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Lukas Heidemann (Feb 15 2021 at 17:11):I'm currently trying to understand Street's orientals better, in order to hopefully adapt the construction from the strict to a weaker setting. In low dimensions, the orientals appear to give a single 0-vertex, a 1-morphism, a "multiplication" 2-morphism, an associator 3-morphism and a pentagonator 4-morphism. Starting with the pentagonator there is an interchange involved; this is innocuous in a strict setting but becomes important in semi-strict models with weak interchange. I would imagine that the choice of arrangements and interchanges could be contractible, and ideally there would be a functorial "mechanical" choice on how to arrange things. Is there any discussion of something like this in the literature somewhere? Since the orientals arise by taking cones iteratively, it would probably also help to understand joins or Gray tensor products in the presence of weak interchange.
Amar Hadzihasanovic (Feb 15 2021 at 21:25):Judging on the abstract of his talk in 2 days at the Australian Category Seminar, you may want to ask Yuki Maehara.