  We revisit Kapranov and Voevodsky's idea of spaces modelled on combinatorial
pasting diagrams, now as a framework for higher-dimensional rewriting and the
basis of a model of weak omega-categories. In the first part, we elaborate on
Steiner's theory of directed complexes as a combinatorial foundation. We
individuate convenient classes of directed complexes and develop the theory of
diagrammatic sets relative to one such class. We study a notion of equivalence
internal to a diagrammatic set, and single out as models of weak
omega-categories those diagrammatic sets whose every composable diagram is
connected by an equivalence to a single cell. We then define a semistrict model
providing algebraic composites and study the embedding of strict
omega-categories into this model. Finally, we prove a version of the homotopy
hypothesis for the infinity-groupoids in the weak model, and exhibit a specific
mistake in a proof by Kapranov and Voevodsky that had previously been refuted
indirectly.
