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Homotopy theoretic models of identity types



Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such as Coq and Agda (cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature.



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[1]Abadi, M., Cardelli, L., Curien, P.–L. and Lévy, J.–J. Explicit substitution. In Proceedings of the 17th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (ACM Press, 1989), 31–46.
[2]Bénabou, J.Fibred categories and the foundations of naive category theory. J. Symbolic Logic 50 (1985), 1037.
[3]Bertot, Y. and Castéran, P.Interactive Theorem Proving and Program Development (Springer-Verlag, 2004).
[4]Bousfield, A. K.Constructions of factorization systems in categories. J. Pure Appl. Algebra 9 (1977), 207220.
[5]Coquand, C. and Coquand, T. Structured type theory. In Proceedings of the Workshop on Logical Frameworks and Meta-Languages (LFM'99), Paris (1999)
[6]Curien, P.-L.Substitution up to isomorphism. Fund. Informaticae 19 (1993), 5186.
[7]Dwyer, W. G. and Spalinski, J. Homotopy theories and model categories. In Handbook of Algebraic Topology, James, I. M., editor (North-Holland, 1995), 73126.
[8]Hofmann, M. On the interpretation of type theory in locally cartesian closed categories. In Computer Science Logic 1994, Tiuryn, J. and Pacholski, Leszek, editors (Springer-Verlag, 1995), 427441.
[9]Hofmann, M. and Streicher, T. The groupoid interpretation of type theory. In Twenty-Five Years of Constructive Type Theory, Sambin, G. and Smith, J., editors (Oxford University Press, 1998), 83111.
[10]Hovey, M.Model Categories. Math. Surv. Monogr. 63 (American Mathematical Society, 1999).
[11]Joyal, A.Quasi-categories and Kan complexes. J. Pure Appl. Algebra 175 (2002), 207222.
[12]Joyal, A. Notes on quasi-categories. Unpublished manuscript (2007).
[13]Lurie, J. Higher topos theory. Unpublished manuscript, available on the arXiv as arXiv:math/0608040 (2007).
[14]Martin-Löf, P. An intuitionistic theory of types: predicative part. In Logic Colloquium '73, Rose, H. E. and Shepherdson, J. C., editors (North-Holland, 1975), 73118.
[15]Morel, F. and Voevodsky, V.A 1-homotopy theory of schemes. Inst. Hautes Etudes Sci Pub. Math. 90 (1999), 45143.
[16]Nordström, B., Petersson, K. and Smith, J. M.Programming in Martin–Löf's Type Theory. An Introduction (Oxford University Press, 1990).
[17]Quillen, D.Homotopical Algebra. Lecture Notes in Mathematics (Springer-Verlag, 1967).
[18]Seely, R. A. G.Locally cartesian closed categories and type theory. Math. Proc. Camb. Phil. Soc. 95 (1984), 3348.
[19]Streicher, T. Investigations into intensional type theory. Habilitationsschrift, Ludwig-Maximilians-Universität München (1993).
[20]Warren, M. A. Homotopy theoretic aspects of constructive type theory. PhD. thesis. Carnegie Mellon University. In preparation.


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