Higher dimensional hypercategories

TL;DR

This paper introduces higher dimensional hypergraphs as a generalization of existing structures, providing new graphical representations and defining strict and weak hypercategories, with applications to known categorical frameworks.

Contribution

It presents a novel combinatorial framework for higher dimensional hypercategories, including graphical representations and the formalization of strict and weak variants.

Findings

Defined strict hypercategories using hypergraphs

Introduced two types of graphical representations of higher cells

Connected usual categories with weak hypercategories

Abstract

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(2) Akira Huguchi and Toru Tsujishita, Strict $n$-hypercategories, after completion of which this manuscript will be withdrawn.