Internal Universes in Models of Homotopy Type Theory
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 CohenCoquandHuberMörtberg (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.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 arXiv:
 arXiv:1801.07664
 Bibcode:
 2018arXiv180107664L
 Keywords:

 Computer Science  Logic in Computer Science;
 F.4.1;
 F.3.2
 EPrint:
 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. 22:122:17, 2018