  We introduce higher dimensional hypergraphs, which is a generalization of
Baez-Dolans's opetopic sets and Hermida-Makkai-Power's multigraphs. This is
based on a simple combinatorial structure called shells and the formal
composites of pasting diagrams based on the closure of open shells. We give two
types of graphical representation of higher dimensional cells which show
effectively the relationship of cells of different dimensions. Using the
hypergraphs, we define strict hypercategories and illustrate its use by taking
Lafont's interaction combinator as an example. We also give a definition of
weak $\omega$-hypercategories and show that usual category is identified with a
special kind of weak hypercategory as an illustration of arguments provided by
our framework
  In the replacement of 9 Aug, an omission of an important condition in the
definition of shells is corrected.
  We are preparing two papers which develop two themes roughly presented in
this preprint: (1) Hiroyuki Miyoshi and Toru Tsujishita, Weak
$\omega$-Categories as $\omega$-Hypergraphs, presented at CT99, International
Category Theory Meeting Category Theory, July 1999, Coimbra, and (2) Akira
Huguchi and Toru Tsujishita, Strict $n$-hypercategories, after completion of
which this manuscript will be withdrawn.

Category Theory;Logic