Abstract
This paper studies the problem of coherence in category theory from a type-theoretic viewpoint. We first show how a Curry-Howard interpretation of a formal proof of normalization for monoids almost directly yields a coherence proof for monoidal categories. Then we formalize this coherence proof in intensional intuitionistic type theory and show how it relies on explicit reasoning about proof objects for intensional equality. This formalization has been checked in the proof assistant ALF.
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P. Aczel. Galois: a theory development project. A report on work in progress for the Turin meeting on the Representation of Logical Frameworks, 1993.
S. Agerholm, I. Beylin, and P. Dybjer. A comparison of HOL and ALF formalizations of a categorical coherence theorema. In Theorem Proving in Higher Order Logic (HOL '96). Springer LNCS, 1996. To appear. Available on http://www.cs.chalmers.se/∼ilya/FMC/hol_alf.ps.gz.
T. Altenkirch, V. Gaspes, B. Nordström, and B. von Sydow. A user's guide to ALF. Draft, January 1994.
T. Altenkirch, M. Hofmann, and T. Streicher. Categorical reconstruction of a reduction free normalization proof. In D. Pitt, D. E. Rydeheard, and P. Johnstone, editors, Springer LNCS 953, Category Theory and Computer Science, 6th International Conference, CTCS '95, Cambridge, UK, August 1995.
U. Berger and H. Schwichtenberg. An inverse to the evaluation functional for typed λ-calculus. In Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science, Amsterdam, pages 203–211, July 1991.
C. Coquand. From semantics to rules: a machine assisted analysis. In E. Börger, Y. Gurevich, and K. Meinke, editors, Proceedings of CSL '93, LNCS 832, 1993.
T. Coquand and P. Dybjer. Intuitionistic model constructions and normalization proofs. Preliminary Proceedings of the 1993 TYPES Workshop, Nijmegen, 1993.
P. Dybjer. Inductive sets and families in Martin-Löf's type theory and their set-theoretic semantics. In Logical Frameworks, pages 280–306. Cambridge University Press, 1991.
P. Dybjer. Internal type theory. In TYPES '95, Types for Proofs and Programs, Lecture Notes in Computer Science. Springer, 1996.
P. Dybjer and V. Gaspes. Implementing a category of sets in ALF. Manuscript, 1993.
Formal proof of coherence theorem. Home page. http://www.cs.chalmers.se/∼ilya/FMC/.
R. Gordon, A. J. Power, and R. Street. Coherence for tricategories. In Memoirs of the American Mathematical Society. To appear.
M. Hedberg. Normalizing the associative law: an experiment with Martin-Löf's type theory. Formal Aspects of Computing, pages 218–252, 1991.
M. Hofmann. Elimination of extensionality and quotient types in Martin-Löf's type theory. In Types for Proofs and Programs, International Workshop TYPES'93, LNCS 806, 1994.
G. Huet. Initiation à la Théorie des Catégories. Notes de cours du DEA Fonctionnalité, Structures de Calcul et Programmation donné à l'Université Paris VII en 1983–84 et 1984–85, 1987.
G. Huet and A. Saibi. Constructive category theory. In Proceedings of the Joint CLICS-TYPES Workshop on Categories and Type Theory, Göteborg, January 1995.
Y. Lafont. Logique, Categories & Machines. Implantation de Langages de Programmation guidée par la Logique Catégorique. PhD thesis, l'Universite Paris VII, January 1988.
S. Mac Lane. Categories for the Working Mathematician. Springer-Verlag, 1971.
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Beylin, I., Dybjer, P. (1996). Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids. In: Berardi, S., Coppo, M. (eds) Types for Proofs and Programs. TYPES 1995. Lecture Notes in Computer Science, vol 1158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61780-9_61
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DOI: https://doi.org/10.1007/3-540-61780-9_61
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