2000/03/23 by Hiroyuki Miyoshi, Miyoshi, Hiroyuki, Toru Tsujishita +2
Decision Sciences · Mathematics · Neuroscience · #03B30 #18D05 #18D10 #Advanced Topology and Set Theory #Category Theory (math.CT) #FOS: Mathematics #Fuzzy and Soft Set Theory #Logic (math.LO) #Neuroinflammation and Neurodegeneration Mechanisms #math.CT #math.LO #msc:03B30 #msc:18D05 #msc:18D10
paper · pdf · doi:10.48550/arxiv.math/0003137
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 in Jul 1999
arxiv created 2000/03/23 · openalex publication_date 2000/03/23 · arxiv updated 2009/11/30 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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.