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Computer Science > Logic in Computer Science

arXiv:1801.07664 (cs)
[Submitted on 23 Jan 2018 (v1), last revised 5 Jul 2018 (this version, v4)]

Title:Internal Universes in Models of Homotopy Type Theory

Authors:Daniel R. Licata, Ian Orton, Andrew M. Pitts, Bas Spitters
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Abstract:We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from which these model are constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-Mörtberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory.
Comments: In H. Kirchner (ed), Proceedings of the 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 108, pp. 22:1-22:17, 2018
Subjects: Logic in Computer Science (cs.LO)
ACM classes: F.4.1; F.3.2
Cite as: arXiv:1801.07664 [cs.LO]
  (or arXiv:1801.07664v4 [cs.LO] for this version)
  https://doi.org/10.48550/arXiv.1801.07664
Journal reference: Leibniz International Proceedings in Informatics (LIPIcs), Vol. 108, pp. 22:1-22:17, 2018
Related DOI: https://doi.org/10.4230/LIPIcs.FSCD.2018.22

Submission history

From: Andrew Pitts [view email]
[v1] Tue, 23 Jan 2018 17:21:55 UTC (29 KB)
[v2] Fri, 26 Jan 2018 11:50:58 UTC (29 KB)
[v3] Mon, 30 Apr 2018 09:42:09 UTC (47 KB)
[v4] Thu, 5 Jul 2018 08:54:10 UTC (47 KB)
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