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On generalized algebraic theories and categories with families

Published online by Cambridge University Press:  18 October 2021

Marc Bezem ,
Thierry Coquand Open the ORCID record for Thierry Coquand [Opens in a new window] ,
Peter Dybjer Open the ORCID record for Peter Dybjer [Opens in a new window]  and
Martín Escardó Open the ORCID record for Martín Escardó [Opens in a new window]
Marc Bezem
Affiliation:
Department of Informatics, University of Bergen, Norway
Thierry Coquand
Affiliation:
Department of Computer Science and Engineering, University of Göteborg, Sweden
Peter Dybjer*
Affiliation:
Department of Computer Science and Engineering, Chalmers University of Technology, Sweden
Martín Escardó
Affiliation:
School of Computer Science, University of Birmingham, UK
*
*Corresponding author. Email: [email protected]
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Abstract

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We give a syntax independent formulation of finitely presented generalized algebraic theories as initial objects in categories of categories with families (cwfs) with extra structure. To this end, we simultaneously define the notion of a presentation Σ of a generalized algebraic theory and the associated category CwFΣ of small cwfs with a Σ-structure and cwf-morphisms that preserve Σ-structure on the nose. Our definition refers to the purely semantic notion of uniform family of contexts, types, and terms in CwFΣ. Furthermore, we show how to syntactically construct an initial cwf with a Σ-structure. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of Martin-Löf type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a small category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual cwfs.

Keywords

Dependent type theorygeneralized algebraic theorycategory with familiesinitial modelinternal categoryMartin-Löf type theory
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Special Issue 9: In Homage to Martin Hofmann , October 2021 , pp. 1006 - 1023
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

The Agda wiki. http://appserv.cs.chalmers.se/users/ulfn/wiki/agda.php.Google Scholar
Aczel, P. (1978). The type theoretic interpretation of constructive set theory. In: Logic Colloquium 1977 (Proc. Conf., Wrocław, 1977), vol. 96. Stud. Logic Foundations Math. Amsterdam-New York, North-Holland, 55–66.10.1016/S0049-237X(08)71989-XCrossRefGoogle Scholar
Aczel, P. (1998). On relating type theories and set theories. In: Altenkirch, T., Reus, B. and Naraschewski, W. (eds.) Types for Proofs and Programs. Springer, 33–46.Google Scholar
Altenkirch, T. and Kaposi, A. (2016). Type theory in type theory using quotient inductive types. In: Bodík, R. and Majumdar, R. (eds.) Proceedings of the 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2016, St. Petersburg, FL, USA, January 20–22, 2016. ACM, 18–29.10.1145/2837614.2837638CrossRefGoogle Scholar
Awodey, S., Frey, J. and Speight, S. (2018). Impredicative encodings of (higher) inductive types. In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2018). New York, NY, USA, Association for Computing Machinery.Google Scholar
Birkhoff, G. (1935). On the structure of abstract algebras. In: Proceeding of the Cambridge Philosophical Society, vol. 31, 433–454.10.1017/S0305004100013463CrossRefGoogle Scholar
Brunerie, G. (2019). A formalization of the initiality conjecture in Agda. Slides from a talk about joint work with Menno de Boer, Peter Lumsdaine, and Anders Mörtberg, at HoTT, Pittsburgh, CMU.Google Scholar
Cartmell, J. (1978). Generalized Algebraic Theories and Contextual Categories. D. Phil., Oxford University.Google Scholar
Cartmell, J. (1986). Generalized algebraic theories and contextual categories. Annals of Pure and Applied Logic 32 209–243.10.1016/0168-0072(86)90053-9CrossRefGoogle Scholar
Cartmell, J. (2018). Generalised algebraic axiomatisations of contextual categories. https://www.researchgate.net/publication/325763538_Generalised_Algebraic_Axiomatisations_of_Contextual_Categories.Google Scholar
Castellan, S., Clairambault, P. and Dybjer, P. (2015). Undecidability of equality in the free locally cartesian closed category. In 13th International Conference on Typed Lambda Calculi and Applications, TLCA 2015, July 1–3, 2015, Warsaw, Poland, 138–152.Google Scholar
Castellan, S., Clairambault, P. and Dybjer, P. (2017). Undecidability of equality in the free locally cartesian closed category (extended version). Logical Methods in Computer Science 13 (4).Google Scholar
Castellan, S., Clairambault, P. and Dybjer, P. (2021). Categories with families: Unityped, simply typed, and dependently typed. In: Casadio, C. and Scott, P. J. (eds.) Joachim Lambek: The interplay of Mathematics, Logic, and Linguistics, Outstanding Contributions to Logic, Editor-in-Chief: Sven-Ove Hansson. vol. 20. Springer.Google Scholar
Clairambault, P. and Dybjer, P. (2011). The biequivalence of locally cartesian closed categories and Martin-Löf type theories. In: Typed Lambda Calculi and Applications - 10th International Conference, TLCA 2011, Novi Sad, Serbia, June 1–3, 2011. Proceedings, 91–106.10.1007/978-3-642-21691-6_10CrossRefGoogle Scholar
Clairambault, P. and Dybjer, P. (2014). The biequivalence of locally cartesian closed categories and Martin-Löf type theories. Mathematical Structures in Computer Science 24 (6).10.1017/S0960129513000881CrossRefGoogle Scholar
Curien, P., Garner, R. and Hofmann, M. (2014). Revisiting the categorical interpretation of dependent type theory. Theoretical Computer Science 546 99–119.10.1016/j.tcs.2014.03.003CrossRefGoogle Scholar
Dybjer, P. (1996). Internal type theory. In: TYPES 1995, Types for Proofs and Programs, vol. 1158. Lecture Notes in Computer Science. Springer, 120–134.10.1007/3-540-61780-9_66CrossRefGoogle Scholar
Forsberg, F. N. and Setzer, A. (2010). Inductive-inductive definitions. In Dawar, A. and Veith, H., editors, Computer Science Logic, 24th International Workshop, CSL 2010, 19th Annual Conference of the EACSL, Brno, Czech Republic, August 23–27, 2010. Proceedings, vol. 6247. Lecture Notes in Computer Science. Springer, 454–468.Google Scholar
Freyd, P. (1964). Abelian Categories. New York, Harper and Row.Google Scholar
Hofmann, M. (1994). On the interpretation of type theory in locally cartesian closed categories. In: Pacholski, L. and Tiuryn, J. (eds.) CSL, vol. 933. Lecture Notes in Computer Science. Springer.Google Scholar
Hofmann, M. (1995). Conservativity of equality reflection over intensional type theory. In: Berardi, S. and Coppo, M. (eds.) Types for Proofs and Programs, International Workshop TYPES 1995, Torino, Italy, June 5–8, 1995, Selected Papers, vol. 1158. Lecture Notes in Computer Science. Springer, 153–164.Google Scholar
Hofmann, M. (1996). Syntax and semantics of dependent types. In: Pitts, A. and Dybjer, P. (eds.) Semantics and Logics of Computation. Cambridge University Press.Google Scholar
Hofmann, M. (1997). Extensional Constructs in Intensional Type Theory . CPHC/BCS distinguished dissertations. Springer.Google Scholar
Hofmann, M. and Streicher, T. (1994). A groupoid model refutes uniqueness of identity proofs. In: LICS 1994, 208–212. IEEE Press.10.1109/LICS.1994.316071CrossRefGoogle Scholar
Kaposi, A., Kovács, A. and Altenkirch, T. (2019). Constructing quotient inductive-inductive types. Proceedings of the ACM on Programming Languages, 3 (POPL) 2:1–2:24.Google Scholar
Martin-Löf, P. (1984). Intuitionistic Type Theory. Bibliopolis.Google Scholar
Martin-Löf, P. (1992). Substitution calculus. Notes from a lecture given in Göteborg.Google Scholar
Palmgren, E. and Vickers, S. J. (2007). Partial Horn logic and cartesian categories. Annals of Pure and Applied Logic 145 (3) 314–353.10.1016/j.apal.2006.10.001CrossRefGoogle Scholar
Reynolds, J. C. (1984). Polymorphism is not set-theoretic. In: Kahn, G., MacQueen, D. B., and Plotkin, G. D., editors, Semantics of Data Types, International Symposium, Sophia-Antipolis, France, June 27–29, 1984, Proceedings, vol. 173. Lecture Notes in Computer Science. Springer, 145–156.10.1007/3-540-13346-1_7CrossRefGoogle Scholar
Reynolds, J. C. and Plotkin, G. D. (1993). On functors expressible in the polymorphic typed lambda calculus. Information and Computation 105 (1) 1–29.10.1006/inco.1993.1037CrossRefGoogle Scholar
Streicher, T. (1991). Semantics of Type Theory. Birkhäuser.10.1007/978-1-4612-0433-6CrossRefGoogle Scholar
Uemura, T. (2019). A general framework for the semantics of type theory. ArXiv, abs/1904.04097.Google Scholar
Voevodsky, V. (2014). Subsystems and regular quotients of C-systems. ArXiv, abs/1406.7413.Google Scholar
Voevodsky, V. (2017). Models, interpretations and the initiality conjectures. Notes from a lecture at the 2017 Logic Colloquium in Stockholm, special session on Category Theory and Type Theory in honor of Per Martin-Lf on his 75th birthday. https://www.math.ias.edu/Voevodsky/files/files-annotated/Dropbox/Unfinished_papers/Type_systems/Notes_on_Type_Systems/2017_LC_Martin-Lof_special_session/BSL_extended_abstract.pdf.Google Scholar