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Coherence for bicategorical cartesian closed structure

Published online by Cambridge University Press:  18 October 2021

Marcelo Fiore Open the ORCID record for Marcelo Fiore [Opens in a new window]  and
Philip Saville Open the ORCID record for Philip Saville [Opens in a new window]
Marcelo Fiore
Affiliation:
Department of Computer Science and Technology, University of Cambridge, Cambridge, UK
Philip Saville*
Affiliation:
School of Informatics, University of Edinburgh, Edinburgh, UK Department of Computer Science, University of Oxford, Oxford, UK
*
*Corresponding author. Email: philip.saville@cs.ox.ac.uk
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Abstract

We prove a strictification theorem for cartesian closed bicategories. First, we adapt Power’s proof of coherence for bicategories with finite bilimits to show that every bicategory with bicategorical cartesian closed structure is biequivalent to a 2-category with 2-categorical cartesian closed structure. Then we show how to extend this result to a Mac Lane-style “all pasting diagrams commute” coherence theorem: precisely, we show that in the free cartesian closed bicategory on a graph, there is at most one 2-cell between any parallel pair of 1-cells. The argument we employ is reminiscent of that used by Čubrić, Dybjer, and Scott to show normalisation for the simply-typed lambda calculus (Čubrić et al., 1998). The main results first appeared in a conference paper (Fiore and Saville, 2020) but for reasons of space many details are omitted there; here we provide the full development.

Keywords

Bicategoriescartesian closedcoherencestrictification

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Special Issue 7: The Power Festschrift , August 2021 , pp. 822 - 849
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Abbott, M. G. (2003). Categories of Containers. PhD thesis, University of Leicester.CrossRefGoogle Scholar
Bénabou, J. (1967). Introduction to bicategories. In: Reports of the Midwest Category Seminar, Berlin, Heidelberg, Springer Berlin Heidelberg, 1–77.CrossRefGoogle Scholar
Bénabou, J. (1985). Fibered categories and the foundations of naive category theory. Journal of Symbolic Logic 50 (1) 10–37.CrossRefGoogle Scholar
Blackwell, R., Kelly, G. M. and Power, A. J. (1989). Two-dimensional monad theory. Journal of Pure and Applied Algebra 59 (1) 1–41.CrossRefGoogle Scholar
Borceux, F. (1994). Bicategories and Distributors, volume 1 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, 281–324.Google Scholar
Carboni, A., Kelly, G. M., Walters, R. F. C. and Wood, R. J. (2008). Cartesian bicategories II. Theory and Applications of Categories 19 (6) 93–124.Google Scholar
Carboni, A., Lack, S. and Walters, R. F. C. (1993). Introduction to extensive and distributive categories. Journal of Pure and Applied Algebra 84 (2) 145–158.CrossRefGoogle Scholar
Carboni, A. and Walters, R. F. C. (1987). Cartesian bicategories I. Journal of Pure and Applied Algebra 49 (1) 11–32.CrossRefGoogle Scholar
Cattani, G. L., Fiore, M. and Winskel, G. (1998). A theory of recursive domains with applications to concurrency. In: Proceedings of the Thirteenth Annual IEEE Symposium on Logic in Computer Science (LICS 1998). IEEE Computer Society Press, 214–225.CrossRefGoogle Scholar
Čubrić, D., Dybjer, P. and Scott, P. (1998). Normalization and the Yoneda embedding. Mathematical Structures in Computer Science. CrossRefGoogle Scholar
Day, B. (1970). On closed categories of functors. In: Mac Lane, S., Applegate, H., Barr, M., Day, B., Dubuc, E., Phreilambud, , Pultr, A., Street, R., Tierney, M. and Swierczkowski, S. (eds.) Reports of the Midwest Category Seminar IV, Berlin, Heidelberg, Springer Berlin Heidelberg, 1–38.CrossRefGoogle Scholar
Day, B. and Street, R. 1997. Monoidal bicategories and Hopf algebroids. Advances in Mathematics 129 (1) 99–157.CrossRefGoogle Scholar
Fiore, M., Gambino, N., Hyland, M. and Winskel, G. 2007. The cartesian closed bicategory of generalised species of structures. Journal of the London Mathematical Society 77 (1) 203–220.CrossRefGoogle Scholar
Fiore, M. and Joyal, A. 2015. Theory of para-toposes. Talk at the Category Theory 2015 Conference. Departamento de Matematica, Universidade de Aveiro (Portugal).Google Scholar
Fiore, M. and Saville, P. 2020. Coherence and normalisation-by-evaluation for bicategorical cartesian closed structure. In: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science. ACM.CrossRefGoogle Scholar
Fiore, T. (2006). Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory. Memoirs of the American Mathematical Society. AMS.Google Scholar
Forest, S. and Mimram, S. (2018). Coherence of Gray categories via rewriting. In: 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.Google Scholar
Frey, J. (2019). A language for closed cartesian bicategories. Category Theory 2019, University of Edinburgh, Edinburgh, UK.Google Scholar
Gambino, N. and Joyal, A. (2017). On operads, bimodules and analytic functors. Memoirs of the American Mathematical Society 249 (1184) 153–192.CrossRefGoogle Scholar
Gordon, R., Power, A. J. and Street, R. (1995). Coherence for tricategories. Memoirs of the American Mathematical Society.CrossRefGoogle Scholar
Gray, J. W. (1974). Formal Category Theory: Adjointness for 2-Categories, vol. 391. Lecture Notes in Mathematics. Springer.CrossRefGoogle Scholar
Gurski, N. (2006). An Algebraic Theory of Tricategories. University of Chicago, Department of Mathematics.Google Scholar
Gurski, N. (2013). Coherence in Three-Dimensional Category Theory. Cambridge University Press.CrossRefGoogle Scholar
Hilken, B. (1996). Towards a proof theory of rewriting: the simply-typed 2λ-calculus. Theoretical Computer Science 170 (1) 407–444.CrossRefGoogle Scholar
Hirschowitz, T. (2013). Cartesian closed 2-categories and permutation equivalence in higher-order rewriting. Logical Methods in Computer Science 9 1–22.CrossRefGoogle Scholar
Houston, R. (2007). Linear Logic without Units. PhD thesis, University of Manchester.Google Scholar
Joyal, A. and Street, R. (1993). Braided tensor categories. Advances in Mathematics 102 (1) 20–78.CrossRefGoogle Scholar
Kelly, G. M. (1982). Basic Concepts of Enriched Category Theory. Number 64 in London Mathematical Society Lecture Note Series. Cambridge University Press.Google Scholar
Lack, S. (2010). A 2-Categories Companion. New York, NY, Springer New York, 105–191.Google Scholar
Lack, S., Walters, R. F. C. and Wood, R. J. (2010). Bicategories of spans as cartesian bicategories. Theory and Applications of Categories 24 (1) 1–24.Google Scholar
Lambek, J. and Scott, P. J. (1986). Introduction to Higher Order Categorical Logic. New York, NY, USA, Cambridge University Press.Google Scholar
Leinster, T. (1998). Basic Bicategories. Available at https://arxiv.org/abs/math/9810017.Google Scholar
Leinster, T. (2004). Higher Operads, Higher Categories. Number 298 in London Mathematical Society Lecture Note Series. Cambridge University Press.CrossRefGoogle Scholar
Mac Lane, S. (1963). Natural associativity and commutativity. Rice University Studies. Google Scholar
Mac Lane, S. (1998). Categories for the Working Mathematician, vol. 5. Graduate Texts in Mathematics, second edition. Springer-Verlag New York.Google Scholar
Mac Lane, S. and Paré, R. (1985). Coherence for bicategories and indexed categories. Journal of Pure and Applied Algebra 37 59–80.CrossRefGoogle Scholar
Makkai, M. (1996). Avoiding the axiom of choice in general category theory. Journal of Pure and Applied Algebra 108 (2) 109–173.CrossRefGoogle Scholar
Ouaknine, J. (1997). A two-dimensional extension of Lambek’s categorical proof theory. Master’s thesis, McGill University.Google Scholar
Paquet, H. (2020). Probabilistic concurrent game semantics. PhD thesis, University of Cambridge.Google Scholar
Power, A. J. (1989a). Coherence for bicategories with finite bilimits I. In: Gray, J. W. and Scedrov, A. (eds.) Categories in Computer Science and Logic: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference Held June 14–20, 1987 with Support from the National Science Foundation, vol. 92. American Mathematical Society, 341–349.CrossRefGoogle Scholar
Power, A. J. (1989b). A general coherence result. Journal of Pure and Applied Algebra 57 (2) 165–173.CrossRefGoogle Scholar
Power, A. J. (1998). 2-categories. BRICS Notes Series. Google Scholar
Saville, P. (2020). Cartesian closed bicategories: type theory and coherence. PhD thesis, University of Cambridge.Google Scholar
Stay, M. (2016). Compact closed bicategories. Theories and Applications of Categories 31 (26) 755–798.Google Scholar
Street, R. (1972). The formal theory of monads. Journal of Pure and Applied Algebra 2 (2) 149–168.CrossRefGoogle Scholar
Street, R. (1980). Fibrations in bicategories. Cahiers de Topologie et Géométrie Différentielle Catégoriques 21 (2) 111–160.Google Scholar