Anyway, these things got trendy a few years back because homotopies in them --- well, dihomotopies --- let you talk about concurrency and deadlock and saftey in a cute, diagrammatic way. But they're another obvious place to start thinking about higher-order algebra. Homotopies in ordinary topological spaces make up omega-groupouids, and "directing" the topology seems to get rid of commutativity in the right way. I seems pretty obvious that a ditopology ought to give rise to a weak omega-category --- I figure this all got worked out by someone, but I haven't really combed the literature very aggressively.
Except for the obvious fact that still nobody's exactly sure what weak omega-categories are. So... I looked at Tom Leinster's survey again, and I think there is maybe still a niche for some sort of ditopological definition.