Abstract
Coherence is demonstrated for categories with binary products and sums, but without the terminal and the initial object, and without distribution. This coherence amounts to the existence of a faithful functor from a free category with binary products and sums to the category of relations on finite ordinals. This result is obtained with the help of proof-theoretic normalizing techniques. When the terminal object is present, coherence may still be proved if of binary sums we keep just their bifunctorial properties. It is found that with the simplest understanding of coherence this is the best one can hope for in bicartesian categories. The coherence for categories with binary products and sums provides an easy decision procedure for equality of arrows. It is also used to demonstrate that the categories in question are maximal, in the sense that in any such category that is not a preorder all the equations between arrows involving only binary products and sums are the same. This shows that the usual notion of equivalence of proofs in nondistributive conjunctive-disjunctive logic is optimally defined: further assumptions would make this notion collapse into triviality.
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References
Cockett, J. R. B., ‘Introduction to distributive categories’, Mathematical Structures in Computer Science 3 (1993), 277-307.
Cockett, J. R. B., and R. A. G. Seely, ‘Finite sum-product logic’, Theory and Application of Categories 8 (2001), 63-99 (preliminary draft, March 2000).
DoŠen, K., and Z. PetriĆ, ‘The maximality of cartesian categories’, Mathematical Logic Quarterly 47 (2001), 137-144 (available at: http://xxx.lanl.gov/math.CT/9911059).
DoŠen, K., and Z. PetriĆ, ‘The maximality of the typed lambda calculus and of cartesian closed categories’, Publications de l'Institut Mathématique (N.S.) 68(82) (2000), 1-19 (available at: http://xxx.lanl.gov/math.CT/9911073).
DoŠen, K., and Z. PetriĆ, ‘Coherent bicartesian and sesquicartesian categories’, R. Kahle et al. (eds.), Proof Theory in Computer Science, Lecture Notes in Computer Science 2183, Springer, Berlin, 2001, pp. 78-92 (available at: http://xxx.lanl.gov/math.CT/0006091).
Eckmann, B., and P. J. Hilton, ‘Group-like structures in general categories I: Multiplication and comultiplication’, Mathematische Annalen 145 (1962), 227-255.
Eilenberg, S., and G. M. Kelly, ‘A generalization of the functorial calculus’, Journal of Algebra 3 (1966), 366-375.
Kelly, G. M., and S. Mac Lane, ‘Coherence in closed categories’, Journal of Pure and Applied Algebra 1 (1971), 97-140, 219.
Kelly, G. M., ‘An abstract approach to coherence’, in [16], pp. 106-147.
Lambek, J., ‘Deductive systems and categories I: Syntactic calculus and residuated categories’, Mathematical Systems Theory 2 (1968), 287-318.
Lambek, J., ‘Deductive systems and categories II: Standard constructions and closed categories’, in Category Theory, Homology Theory and their Applications I, Lecture Notes in Mathematics 86, Springer, Berlin 1969, pp. 76-122.
Lambek, J., ‘Deductive systems and categories III: Cartesian closed categories, intuitionist propositional calculus, and combinatory logic’, in F. W. Lawvere (ed.), Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics 274, Springer, Berlin 1972, pp. 57-82.
Lambek, J., and P. J. Scott, Introduction to Higher-Order Categorical Logic, Cambridge University Press, Cambridge 1986.
Lambek, F. W., and S. H. Schanuel, Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, Cambridge 1997 (first edition: Buffalo Workshop Press, Buffalo 1991).
Mac Lane, S., ‘Natural associativity and commutativity’, Rice University Studies, Papers in Mathematics 49 (1963), 28-46.
Mac Lane, S., (ed.), Coherence in Categories, Lecture Notes in Mathematics 281, Springer, Berlin 1972.
Mints, G. E., ‘Category theory and proof theory (in Russian)’, in Aktual'nye voprosy logiki i metodologii nauki, Naukova Dumka, Kiev 1980, pp. 252-278 (English translation, with permuted title, in G. E. Mints, Selected Papers in Proof Theory, Bibliopolis, Naples 1992).
PetriĆ, Z., ‘Coherence in substructural categories’, Studia Logica 70 (2002), 271-296.
Rebagliato, J., and V. VerdÚ, ‘On the algebraization of some Gentzen systems’, Fundamenta Informaticae 18 (1993), 319-338.
Simpson, A. K., ‘Categorical completeness results for the simply-typed lambdacalculus’, in M. Dezani-Ciancaglini and G. Plotkin (eds.), Typed Lambda Calculi and Applications (Edinburgh, 1995), Lecture Notes in Computer Science 902, Springer, Berlin 1995, pp. 414-427.
Szabo, M. E., ‘A categorical equivalence of proofs’, Notre Dame Journal of Formal Logic 15 (1974), 177-191.
Szabo, M. E., ‘A counter-example to coherence in cartesian closed categories’, Canadian Mathematical Bulletin 18 (1975), 111-114.
Szabo, M. E., Algebra of Proofs, North-Holland, Amsterdam 1978.
Szabo, M. E., ‘Coherence in cartesian closed categories and the generality of proofs’, Studia Logica 48 (1989), 285-297.
Troelstra, A. S., and H. Schwichtenberg, Basic Proof Theory, Cambridge University Press, Cambridge 1996.
Zach, R., ‘Completeness before Post: Bernays, Hilbert and the development of propositional logic’, The Bulletin of Symbolic Logic 5 (1999), 331-366.
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Došen, K., Petrić, Z. Bicartesian Coherence. Studia Logica 71, 331–353 (2002). https://doi.org/10.1023/A:1020568830946
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DOI: https://doi.org/10.1023/A:1020568830946