  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-M\"ortberg (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.
