Internal Universes in Models of Homotopy Type Theory

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-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 a completely internal development of models of homotopy type theory within what we call crisp type theory.