Skip to main content Accessibility help
×
Home
Hostname: page-component-5d6d958fb5-9cwrl Total loading time: 0.561 Render date: 2022-11-28T15:11:11.272Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

A homotopy-theoretic model of function extensionality in the effective topos

Published online by Cambridge University Press:  10 September 2018

DAN FRUMIN
Affiliation:
Institute for Computing and Information Sciences, Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands Email: dfrumin@cs.ru.nl
BENNO VAN DEN BERG
Affiliation:
Institute for Logic, Language and Computation, Universiteit van Amsterdam, P.O. Box 94242, 1090 GE Amsterdam, The Netherlands Email: B.vandenBerg3@uva.nl

Abstract

We present a way of constructing a Quillen model structure on a full subcategory of an elementary topos, starting with an interval object with connections and a certain dominance. The advantage of this method is that it does not require the underlying topos to be cocomplete. The resulting model category structure gives rise to a model of homotopy type theory with identity types, Σ- and Π-types, and functional extensionality. We apply the method to the effective topos with the interval object ∇2. In the resulting model structure we identify uniform inhabited objects as contractible objects, and show that discrete objects are fibrant. Moreover, we show that the unit of the discrete reflection is a homotopy equivalence and the homotopy category of fibrant assemblies is equivalent to the category of modest sets. We compare our work with the path object category construction on the effective topos by Jaap van Oosten.

Type
Paper
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Angiuli, C., Harper, R. and Wilson, T. (2016). Computational higher type theory I: Abstract cubical realizability. arXiv:1604.08873.Google Scholar
Awodey, S. and Warren, M. A. (2009). Homotopy theoretic models of identity types. Mathematical Proceedings of the Cambridge Philosophical Society 146 (1) 4555.CrossRefGoogle Scholar
Bezem, M., Coquand, T. and Huber, S. (2014). A model of type theory in cubical sets. In: Proceedings of the 19th International Conference on Types for Proofs and Programs (TYPES 2013), vol. 26, 107–128.Google Scholar
Bourke, J. and Garner, R. (2016). Algebraic weak factorisation systems I: Accessible AWFS. Journal of Pure and Applied Algebra 220 (1) 108147.CrossRefGoogle Scholar
Cisinski, D.-C. (2002). Théories homotopiques dans les topos. Journal of Pure and Applied Algebra 174 (1) 4382.CrossRefGoogle Scholar
Cohen, C., Coquand, T., Huber, S. and Mörtberg, A. (2018). Cubical type theory: A constructive interpretation of the univalence axiom. In: Proceedings of the 21st International Conference on Types for Proofs and Programs (TYPES 2015), Leibniz International Proceedings in Informatics (LIPIcs), 5:1–5:34, vol. 69, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany.Google Scholar
Curien, P.-L. (1993). Substitution up to isomorphism. Fundamenta Informaticae 19 (1–2) 5185.Google Scholar
Frumin, D. (2016) Weak Factorisation Systems in the Effective Topos, Master's thesis, University of Amsterdam, The Netherlands. Available at https://www.illc.uva.nl/Research/Publications/Reports/MoL-2016-13.text.pdf.Google Scholar
Gambino, N. and Sattler, C. (2017). The Frobenius condition, right properness, and uniform fibrations. Journal of Pure and Applied Algebra 221 (12) 30273068.CrossRefGoogle Scholar
Hofmann, M. (1995). Extensional Concepts in Intensional Type Theory, University of Edinburgh, College of Science and Engineering, School of Informatics.Google Scholar
Hofmann, M. and Streicher, T. (1998). The groupoid interpretation of type theory. In: Twenty-Five Years of Constructive Type Theory (Venice, 1995) 83111, Oxford Logic Guides, vol. 36, Oxford University Press, New York.Google Scholar
Hyland, M. (1982). The effective topos. In: The L.E.J. Brouwer Centenary Symposium, North-Holland, 165–216.CrossRefGoogle Scholar
Hyland, M. (1988). A small complete category. Annals of Pure and Applied Logic 40 (2) 135165.CrossRefGoogle Scholar
Joyal, A. and Tierney, M. (2008) Notes on simplicial homotopy. Available at http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern47.pdf.Google Scholar
Kapulkin, C. and Lumsdaine, P.L. (2016). The simplicial model of univalent foundations (after Voevodsky). arXiv:1211.2851.Google Scholar
Kleene, S.C. (1945). On the interpretation of intuitionistic number theory. Journal of Symbolic Logic 10 109124.CrossRefGoogle Scholar
Lumsdaine, P.L. (2010). Weak ω-categories from intensional type theory. Logical Methods in Computer Science 6 (3:24) pp. 119.CrossRefGoogle Scholar
Lumsdaine, P. L. and Warren, M. A. (2015). The local universes model: An overlooked coherence construction for dependent type theories. ACM Transactions on Computational Logic 16 (3) Art no. 23.CrossRefGoogle Scholar
van Oosten, J. (2008). Realizability: An Introduction to Its Categorical Side, Studies in Logic, vol. 152, Elsevier Science, San Diego, USA.Google Scholar
van Oosten, J. (2015). A notion of homotopy for the effective topos. Mathematical Structures in Computer Science 25 (05) 11321146.CrossRefGoogle Scholar
Orton, I. and Pitts, A.M. (2016). Axioms for modelling cubical type theory in a topos. In: nad, J.-M. Talbot, L.R. (ed.) 25th EACSL Annual Conference on Computer Science Logic (CSL 2016), Leibniz International Proceedings in Informatics (LIPIcs) 24:1–24:19, vol. 62, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany.Google Scholar
Riehl, E. (2011). Algebraic model structures. New York Journal of Mathematics 17 173231.Google Scholar
Riehl, E. (2014). Categorical Homotopy Theory, Cambridge University Press.CrossRefGoogle Scholar
Rosolini, G. (1986) Continuity and effectiveness in topoi. PhD thesis, Carnegie Mellon University. Available at ftp://ftp.disi.unige.it/pub/person/RosoliniG/papers/coneit.ps.gz.Google Scholar
Seely, R. A. G. (1984). Locally cartesian closed categories and type theory. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 95, Cambridge University Press, 33–48.CrossRefGoogle Scholar
Shulman, M. (2015). Univalence for inverse diagrams and homotopy canonicity. Mathematical Structures in Computer Science 25 (5) 12031277.CrossRefGoogle Scholar
Streicher, T. (1991). Semantics of Type Theory, Correctness, Completeness and Independence Results, Progress in Theoretical Computer Science, Birkhäuser.Google Scholar
Streicher, T. (2007/2008) Realizability. Lecture notes. Available at http://www.mathematik.tu-darmstadt.de/~streicher/REAL/REAL.pdf.Google Scholar
Swan, A. (2015) Identity Types in an Algebraic Model Structure. Draft. Available at http://sites.google.com/site/wakelinswan/idams.pdf.Google Scholar
The Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics, Institute for Advanced Study. Available at https://homotopytypetheory.org/book.Google Scholar
van den Berg, B. (2016). Path categories and propositional identity types. ACM Transactions on Computational Logic. arXiv:1604.06001.Google Scholar
van den Berg, B. and Garner, R. (2011). Types are weak ω-groupoids. Proceedings of the London Mathematical Society 102 (2) 370394.CrossRefGoogle Scholar
van den Berg, B. and Garner, R. (2012). Topological and simplicial models of identity types. ACM Transactions on Computational Logic 13 (1), 3:13:44.CrossRefGoogle Scholar
Voevodsky, V. (2010). Univalent foundations project. A modified version of an NSF grant application. Available at http://www.math.ias.edu/vladimir/files/univalent_foundations_project.pdf.Google Scholar
1
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A homotopy-theoretic model of function extensionality in the effective topos
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

A homotopy-theoretic model of function extensionality in the effective topos
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

A homotopy-theoretic model of function extensionality in the effective topos
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *