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Mar 23 at 3:02 vote accept მამუკა ჯიბლაძე
Mar 22 at 22:48 comment added Dmitri Pavlov @მამუკაჯიბლაძე: I think the answer is negative already for 1-categories: there are many nonisomorphic braidings on a monoidal category that induce a commutative monoid structure on isomorphism classes of objects.
Mar 22 at 17:41 comment added მამუკა ჯიბლაძე A side question - is not every homotopy commutative group equivalent to a braided one, unique up to homotopy?
Mar 22 at 17:18 history answered Dmitri Pavlov CC BY-SA 4.0