Abstract
A coherence theorem states that the arrows between two particular objects in free categories are unique. In this paper, we give a direct proof for cartesian closed categories (CCC's) without passing to typed lambda calculus. We first derive categorical combinators for CCC's together with their equations directly from the adjoint functors defining CCC's. Then categorical interpretation of the intuitionistic sequent calculus is regarded as the construction of free CCC's. Each arrow is generated along the proof derivation in form of categorical combinators. The system enjoys the cut-elimination theorem and we can use various proof-theoretic techniques such as Kleene's permutability theorem. The coherence is proved by showing that the reconstruction of derivations for the given class of arrows is deterministic and unique up to equivalence.
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A.A. Babaev and S.V. Soloviev. A coherence theorem for canonical maps in cartesian closed categories. Zapisiki Nauchnykh Seminarov LOMI, 88, 1979. Russian with English summary. English translation appears in J. of Soviet Math., 20, 1982.
Michael Barr and Charles Wells. Category Theory for Computing Science. International Series in Computer Science. Prentice Hall, 1990.
[Cousineau et al., 1985] Guy Cousineau, P.-L. Curien, and Michael Mauny. The Categorical Abstract Machines. Lecture Notes in Computer Science, 201:50–64, 1985.
P.-L. Curien. Categorical combinators, Sequential algorithms and Functional programming. Research Notes in Theoretical Computer Science. Pitman, 1986. The revised edition is published from Birkhäuser, in the series of Progress in Theoretical Computer Science, 1993.
Michael Dummett. Elements of Intuitionism, volume 2 of Oxford Logic Guides. Oxford University Press, 1977.
J. Gallier. Constructive logics, part I: A tutorial on proof systems and typed λ-calculi. Theor. Comput. Sci., 110:249–339, 1993.
[Girard et al., 1989] J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.
Collin Barry Jay. The structure of free closed categories. J. Pure and Applied Algebra, 66:271–285, 1990.
G.M. Kelly and Saunders Mac Lane. Coherence in closed categories. J. Pure and Applied Algebra, 1(1):97–140, 1972.
S.C. Kleene. Permutability of inferences in Gentzen's calculi LK and LJ. Memoirs of the American Mathematical Society, 10:1–26, 1952.
J. Lambek and P.J. Scott. Introduction to Higher Order Categorical Logic. Cambridge University Press, 1986.
J. Lambek. Deductive systems and categories II: Standard constructions and closed categories. Lecture Notes in Mathematics, 86:76–122, 1969.
J. Lambek. Multicategories revisited. In J.W Gray and A. Scedrov, editors, Categories in Computer Science and Logic, pages 217–239. American Mathematical Society, 1989. Contemporary Mathematics Vol.92.
F.W. Lawvere. Adjointness in foundations. Dialectica, 23:281–296, 1969.
F.W. Lawvere. Equality in hyperdoctorines and the comprehension schema as an adjoint functor. In Proc. NY Symposium on Applications of Categorical Logic, pages 1–14. American Mathematical Society, 1970.
Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer-Verlag, 1971.
Colin McLarty. Elementary Categories, Elementary Toposes, volume 21 of Oxford Logic Guides. Oxford University Press, 1992.
[Miller et al., 1991] Dale Miller, Gopalan Nadathur, Frank Pfenning, and Andre Scedrov. Uniform proofs as a foundation for logic programming. Ann. Pure and Applied Logic, 51:125–157, 1991.
Dale Miller. A logic programming language with lambda-abstraction, function variables, and simple unification. J. of Logic and Computation, 1(4):497–536, 1991.
Gregorii E. Mints. Proof theory and category theory. In Selected Papers in Proof Theory, chapter 10, pages 183–212. Bibliopolis/North-Holland, 1992.
Gregorii E. Mints. A simple proof of the coherence theorem for cartesian closed categories. In Selected Papers in Proof Theory, chapter 11, pages 213–220. Bibliopolis/North-Holland, 1992.
M.E. Szabo. A counter-example to coherence in cartesian closed categories. Canad.Math.Bull., 18(1):111–114, 1975.
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Mori, A., Matsumoto, Y. (1995). Coherence for cartesian closed categories: A sequential approach. In: Dershowitz, N., Lindenstrauss, N. (eds) Conditional and Typed Rewriting Systems. CTRS 1994. Lecture Notes in Computer Science, vol 968. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60381-6_17
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DOI: https://doi.org/10.1007/3-540-60381-6_17
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