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The Maximality of Cartesian Categories
Author(s) -
Došen Kosta,
Petrić Zoran
Publication year - 2001
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/1521-3870(200101)47:1<137::aid-malq137>3.0.co;2-f
Subject(s) - cartesian closed category , mathematics , cartesian coordinate system , normalization (sociology) , cartesian product , equivalence (formal languages) , closed category , congruence (geometry) , pure mathematics , calculus (dental) , discrete mathematics , algebra over a field , geometry , functor , medicine , dentistry , sociology , anthropology
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.