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Xuanrui Qi (Aug 15 2023 at 09:57):It's easy to find symmetric monoidal categories out in the wild, but less so braided (non-symmetric) monoidal categories. I can of course think of, eh, the braid category, and examples arising from representations of quantum groups. But I don't know of many else, besides vaguely understanding that braided MCs are related to TQFT somehow.
Are there any other "natural" examples, preferably from algebra or geometry?
Philip Saville (Aug 15 2023 at 10:34):there's a few examples in Joyal and Street's Braided tensor categories, some of which you might consider natural. There's two versions of the paper, the one in Macquarie mathematics reports has more examples: link
Mike Shulman (Aug 15 2023 at 11:17):The category of endomorphisms of any identity morphism in a (weak) 3-category is braided monoidal, and generally not symmetric. So, for example, for any 2-category (an object of the 3-category ), the category of natural transformations is braided monoidal, sometimes called the "center of ". In particular, if has one object, this is the [[Drinfeld center]] of a monoidal category, which I guess is a quantum-groupy construction so this is related to one you already mentioned.
Simon Burton (Aug 15 2023 at 17:07):@Mike Shulman i'm thinking theres a really good string diagram to draw here, but i'm not quite seeing it.. this is something i lifted from a paper on "anyons":
image.png
Mike Shulman (Aug 15 2023 at 17:23):Well, it's basically the Eckmann-Hilton argument.
Christopher Tapo (Aug 22 2023 at 22:09):Although braided fusion categories and modular tensor categories are examples of braided monoidal categories that usually relate to TQFT, CFT, and quantum groups, there are papers like this that show a relationship between modular tensor categories and Seifert fibered spaces. Even though there is still a focus on the application to physics, the relationship between Seifert manifolds and braided monoidal categories is, I think, interesting in its own right.