Internal Universes in Models of Homotopy Type Theory

@article{Licata2018InternalUI,
  title={Internal Universes in Models of Homotopy Type Theory},
  author={Daniel R. Licata and Ian Orton and Andrew M. Pitts and Bas Spitters},
  journal={ArXiv},
  year={2018},
  volume={abs/1801.07664}
}
We show that universes of fibrations in various models of homotopy type theory have an essentially global character: they cannot be described in the internal language of the presheaf topos from which the model is 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-M\"ortberg (CCHM) notion of fibration from their… 

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