  Bezem, Coquand, and Huber have recently given a constructively valid model of
higher type theory in a category of nominal cubical sets satisfying a novel
condition, called the uniform Kan condition (UKC), which generalizes the
standard cubical Kan condition (as considered by, for example, Williamson in
his survey of combinatorial homotopy theory) to admit phantom "additional"
dimensions in open boxes. This note, which represents the authors' attempts to
fill in the details of the UKC, is intended for newcomers to the field who may
appreciate a more explicit formulation and development of the main ideas. The
crux of the exposition is an analogue of the Yoneda Lemma for co-sieves that
relates geometric open boxes bijectively to their algebraic counterparts, much
as its progenitor for representables relates geometric cubes to their algebraic
counterparts in a cubical set. This characterization is used to give a
formulation of uniform Kan fibrations in which uniformity emerges as naturality
in the additional dimensions.
