We consider certain applications of proof theory to the study of algebraic categories, The case usually studied in the literature is that of free categories with an additional structure. In this paper, we consider several problems in nonfree categories, such as the problem of full coherence, the problem of dependency of diagrams, the problem of description of arbitrary natural transformations, which show that the applications of proof theory to categories may go much farther. Bibliography: 18 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Dosen and Z. Petric, “The maximality of the typed lambda calculus and of cartesian closed categories," Publ. Inst. Math. (Beograd) (N. S.), 68 (82), 1–19 (2000).
S. Eilenberg and G. M. Kelly, “A generalization of the functorial calculus,”’ J. Algebra, 3, 366–375 (1966).
G.-Y. Girard and Y. Lafont, “Linear logic and lazy computation," in: Proceedings of TAPSOFT 87 (Pisa), Vol. 2, Lect. Notes Comput. Sci., 250 (1987), pp. 52–66.
J. Golan, Semirings and Their Applications, Kluwer Acad. Publishers, Dordrecht (1999).
G. M. Kelly and S. Mac Lane, “Coherence in closed categories," J. Pure Appl. Algebra, 1, No. 1, 97–140 (1971).
G. M. Kelly, “A cut-elimination theorem,’” Lect. Notes Math., 281, 196–213 (1972).
J. Lambek, “Deductive systems and categories. l," Math. Systems Theory, 2, 287–318 (1968).
J. Lambek, "Deductive systems and categories. ll," Lect. Notes Math., 86, 76–122 (1969).
J. Lambek, “Deductive systems and categories. III,”’ Lect. Notes Math., 274, 57–82 (1972).
L. Mehats and S. Soloviev, “Coherence in SMCCs and equivalences on derivations in IMLL with unit," Ann. Pure Appl. Logic, 147, No. 3, 127–179 (2007).
G. E. Mints, “Closed categories and proof theory,” J. Sov. Math., 15, 45–62 (1981).
G. E. Mints. "Category theory and proof theory,” in: Aktualnye Voprosy Logiki i Metodologii Nauki, Naukova Dumka, Kiev (1980), pp. 252–278. English translation, with modified title, in: G. E. Mints, Selected Papers in Proof Theory, Bibliopolis, Naples (1992).
S. Soloviev, “On natural transformations of distinguished functors and their superpositions in certain closed categories.” J. Pure Appl. Algebra, 47, 181–204 (1987).
S. Soloviev, “On the conditions of full coherence in closed categories," J. Pure Appl. Algebra, 69, 301–329 (1990).
S. Soloviev, “Proof of a conjecture of S. Mac Lane,” Ann. Pure Appl. Logic, 90, 101–162 (1997).
R. Cockett, M. Hyland, and S. Soloviev, “Natural transformations between tensor powers in the presence of direct sums," Rapport de Recherche 01-12-R, IRIT (2001).
S. Soloviev and V. Orevkov, “On categorical equivalence of Gentzen-style derivations in IMLL," Theoret. Comput. Sci., 303, 245–260 (2003).
R. Voreadou, “Coherence and non-commutative diagrams in closed categories.” Mem. Amer. Math. Soc., 9, No. 182 (1977).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 373, 2009, pp. 318–344.
Rights and permissions
About this article
Cite this article
El Khoury, A., Soloviev, S., Méhats, L. et al. Categorical interpretation of logical derivations and its applications in algebra. J Math Sci 168, 491–503 (2010). https://doi.org/10.1007/s10958-010-0002-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-010-0002-2