  Developing an idea of Kapranov and Voevodsky, we introduce a model of weak
omega-categories based on directed complexes, combinatorial presentations of
pasting diagrams. We propose this as a convenient framework for
higher-dimensional rewriting.
  We define diagrammatic sets to be presheaves on a category of directed
complexes presenting pasting diagrams with spherical boundaries. Diagrammatic
sets have structural face and degeneracy operations, but no structural
composition. We define a notion of equivalence cell in a diagrammatic set, and
say that a diagrammatic set is representable if all pasting diagrams with
spherical boundaries are connected to individual cells -- their weak composites
-- by a higher-dimensional equivalence cell. We develop the basic theory of
representable diagrammatic sets (RDSs), and prove that equivalence cells
satisfy the expected properties in an RDS. We study nerves of strict
omega-categories as RDSs and prove that 2-truncated RDSs are equivalent to
bicategories.
  Finally, we connect the combinatorics of diagrammatic sets and simplicial
sets, and prove a version of the homotopy hypothesis for "groupoidal" RDSs:
there exists a geometric realisation functor for diagrammatic sets, inducing
isomorphisms between combinatorial and classical homotopy groups, which has a
homotopical right inverse.

Category Theory