Proof nets provide permutation-independent representations of proofs and are used to investigate coherence problems for monoidal categories. We investigate a coherence problem concerning Second Order Multiplicative Linear Logic (MLL2), that is, the one of characterizing the equivalence over proofs generated by the interpretation of quantifiers by means of ends and coends.We provide a compact representation of proof nets for a fragment of MLL2 related to the Yoneda isomorphism. By adapting the "rewiring approach" used in coherence results for * -autonomous categories, we define an equivalence relation over proof nets called "rewitnessing". We prove that this relation characterizes, in this fragment, the equivalence generated by coends.
IntroductionProof nets are usually investigated as canonical representations of proofs. For the proof-theorist, the adjective "canonical" indicates a representation of proofs insensitive to admissible permutations of rules; for the category-theorist, it indicates a faithful representation of arrows in free monoidal categories (e.g.* -autonomous categories), by which coherence results can be obtained.This twofold approach has been developed extensively in the case of Multiplicative Linear Logic (see for instance [5,6]). The use of MLL proof nets to investigate coherence problems relies on the correspondence between proof nets and a particular class of dinatural transformations (see [5]). As dinatural transformations provide a well-known interpretation of parametric polymorphism (see [1,16]), it is natural to consider the extension of this correspondence to second order Multiplicative Linear Logic MLL2. This means investigating the "coherence problem" generated by the interpretation of quantifiers as ends/coends, that is, to look for a faithful proof net representation of coends over a * -autonomous category.The main difficulty of this extension is that, as is well-known, dinaturality does not scale to second order (e.g. System F, see [26]): the dinatural interpretation of proofs generates an equivalence over proofs which strictly extends the equivalence generated by β and η conversions. In particular, coends induce "generalized permutations" of rules ([36]) to which neither System F proofs nor standard proof nets for MLL2 are insensitive. For instance, the interpretation of quantifiers as ends/coends (whose definition is recalled in appendix A) equates the distinct System F derivations in fig. 1a as well as the distinct proof nets in fig. 1b. From these examples it can be seen that such generalized permutations do not preserve the witnesses of existential quantification (or, equivalently, of the elimination of universal quantification).Several well-known issues in the System F representation of categorial structures can be related to this phenomenon. For instance, the failure of universality for the "Russell-Prawitz" translation of connectives (e.g. the failure of the isomorphism A ⊗ B ≃ ∀X ((A ⊸ B ⊸ X ) ⊸ X )), and the failure of initiality for the System F representation of initial...