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Homotopy limits in type theory

Published online by Cambridge University Press:  19 January 2015

JEREMY AVIGAD ,
KRZYSZTOF KAPULKIN  and
PETER LEFANU LUMSDAINE
JEREMY AVIGAD
Affiliation:
Department of Philosophy and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A. Email: avigad@cmu.edu
KRZYSZTOF KAPULKIN
Affiliation:
Department of Mathematics, University of Pittsburgh, Pennsylvania, U.S.A. E-mail: k.kapulkin@gmail.com
PETER LEFANU LUMSDAINE
Affiliation:
Institute for Advanced Study, Princeton, New Jersey, U.S.A. Email: p.l.lumsdaine@gmail.com
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Abstract

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Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to the formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 25 , Special Issue 5: From type theory and homotopy theory to Univalent Foundations of Mathematics , June 2015 , pp. 1040 - 1070
Copyright
Copyright © Cambridge University Press 2015 

References

Awodey, S., Gambino, N. and Sojakova, K. (2012) Inductive types in homotopy type theory. Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science, 95–104.Google Scholar
Ahrens, B., Kapulkin, K. and Shulman, M. (2014) Univalent categories and Rezk completion. Mathematical Structures in Computer Science doi:10.1017/S0960129514000486.CrossRefGoogle Scholar
Awodey, S. and Warren, M. A. (2009) Homotopy theoretic models of identity types. Mathematical Proceedings of the Cambridge Philosophical Society 146 (1) 45–55. (arXiv:0709.0248)Google Scholar
Baues, H. J. (1989) Algebraic homotopy, Cambridge Studies in Advanced Mathematics volume 15, Cambridge University Press. Available at: http://dx.doi.org/10.1017/CBO9780511662522.Google Scholar
Bousfield, A. K. and Kan, D. M. (1972) Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics volume 304, Springer-Verlag.Google Scholar
Brown, K. S. (1973) Abstract homotopy theory and generalized sheaf cohomology. Transactions of the American Mathematical Society 186 419–458.Google Scholar
Dwyer, W. G., Hirschhorn, P. S., Kan, D. M. and Smith, J. H. (2004) Homotopy limit functors on model categories and homotopical categories, Mathematical Surveys and Monographs volume 113, American Mathematical Society.Google Scholar
Gonthier, G., Asperti, A., Avigad, J., Bertot, Y., Cohen, C., Garillot, F., Roux, S. L., Mahboubi, A., O'Connor, R. Biha, S. O., Pasca, I., Rideau, L., Solovyev, A., Tassi, E. and Théry, L. (2013) A machine-checked proof of the odd order theorem. In: Blazy, S., Paulin-Mohring, C., and Pichardie, D. (eds.) Interactive Theorem Proving – Proceedings 4th International Conference, ITP 2013, Rennes, France, July 22–26. Springer Lecture Notes in Computer Science 7998 163–179.Google Scholar
Gambino, N. and Garner, R. (2008) The identity type weak factorisation system. Theoretical Computer Science 409 (1) 94–109. (arXiv:0803.4349)Google Scholar
HoTT group. (2013) Homotopy type theory repository. Ongoing Coq development. Available at: https://github.com/HoTT/coq.Google Scholar
Hofmann, M. and Streicher, T. (1998) The groupoid interpretation of type theory. In: Twenty-five Years of Constructive Type Theory (Venice, 1995), Oxford Logic Guides volume 36, Oxford University Press 83–111.Google Scholar
Kapulkin, K., Lumsdaine, P. L. and Voevodsky, V. (2012) The simplicial model of univalent foundations. (arXiv:1211.2851)Google Scholar
Lumsdaine, P. L. (2009) Weak ω-categories from intensional type theory (conference version). In: Typed Lambda Calculi and Applications (Berlin), Lecture Notes in Computer Science 5608 172–187.Google Scholar
Lumsdaine, P. L. (2011) Model structures from higher inductive types. Unpublished note, December 2011. Available at: //www.mathstat.dal.ca/~p.l.lumsdaine/research/Lumsdaine-Model-strux-from-HITs.pdf.Google Scholar
Lurie, J. (2009) Higher topos theory. Annals of Mathematics Studies volume 170, Princeton University Press.Google Scholar
Mather, M. (1976) Pull-backs in homotopy theory. Canadian Journal of Mathematics 28 (2) 225–263.Google Scholar
Martin-Lóf, P. (1984) Intuitionistic type theory. Studies in Proof Theory, Lecture Notes volume 1, Bibliopolis, Naples.Google Scholar
Pelayo, Á. and Warren, M. (2012) Homotopy type theory and Voevodsky's univalent foundations. (arXiv:1210.5658)Google Scholar
Radulescu-Banu, A. (2006) Cofibrations in homotopy theory. (arXiv:0610009)Google Scholar
Rijke, E. and Spitters, B. (2014) Sets in homotopy type theory. Mathematical Structures in Computer Science doi:10.1017/S0960129514000486.Google Scholar
Shulman, M. (2012) The univalence axiom for inverse diagrams. (arXiv:1203.3253)Google Scholar
Streicher, T. (1991) Semantics of type theory. Progress in Theoretical Computer Science, Birkháuser Boston Inc., Boston, MA. (Correctness, completeness and independence results. With a foreword by Martin Wirsing.)Google Scholar
The Univalent Foundations Program (2013) Homotopy type theory: Univalent foundations of mathematics, Technical report, Institute for Advanced Study. Available at: http://homotopytypetheory.org/book/.Google Scholar
van den Berg, B. and Garner, R. (2011) Types are weak ω-groupoids. Proceedings of the London Mathematical Society 102 (2–3) 370–394. (arXiv:0812.0298)Google Scholar
van den Berg, B. and Garner, R. (2012) Topological and simplicial models of identity types. ACM Transactions on Computational Logic 13 (1) Art. 3, 44. (arXiv:1007.4638v1)Google Scholar
Voevodsky, V. (2013) Univalent foundations repository. Ongoing Coq development. Available at: https://github.com/vladimirias/Foundations.Google Scholar
Voevodsky, V. (2006) A very short note on homotopy λ-calculus. Notes from seminars given at Stanford University. Available at: http://math.ucr.edu/home/baez/Voevodsky_note.ps.Google Scholar
Voevodsky, V. (2010) Notes on type systems. Ongoing unpublished manuscript. Available at: http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/expressions_current.pdf.Google Scholar
Warren, M. A. (2008) Homotopy Theoretic Aspects of Constructive Type Theory, Ph.D. thesis, Carnegie Mellon University.Google Scholar
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