Hostname: page-component-6d856f89d9-nr6nt Total loading time: 0 Render date: 2024-07-16T07:14:20.808Z Has data issue: false hasContentIssue false
  • English
  • Français

Brouwer's fixed-point theorem in real-cohesive homotopy type theory

Published online by Cambridge University Press:  17 August 2017

MICHAEL SHULMAN
MICHAEL SHULMAN*
Affiliation:
Department of Mathematics, University of San Diego, San Diego, CA 92110, USA Email: shulman@sandiego.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We combine homotopy type theory with axiomatic cohesion, expressing the latter internally with a version of ‘adjoint logic’ in which the discretization and codiscretization modalities are characterized using a judgemental formalism of ‘crisp variables.’ This yields type theories that we call ‘spatial’ and ‘cohesive,’ in which the types can be viewed as having independent topological and homotopical structure. These type theories can then be used to study formally the process by which topology gives rise to homotopy theory (the ‘fundamental ∞-groupoid’ or ‘shape’), disentangling the ‘identifications’ of homotopy type theory from the ‘continuous paths’ of topology. In a further refinement called ‘real-cohesion,’ the shape is determined by continuous maps from the real numbers, as in classical algebraic topology. This enables us to reproduce formally some of the classical applications of homotopy theory to topology. As an example, we prove Brouwer's fixed-point theorem.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Issue 6 , June 2018 , pp. 856 - 941
Copyright
Copyright © Cambridge University Press 2017 

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

Awodey, S. and Birkedal, L. (2003). Elementary axioms for local maps of toposes. Journal of Pure and Applied Algebra 177 (3) 215–230.Google Scholar
Awodey, S., Birkedal, L. and Scott, D. S. (1999). Local realizability toposes and a modal logic for computability: (Extended Abstract). Electronic Notes in Theoretical Computer Science 23 (1) 13–26.Google Scholar
Awodey, S. and Warren, M.A. (2009). Homotopy theoretic models of identity types. Mathematical Proceedings of the Cambridge Philosophical Society 146 (45) 45–55.Google Scholar
Baez, J.C. and Hoffnung, A.E. (2011). Convenient categories of smooth spaces. Transactions of the American Mathematical Society 363 (11) 5789–5825.CrossRefGoogle Scholar
Bauer, A. and Lešnik, D. (2012). Metric spaces in synthetic topology. Annals of Pure and Applied Logic 163 (2) 87–100.Google Scholar
Carchedi, D. (2016). On the homotopy type of higher orbifolds and Haefliger classifying spaces. Advances in Mathematics, 294, 756–818. arXiv:1504.02394.CrossRefGoogle Scholar
Dubuc, E.J. (1979). Concrete quasitopoi. In: Fourman, M., Mulvey, C. and Scott, D. (eds.) Applications of Sheaves (Proceedings, Durham 1977). Lecture Notes in Mathematics, Springer-Verlag, 239–254.Google Scholar
Dubuc, E.J. and Español, L. (2006). Quasitopoi over a base category. arXiv:math/0612727.Google Scholar
Dubuc, E.J. and Penon, J. (1986). Objets compacts dans les topos. Journal of the Australian Mathematical Society (Series A) 40 (2) 203–217.CrossRefGoogle Scholar
Dugger, D. (2001). Universal homotopy theories. Advances in Mathematics 164 (1) 144–176.CrossRefGoogle Scholar
Escardó, M. (2004a). Synthetic topology of data types and classical spaces. In: Electronic Notes in Theoretical Computer Science. Available at http://www.cs.bham.ac.uk/~mhe/papers/entcs87.pdf. Elsevier, p. 2004.CrossRefGoogle Scholar
Escardó, M. (2004b). Topology via higher-order intuitionistic logic. Unfinished draft. Available at http://www.cs.bham.ac.uk/~mhe/papers/index.html.Google Scholar
Escardó, M.H. and Streicher, T. (2016). The intrinsic topology of Martin-Löf universes. Annals of Pure and Applied Logic 167 (9), 794–805.CrossRefGoogle Scholar
Frank, M. (2017). Interpolating between choices for the approximate intermediate value theorem. arXiv:1701.02227.Google Scholar
Gepner, D. and Kock, J. (2012). Univalence in locally cartesian closed (∞, 1)-categories. arXiv:1208.1749.Google Scholar
Goodwillie, T.G. (2003). Calculus. III. Taylor series. Geometry & Topology 7 645–711.CrossRefGoogle Scholar
HoTT Project (2015). The homotopy type theory coq library. Available at http://github.com/HoTT/HoTT/.Google Scholar
Johnstone, P.T. (1979). On a topological topos. Proceedings of the London Mathematical Society (3) 38 (2) 237–271.CrossRefGoogle Scholar
Johnstone, P.T. (2002). Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2. Oxford Logic Guides, vol. 43, Oxford Science Publications.Google Scholar
Johnstone, P.T. (2011). Remarks on punctual local connectedness. Theory and Applications of Categories 25 (3) 51–63.Google Scholar
Joyal, A. (2008). Notes on logoi. Available at http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf.Google Scholar
Kapulkin, C. and Lumsdaine, P.L. (2012). The simplicial model of univalent foundations (after Voevodsky). arXiv:1211.2851.Google Scholar
Kraus, N. (2016). Constructions with non-recursive higher inductive types. In: LICS'16.Google Scholar
Lawvere, F.W. (1970). Equality in hyperdoctrines and comprehension schema as an adjoint functor. In: Applications of Categorical Algebra, Providence, R.I.: Amer. Math. Soc., pp. 1–14.Google Scholar
Lawvere, F.W. (2007). Axiomatic cohesion. Theory and Applications of Categories 19 (3) 41–49.Google Scholar
Lawvere, F.W. and Menni, M. (2015). Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness. Theory Applications of Categories 30 (26) 909–932.Google Scholar
Licata, D. and Finster, E. (2014). Eilenberg–MacLane spaces in homotopy type theory. In: LICS. Available at http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf.Google Scholar
Licata, D.R. and Shulman, M. (2013). Calculating the fundamental group of the circle in homotopy type theory. In: LICS'13. eprint: arXiv:1301.3443.Google Scholar
Licata, D. and Shulman, M. (2016). Adjoint logic with a 2-category of modes. In: LFCS '16. Available at http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint.pdf.Google Scholar
Licata, D.R., Shulman, M. and Riley, M. (2017). A fibrational framework for substructural and modal logics. To appear in FSCD '17.Google Scholar
Lin, Z. (2014). Answer to MathOverflow question ‘The real numbers object in Sh(Top).’ Available at http://mathoverflow.net/a/186165/49.Google Scholar
Lumsdaine, P.L. and Shulman, M. (2017). Semantics of higher inductive types. arXiv:1705.07088.Google 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) 23:1–23:31. arXiv:1411.1736.Google Scholar
Lurie, J. (2009). Higher Topos Theory. Annals of Mathematics Studies, vol. 170, Princeton University Press. arXiv:math.CT/0608040.Google Scholar
Lurie, J. (2014). Higher algebra. Available at http://www.math.harvard.edu/~lurie/.Google Scholar
Mac Lane, S. and Moerdijk, I. (1994). Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Universitext. Corrected reprint of the 1992 edition. New York: Springer-Verlag.Google Scholar
Menni, M. (2014). Continuous cohesion over sets. Theory and Applications of Categories 29 (20) 542–568.Google Scholar
Palmgren, E. (2007a). A constructive and functorial embedding of locally compact metric spaces into locales. Topology and its Applications 154 (9) 1854–1880.Google Scholar
Palmgren, E. (2007b). Resolution of the Uniform Lower Bound Problem in Constructive Analysis. Tech. rep. 11. Uppsala University Department of Mathematics.Google Scholar
Penon, J. (1985). De l'infinitésimal au local (Thése de Doctorat d'État). In: Diagrammes S13. Available at http://www.numdam.org/item?id=DIA_1985_S13_1_0, pp. 1–191.Google Scholar
Pfenning, F. and Davies, R. (2001). A judgmental reconstruction of modal logic. Mathematical Structures in Computer Science 11 (4) 511–540.Google Scholar
Reed, J. (2009). A judgmental deconstruction of modal logic. Available at http://www.cs.cmu.edu/~jcreed/papers/jdml.pdf.Google Scholar
Rezk, C. (2014). Global homotopy theory and cohesion. Available at http://www.math.uiuc.edu/~rezk/global-cohesion.pdf.Google Scholar
Rijke, E. (2017). The join construction. arXiv:1701.07538.Google Scholar
Rijke, E., Shulman, M. and Spitters, B. (2017). Modalities in homotopy type theory. arXiv:1706.07526.Google Scholar
Schreiber, U. (2013). Differential cohomology in a cohesive (∞, 1)-topos. Available at http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos; arXiv:1310.7930.Google Scholar
Schreiber, U. and Shulman, M. (2012). Quantum gauge field theory in cohesive homotopy type theory. In: QPL'12. Available at http://ncatlab.org/schreiber/files/QFTinCohesiveHoTT.pdf.Google Scholar
Shulman, M. (2011a). Internalizing the external, or the joys of codiscreteness. Available at https://golem.ph.utexas.edu/category/2011/11/internalizing_the_external_or.html.Google Scholar
Shulman, M. (2011b). Reflective subfibrations, factorization systems, and stable units. Available at https://golem.ph.utexas.edu/category/2011/12/reflective_subfibrations_facto.html.Google Scholar
Streicher, T. (1991). Semantics of Type Theory: Correctness, Completeness, and Independence Results. Progress in Theoretical Computer Science, Birkhäuser.Google Scholar
Taylor, P. (2010). A lambda calculus for real analysis. Journal of Logic & Analysis 2 (5) 1–115.Google Scholar
Troelstra, A.S. and van Dalen, D. (1988). Constructivism in Mathematics. Vol. I. Studies in Logic and the Foundations of Mathematics, vol. 121. Amsterdam: North-Holland Publishing Co., pp. xx+342+XIV.Google Scholar
Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics. first edition. Available at http://homotopytypetheory.org/book/Google Scholar
van Doorn, F. (2016). Constructing the propositional truncation using non-recursive HITs. In: Certified Programs and Proofs '16. arXiv:1512.02274.Google Scholar
Wyler, O. (1991). Lecture Notes on Topoi and Quasitopoi. World Scientific.Google Scholar