmetric monoidal closed categories.