Abstract
The main aim of this article is to suggest a translation of the simplest concepts of category theory into the language of (structural) theory of proofs, to use this translation to simplify the proofs of some known results [1], and to obtain two new ones: the coherence theorem for canonical morphisms in (nonmonoidal, nonsymmetric) closed categories [2], and the solution of the problem of equality of canonical morphisms. Extensions of these results to monoidal closed, symmetric closed, and monoidal symmetric closed categories are sketched. The decision procedure is obtained by means of a correct and faithful translation of canonical morphisms into an expansion of the λ-language, which has the tools for a special account of “superfluous” premises of implications (the thinning rule). The expansions of the λ-language which have so far appeared in the literature have not possessed this facility.
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Literature cited
G. Kelly and S. MacLane, “Coherence in closed categories,” J. Pure Appl. Algebra,1, No. 1, 97–140 (1971).
S. Eilenberg and G. Kelly, “Closed categories,” Proc. Conf. on Categorical Algebra La Jolla, 1965, Springer-Verlag (1966), 421–562.
G. E. Minc, Proof Theory and Category Theory [in Russian].
G. E. Minc, “Normalization of proofs,” Appendix to book by S. C. Kleene, Mathematical Logic [in Russian], Moscow (1973).
D. Prawitz, Natural Deduction, a Proof Theoretical Study, Stockholm (1965).
G. E. Minc, Proof Theory (Arithmetic and Analysis). Algebra, Topology, Geometry [in Russian], Vol. 13, VINITI, Moscow (1975), pp. 5–49.
J. Lambek, “Deductive systems and categories. I,” Math. Systems Theory,2, No. 4, 287–318 (1968).
J. Lambek, “Deductive systems and categories. II,” Lect. Notes Math.,86, 76–122 (1969).
J. Lambek, “Deductive systems and categories. III,” Lect. Notes Math.,274, 57–82 (1972).
D. Prawitz, “Ideas and Results in Proof Theory,” Proc. 2 Scand. Logic Sympos., Amsterdam (1971), pp. 235–307.
G. Kreisel, “A survey of proof theory. II,” Proc. 2 Scand. Logic Sympos., Amsterdam (1971), pp. 109–170.
K. Schütte, Beweistheorie, Berlin (1960).
S. MacLane, “Topology and logic as a source of algebra,” Bull. Am. Math. Soc.,82, No. 1, 1–40 (1976).
J. Zucker, “The correspondence between cut-elimination and normalization,” Ann. Math. Log.,7, No. 1, 1–112 (1974).
M. Szabo, “A categorical equivalence of proofs,” Notre Dame J. Form. Log.,15, No. 2, 171–191 (1974).
M. Szabo, Addendum (to [15]), Notre Dame J. Form. Log.
C. Mann, “The connection between equivalence of proofs and cartesian closed categories,” Proc. London Math. Soc.,31, No. 3, 289–310 (1975).
P. Martin-Löf, “Infinite terms and a system of natural deduction,” Compos. Math.,24, No. 1, 93–103 (1972).
G. Kelly, “An abstract approach to coherence,” Lect. Notes Math.,281, 106–147 (1972).
S. Eilenberg and G. Kelly, “A generalization of the functorial calculus,” J. Algebra,3, No. 3, 366–375 (1966).
A. Troelstra, “Metamathematical investigation of intuitionistic arithmetic and analysis,” Lect. Notes Math.,344 (1974).
G. E. Minc, “A theorem on cut-elimination for relevant logic,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,32, 90–97 (1972).
G. E. Minc, “The independence of postulates of natural calculuses,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,8, 192–195 (1968).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute, im. V. A. Steklova AN SSSR, Vol. 68, pp. 83–114, 1977.
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Mints, G.E. Closed categories and the theory of proofs. J Math Sci 15, 45–62 (1981). https://doi.org/10.1007/BF01404107
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DOI: https://doi.org/10.1007/BF01404107