  This is a book on higher-categorical diagrams, including pasting diagrams. It
aims to provide a thorough and modern reference on the subject, collecting,
revisiting and expanding results scattered across the literature, informed by
recent advances and practical experience with higher-dimensional diagram
rewriting.
  We approach the subject as a kind of directed combinatorial topology: a
diagram is a map from a "directed cell complex", encoded combinatorially as a
face poset together with orientation data. Unlike previous expositions, we
adopt from the beginning a functorial viewpoint, focussing on morphisms and
categorical constructions. We do not tie ourselves to a specific model of
higher categories, and instead treat diagrams as independent combinatorial
structures that admit functorial interpretations in various contexts.
  Topics covered include the theory of layerings of diagrams; acyclicity
properties and their consequences; constructions including Gray products,
suspensions, and joins; special shapes such as globes, oriented simplices,
cubes, and positive opetopes; the interpretation of diagrams in strict
omega-categories and their geometric realisation as simplicial and CW
complexes; and Steiner's theory of directed chain complexes.
