  Networks are often studied as graphs, where the vertices stand for entities
in the world and the edges stand for connections between them. While relatively
easy to study, graphs are often inadequate for modeling real-world situations,
especially those that include contexts of more than two entities. For these
situations, one typically uses hypergraphs or simplicial complexes.
  In this paper, we provide a precise framework in which graphs, hypergraphs,
simplicial complexes, and many other categories, all of which model higher
graphs, can be studied side-by-side. We show how to transform a hypergraph into
its nearest simplicial analogue, for example. Our framework includes many new
categories as well, such as one that models broadcasting networks. We give
several examples and applications of these ideas.
