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Written by Trebor on 2024-12-26 at 11:01

Is there a low-tech proof that braided monoidal categories are the same as doubly monoidal categories? It seems to me that, in a doubly monoidal category, you get a natural isomorphism between the two monidal products, but not canonically so. So there is an extra choice of taking the isomorphism clockwise or counterclockwise, independent of the braiding. Isn't that more data?

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Written by Naïm Camille Favier on 2024-12-26 at 16:53

@trebor Isn't the choice "external"? I.e. depending on that binary choice you get one of two possible equivalences between doubly monoidal categories and braided monoidal categories, or in other words an automorphism on braided monoidal categories (that flips the braiding).

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Written by Trebor on 2024-12-27 at 02:54

@ncf Actually no, the two things are not equivalent https://arxiv.org/abs/0706.2307

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Written by Naïm Camille Favier on 2024-12-27 at 10:19

@trebor ugh...

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