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

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

2000-03-23

arxiv.org/abs/math/0003137v1

Mathematics

Category Theory

26 pages, 8 figures, written in Nov 1999 and adjusted to arXiv.org in Mar 2000; it is based on the first author's talk at CT99

Scientific paper

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.

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