Weak $ω$-categories as $ω$-hypergraphs

Abstract

In this paper, firstly, we introduce a higher-dimensional analogue of hypergraphs, namely $ω$-hypergraphs. This notion is thoroughly flexible because unlike ordinary $ω$-graphs, an n-dimensional edge called an n-cell has many sources and targets. Moreover, cells have polarity, with which pasting of cells is implicitly defined. As examples, we also give some known structures in terms of $ω$-hypergraphs. Then we specify a special type of $ω$-hypergraph, namely directed $ω$-hypergraphs, which are made of cells with direction. Finally, besed on them, we construct our weak $ω$-categories. It is an $ω$-dimensional variant of the weak n-categoreis given by Baez and Dolan. We introduce $ω$-identical, $ω$-invertible and $ω$-universal cells instead of universality and balancedness of Baez-Dolan. The whole process of our definition is in parallel with the way of regarding categories as graphs with composition and identities.