2010/08/10 by Iain R. Aitchison, Aitchison, Iain R. · 2 citations
Mathematics · Physics and Astronomy · #18G50 #2010 Primary: 18D05 #52B22 Secondary:57Q15 #52C35 #68Q42 #68Q85 #68R05 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #math-ph #math.CO #math.CT #math.GT #math.MP #msc:18D05 #msc:18G50 #msc:2010 #msc:52B22 #msc:52C35 #msc:68Q42 #msc:68Q85 #msc:68R05
paper · pdf · doi:10.48550/arxiv.1008.1714
53 pages, 197 figures. The output pdf file might best be viewed in such as TeX rather than Acrobat
arxiv created 2010/08/10 · openalex publication_date 2010/08/10 · arxiv updated 2010/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This reports on the fundamental objects revealed by Ross Street, which he called `orientals'. Street's work was in part inspired by Robert's attempts to use N-category ideas to construct nets of C*-algebras in Minkowski space for applications to relativistic quantum field theory: Roberts' additional challenge was that `no amount of staring at the low dimensional cocycle conditions would reveal the pattern for higher dimensions'. This report takes up this challenge, presenting a natural inductive construction of explicit cubical cocyle conditions, and gives three ways in which the simplicial ones can be derived from these. (A dual string-diagram version of this work, giving rise to a Pascal's triangle of diagrams for cocycle conditions, has been described elsewhere by Street). A consequence of this work is that the Yang-Baxter equation, the `pentagon of pentagons', and higher simplex equations, are in essence different manifestations of the same underlying abstract structure. There has been recent interest in higher-categories, by computer scientists investigating concurrency theory, as well as by physicists, among others. The dual `string' version of this paper makes clear the relationship with higher-dimensional simplex equations in physics. Much work in this area has been done since these notes were written: no attempt has been made to update the original report. However, all diagrams have been redrawn by computer, replacing all original hand-drawn pictures.