Abstract
The glueing construction, defined as a certain comma category, is an important tool for reasoning about type theories, logics, and programming languages. Here we extend the construction to accommodate ‘2-dimensional theories’ of types, terms between types, and rewrites between terms. Taking bicategories as the semantic framework for such systems, we define the glueing bicategory and establish a bicategorical version of the well-known construction of cartesian closed structure on a glueing category. As an application, we show that free finite-product bicategories are fully complete relative to free cartesian closed bicategories, thereby establishing that the higher-order equational theory of rewriting in the simply-typed lambda calculus is a conservative extension of the algebraic equational theory of rewriting in the fragment with finite products only.
Chapter PDF
Similar content being viewed by others
Keywords
References
Abbott, M.G.: Categories of containers. Ph.D. thesis, University of Leicester (2003)
Abramsky, S., Jagadeesan, R.: Games and full completeness for multiplicative linear logic. Journal of Symbolic Logic 59(2), 543–574 (1994). https://doi.org/10.2307/2275407
Alimohamed, M.: A characterization of lambda definability in categorical models of implicit polymorphism. Theoretical Computer Science 146(1-2), 5–23 (1995). https://doi.org/10.1016/0304-3975(94)00283-O
Balat, V., Di Cosmo, R., Fiore, M.: Extensional normalisation and typed-directed partial evaluation for typed lambda calculus with sums. In: Proceedings of the 31st Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages. pp. 64–76 (2004)
Bénabou, J.: Introduction to bicategories. In: Reports of the Midwest Category Seminar. pp. 1–77. Springer Berlin Heidelberg, Berlin, Heidelberg (1967)
Bloom, S.L., Ésik, Z., Labella, A., Manes, E.G.: Iteration 2-theories. Applied Categorical Structures 9(2), 173–216 (2001). https://doi.org/10.1023/a:1008708924144
Borceux, F.: Bicategories and distributors, Encyclopedia of Mathematics and its Applications, vol. 1, pp. 281–324. Cambridge University Press (1994). https://doi.org/10.1017/CBO9780511525858.009
Carboni, A., Kelly, G.M., Walters, R.F.C., Wood, R.J.: Cartesian bicategories II. Theory and Applications of Categories 19(6), 93–124 (2008), http://www.tac.mta.ca/tac/volumes/19/6/19-06abs.html
Carboni, A., Lack, S., Walters, R.F.C.: Introduction to extensive and distributive categories. Journal of Pure and Applied Algebra 84(2), 145–158 (1993). https://doi.org/10.1016/0022-4049(93)90035-r
Carboni, A., Walters, R.F.C.: Cartesian bicategories I. Journal of Pure and Applied Algebra 49(1), 11–32 (1987). https://doi.org/10.1016/0022-4049(87)90121-6
Castellan, S., Clairambault, P., Rideau, S., Winskel, G.: Games and strategies as event structures. Logical Methods in Computer Science 13 (2017)
Crole, R.L.: Categories for Types. Cambridge University Press (1994). https://doi.org/10.1017/CBO9781139172707
Dagand, P.E., McBride, C.: A categorical treatment of ornaments. In: Proceedings of the 28th Annual ACM/IEEE Symposium on Logic in Computer Science. pp. 530–539. IEEE Computer Society, Washington, DC, USA (2013). https://doi.org/10.1109/LICS.2013.60
Fiore, M.: Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertations in Computer Science, Cambridge University Press (1996)
Fiore, M.: Semantic analysis of normalisation by evaluation for typed lambda calculus. In: Proceedings of the 4th ACM SIGPLAN International Conference on Principles and Practice of Declarative Programming. pp. 26–37. ACM, New York, NY, USA (2002). https://doi.org/10.1145/571157.571161
Fiore, M., Di Cosmo, R., Balat, V.: Remarks on isomorphisms in typed lambda calculi with empty and sum types. In: Proceedings of the 28th Annual IEEE Symposium on Logic in Computer Science. pp. 147–156. IEEE Computer Society Press (2002). https://doi.org/10.1109/LICS.2002.1029824
Fiore, M., Gambino, N., Hyland, M., Winskel, G.: The cartesian closed bicategory of generalised species of structures. Journal of the London Mathematical Society 77(1), 203–220 (2007). https://doi.org/10.1112/jlms/jdm096
Fiore, M., Gambino, N., Hyland, M., Winskel, G.: Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures. Selecta Mathematica New Series (2017)
Fiore, M., Joyal, A.: Theory of para-toposes. Talk at the Category Theory 2015 Conference. Departamento de Matematica, Universidade de Aveiro (Portugal)
Fiore, M., Saville, P.: A type theory for cartesian closed bicategories. In: Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science (2019). https://doi.org/10.1109/LICS.2019.8785708
Fiore, M., Saville, P.: Coherence and normalisation-by-evaluation for bicategorical cartesian closed structure. Preprint (2020)
Fiore, M., Simpson, A.: Lambda definability with sums via Grothendieck logical relations. In: Girard, J.Y. (ed.) Typed lambda calculi and applications: 4th international conference. pp. 147–161. Springer Berlin Heidelberg, Berlin, Heidelberg (1999)
Freyd, P.: Algebraically complete categories. In: Lecture Notes in Mathematics, pp. 95–104. Springer Berlin Heidelberg (1991). https://doi.org/10.1007/bfb0084215
Freyd, P.J., Scedrov, A.: Categories, Allegories. Elsevier North Holland (1990)
Gambino, N., Joyal, A.: On operads, bimodules and analytic functors. Memoirs of the American Mathematical Society 249(1184), 153–192 (2017)
Gambino, N., Kock, J.: Polynomial functors and polynomial monads. Mathematical Proceedings of the Cambridge Philosophical Society 154(1), 153–192 (2013). https://doi.org/10.1017/S0305004112000394
Ghani, N.: Adjoint rewriting. Ph.D. thesis, University of Edinburgh (1995)
Gibbons, J.: Conditionals in distributive categories. Tech. rep., University of Oxford (1997)
G.L. Cattani, Fiore, M., Winskel, G.: A theory of recursive domains with applications to concurrency. In: Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science. pp. 214–225. IEEE Computer Society (1998)
Gurski, N.: An Algebraic Theory of Tricategories. University of Chicago, Department of Mathematics (2006)
Hasegawa, M.: Logical predicates for intuitionistic linear type theories. In: Girard, J.Y. (ed.) Typed lambda calculi and applications: 4th international conference. pp. 198–213. Springer Berlin Heidelberg, Berlin, Heidelberg (1999)
Hilken, B.: Towards a proof theory of rewriting: the simply typed 2\(\lambda \)-calculus. Theoretical Computer Science 170(1), 407–444 (1996). https://doi.org/10.1016/S0304-3975(96)80713-4
Hirschowitz, T.: Cartesian closed 2-categories and permutation equivalence in higher-order rewriting. Logical Methods in Computer Science 9, 1–22 (2013)
Jay, C.B., Ghani, N.: The virtues of eta-expansion. Journal of Functional Programming 5(2), 135–154 (1995). https://doi.org/10.1017/S0956796800001301
Johann, P., Polonsky, P.: Higher-kinded data types: Syntax and semantics. In: 34th Annual ACM/IEEE Symposium on Logic in Computer Science. IEEE (2019). https://doi.org/10.1109/lics.2019.8785657
Jung, A., Tiuryn, J.: A new characterization of lambda definability. In: Bezem, M., Groote, J.F. (eds.) Typed Lambda Calculi and Applications. pp. 245–257. Springer Berlin Heidelberg, Berlin, Heidelberg (1993)
Lack, S.: A 2-Categories Companion, pp. 105–191. Springer New York, New York, NY (2010)
Lack, S., Walters, R.F.C., Wood, R.J.: Bicategories of spans as cartesian bicategories. Theory and Applications of Categories 24(1), 1–24(2010)
Lafont, Y.: Logiques, catégories et machines. Ph.D. thesis, UniversitéParis VII (1987)
Lambek, J., Scott, P.J.: Introduction to Higher Order Categorical Logic. Cambridge University Press, New York, NY, USA (1986)
Leinster, T.: Basic bicategories (May 1998), https://arxiv.org/pdf/math/9810017.pdf
Leinster, T.: Higher operads, higher categories. No. 298 in London Mathematical Society Lecture Note Series, Cambridge University Press (2004)
Ma, Q.M., Reynolds, J.C.: Types, abstraction, and parametric polymorphism, part 2. In: Brookes, S., Main, M., Melton, A., Mislove, M., Schmidt, D. (eds.) Mathematical Foundations of Programming Semantics. pp. 1–40. Springer Berlin Heidelberg, Berlin, Heidelberg (1992)
Mac Lane, S.: Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5. Springer-Verlag New York, second edn. (1998). https://doi.org/10.1007/978-1-4757-4721-8
Mac Lane, S., Paré, R.: Coherence for bicategories and indexed categories. Journal of Pure and Applied Algebra 37, 59–80 (1985). https://doi.org/10.1016/0022-4049(85)90087-8
Marmolejo, F., Wood, R.J.: Kan extensions and lax idempotent pseudomonads. Theory and Applications of Categories 26(1), 1–29 (2012)
Mitchell, J.C., Scedrov, A.: Notes on sconing and relators. In: Börger, E.,J., G., Kleine Büning, H., Martini, S., Richter, M.M. (eds.) Computer Science Logic. pp. 352–378. Springer Berlin Heidelberg, Berlin, Heidelberg (1993)
Ouaknine, J.: A two-dimensional extension of Lambek’s categorical proof theory. Master’s thesis, McGill University (1997)
Paquet, H.: Probabilistic concurrent game semantics. Ph.D. thesis, University of Cambridge (2020)
Plotkin, G.D.: Lambda-definability and logical relations. Tech. rep., University of Edinburgh School of Artificial Intelligence (1973), memorandum SAI-RM-4
Power, A.J.: An abstract formulation for rewrite systems. In: Pitt, D.H., Rydeheard, D.E., Dybjer, P., Pitts, A.M., Poigné, A. (eds.) Category Theory and Computer Science. pp. 300–312. Springer Berlin Heidelberg, Berlin, Heidelberg (1989)
Power, A.J.: Coherence for bicategories with finite bilimits I. In: Gray, J.W., Scedrov, A. (eds.) Categories in Computer Science and Logic: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference, vol. 92, pp. 341–349. AMS (1989)
Power, A.J.: A general coherence result. Journal of Pure and Applied Algebra 57(2), 165–173 (1989). https://doi.org/10.1016/0022-4049(89)90113-8
Rydeheard, D.E., Stell, J.G.: Foundations of equational deduction: A categorical treatment of equational proofs and unification algorithms. In: Pitt, D.H., Poigné, A., Rydeheard, D.E. (eds.) Category Theory and Computer Science. pp. 114–139. Springer Berlin Heidelberg, Berlin, Heidelberg (1987)
Saville, P.: Cartesian closed bicategories: type theory and coherence. Ph.D. thesis, University of Cambridge (Submitted)
Seely, R.A.G.: Modelling computations: A 2-categorical framework. In: Gries, D.(ed.) Proceedings of the 2nd Annual IEEE Symposium on Logic in Computer Science. pp. 65–71. IEEE Computer Society Press (June 1987)
Statman, R.: Logical relations and the typed \(\lambda \)-calculus. Information and Control 65, 85–97 (1985)
Stell, J.: Modelling term rewriting systems by sesqui-categories. In: Proc. Catégories, Algèbres, Esquisses et Néo-Esquisses (1994)
Street, R.: Fibrations in bicategories. Cahiers de Topologie et Géométrie Différentielle Catégoriques 21(2), 111–160 (1980), https://eudml.org/doc/91227
Street, R.: Categorical structures. In: Hazewinkel, M. (ed.) Handbook of Algebra, vol. 1, chap. 15, pp. 529–577. Elsevier (1995)
Tabareau, N.: Aspect oriented programming: A language for 2-categories. In: Proceedings of the 10th International Workshop on Foundations of Aspect-oriented Languages. pp. 13–17. ACM, New York, NY, USA (2011).https://doi.org/10.1145/1960510.1960514
Taylor, P.: Practical Foundations of Mathematics, Cambridge Studies in Advanced Mathematics, vol. 59. Cambridge University Press (1999)
Troelstra, A.S., Schwichtenberg, H.: Basic proof theory. No. 43 in Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, second edn. (2000)
Verity, D.: Enriched categories, internal categories and change of base. Ph.D. thesis, University of Cambridge (1992), TAC reprint available at http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html
Weber, M.: Yoneda structures from 2-toposes. Applied Categorical Structures 15(3), 259–323 (2007). https://doi.org/10.1007/s10485-007-9079-2
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Copyright information
© 2020 The Author(s)
About this paper
Cite this paper
Fiore, M., Saville, P. (2020). Relative Full Completeness for Bicategorical Cartesian Closed Structure. In: Goubault-Larrecq, J., König, B. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2020. Lecture Notes in Computer Science(), vol 12077. Springer, Cham. https://doi.org/10.1007/978-3-030-45231-5_15
Download citation
DOI: https://doi.org/10.1007/978-3-030-45231-5_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-45230-8
Online ISBN: 978-3-030-45231-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
