When I'm mapping out a narrative arc, sometimes I will want to indicate that some action is monotone or idempotent, or that it is isomorphic or reversible from some other state. Those properties end up being path equivalences.

Similarly, sometimes there can be multiple social relations between characters within a narrative setting; somebody might be both a friend and a parent, for example. Properly modelling this requires allowing multiple distinct edges between vertices.

And, finally, technically any hypergraph with path equivalence is a category.

When I'm doing narrative analysis on a whiteboard, the diagrams that I draw are category-theoretic, not graph-theoretic, and I'd like to be able to bring that richness into my narrative-mapping tool without having to flatten everything down into annotated graphs.