  This paper considers the difficulty in the set-system approach to
generalizing graph theory. These difficulties arise categorically as the
category of set-system hypergraphs is shown not to be cartesian closed and
lacks 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 be
studied by their locally graphic behavior via homomorphisms that allow an edge
of 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 incidence
hypergraphs via a logical functor that is the inverse image of an essential
geometric morphism between the topoi. Consequently, the quiver exponential is
shown to be simply represented using incidence hypergraph homomorphisms.
