On the 2-Categorical View of Proofs

Abstract

The relationship between logic and Category Theory was initiated by Lambek when viewing deductive systems as free categories with deductions as morphisms, formulas as objects and the cut-rule as composition. MacLane coherence theorems on monoidal categories showed how equality between morphism in a category resembles equality between proofs in a system with a kind of cut-elimination theorem.This raised what is nowadays known as categorical logic. Intuitionistic Natural Deduction systems are mapped into suitable categories according to formula-as-objects and proofs-as-morphisms notions of interpretation, extended to include functors as the categorical counterpart of the logical connectives. On the Proof-Theoretical side, Prawitz reductionistic conjecture plays a main role on dening an identity criteria between logical derivations. Reductions between proofs are worth knowing and representing whenever a deeper understanding of equality is present. From the 1-categorical point of view, morphisms are compared only by means of equations. This brings asymmetries into the proof-theoretical and categorical relationship. In the 70s Seely considered a 2-categorical interpretation as a solution to this problem. This article details Seely’s proposal and shows how even under this broader interpretation Prawitz based identity criteria cannot be completely supported. The article also considers the recent use of structural reductions, a kind of global reduction between proofs, as a help for supporting Prawitz based identity criteria.