The maximality of cartesian categories

Abstract

It is proved that equations between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equation in the language of free cartesian categories collapses a cartesian category into a preorder. An analogous result holds for categories with binary products, which may lack a terminal object. The proof is based on a coherence result for cartesian categories, which is related to model-theoretic methods of normalization. This maximality of cartesian categories, which is analogous to Post completeness, shows that the usual equivalence between deductions in conjunctive logic induced by βη normalization in natural deduction is chosen optimally.