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I know the term "modular category" only from representations of quantum groups, TQFTs and fusion (finitely semisimple linear) categories. There, a modular category is a ribbon fusion category where a certain matrix of braidings is invertible, i.e. the braiding is "maximally nonsymmetric".

One can show that a ribbon fusion category is modular if the only object that braids trivially with any other object is the monoidal identity. It seems we haven't really used the linear or abelian structure on the category. Also, to define dualities, braidings and ribbon structures, it doesn't seem like you need the linear structure. So I'm asking:

Can we define a modular category to be a ribbon category (not necessarily linear or abelian) where only the monoidal identity braids trivially? What interesting examples are there except the abelian ones?

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The notion you want is that of a modular object. These are the sorts of things an oriented 3D TFT assigns to a circle.

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