Dialectica Categories for the Lambek Calculus

We revisit the old work of de Paiva on the models of the Lambek Calculus in dialectica models making sure that the syntactic details that were sketchy on the first version got completed and verified. We extend the Lambek Calculus with a \kappa modality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality !, as Girard did, to re-introduce weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present the categorical semantics, proved sound and complete. Finally we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before

Coherence in category theory and the Church-Rosser property

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Coherence in Category Theory and the Church-Rosser Property

Szabo's derivation systems on sequent calculi with exchange and product are not Church-Rosser. Thus his coherence results for categories having a symmetric product (either monoidal or cartesian) are false. 1 Introduction Gentzen's sequent calculi [9] have been applied extensively in category theory, e.g [2, 3, 4, 6, 7, 8]. Sequents correspond to morphisms of a category, and the rules of the calculus correspond to categorical structures (e.g. having an associative tensor product). Cut-elimination was then used to put bounds on the complexity of these structures, e.g. to produce exhaustive lists (perhaps with duplications) of the canonical natural transformations between given functors. For symmetric, monoidal closed categories it was shown in [12] how to decide in principle whether two such transformations are equal, while an effective, linear-time decision procedure was given in [1]. Derivation systems (reduction rules) can be used to eliminate some duplicates in the list of cut-free ..