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This paper considers the difficulty in the set-system approach togeneralizing graph theory. These difficulties arise categorically as thecategory of set-system hypergraphs is shown not to be cartesian closed andlacks enough projective objects, unlike the category of directed multigraphs(i.e. quivers). The category of incidence hypergraphs is introduced as a"graph-like" remedy for the set-system issues so that hypergraphs may bestudied by their locally graphic behavior via homomorphisms that allow an edgeof the domain to be mapped into a subset of an edge in the codomain. Moreover,it is shown that the category of quivers embeds into the category of incidencehypergraphs via a logical functor that is the inverse image of an essentialgeometric morphism between the topoi. Consequently, the quiver exponential isshown to be simply represented using incidence hypergraph homomorphisms.

Incidence hypergraphs: The categorical inconsistency of set-systems and a characterization of quiver exponentials