  The geometric and algebraic properties of Gray categories with duals are
investigated by means of a diagrammatic calculus. The diagrams are
three-dimensional stratifications of a cube, with regions, surfaces, lines and
vertices labelled by Gray category data. These can be viewed as a
generalisation of ribbon diagrams. The Gray categories present two types of
duals, which are extended to functors of strict tricategories with natural
isomorphisms, and correspond directly to symmetries of the diagrams. It is
shown that these functors can be strictified so that the symmetries of a cube
are realised exactly. A new condition on Gray categories with duals called the
spatial condition is defined. A class of diagrams for which the evaluation for
spatial Gray categories is invariant under homeomorphisms is exhibited. This
relation between the geometry of the diagrams and structures in the Gray
categories proves useful in computations and has potential applications in
topological quantum field theory.
