  In this paper, firstly, we introduce a higher-dimensional analogue of
hypergraphs, namely $\omega$-hypergraphs. This notion is thoroughly flexible
because unlike ordinary $\omega$-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 $\omega$-hypergraphs. Then we specify a special type of
$\omega$-hypergraph, namely directed $\omega$-hypergraphs, which are made of
cells with direction. Finally, besed on them, we construct our weak
$\omega$-categories. It is an $\omega$-dimensional variant of the weak
n-categoreis given by Baez and Dolan. We introduce $\omega$-identical,
$\omega$-invertible and $\omega$-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.
