  The tensor product of props was defined by Hackney and Robertson as an
extension of the Boardman-Vogt product of operads to more general monoidal
theories. Theories that factor as tensor products include the theory of
commutative monoids and the theory of bialgebras. We give a topological
interpretation (and vast generalisation) of this construction as a
low-dimensional projection of a "smash product of pointed directed spaces".
Here directed spaces are embodied by combinatorial structures called
diagrammatic sets, while Gray products replace cartesian products. The
correspondence is mediated by a web of adjunctions relating diagrammatic sets,
pros, probs, props, and Gray-categories. The smash product applies to
presentations of higher-dimensional theories and systematically produces
higher-dimensional coherence cells.
