 
Summary: HCategories and Graphs
Aaron D. Ames and Paulo Tabuada
Abstract
Hcategories are an essential ingredient of categorical models for hybrid systems. In this note,
we consider oriented Hcategories and form the category of these categories, Hcat. The main
result is that there is an isomorphism of categories:
Hcat = Grph.
The proof of this fact is constructive in nature, i.e., it is shown how to obtain a graph from and
Hcategory and an Hcategory from a graph.
1. Introduction
Small categories play a critical role in the study of diagrams in a category, i.e., one
often considerers the the functor category TC whose objects are all functors FFF : C
T, for a small category C. Similarly, diagrams can also be defined by a graph whose
vertices index a collection of objects in T and edges index a collection of morphisms
in T. These two notions of "diagrams" in a category are related via functors from
Grph to Cat and Cat to Grph which form an adjoint pair [7]. The disadvantage to
this construction is that the creation of a small category from a graph is achieved
by adding information to the graph (in the form of paths in the graph); it may be
the case that this added information is unwanted and/or unneeded. This motivates
the creation of a small category from a graph in which the resulting category is the
