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If symmetry in a monoidal category corresponds to exchange when interpreting it as logical inference, is there some similar analogue to braiding?
Ah just found this https://www.seas.upenn.edu/~sweirich/types/archive/1992/msg00068.html
Paul-andré mélliès has been working around these ribbon structure for some time e.g. https://www.irif.fr/~mellies/papers/lics2018-ribbon-tensorial-logic.pdf