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at braided monoidal 2-categiry the following query box was sitting, which I hereby move from there to here
+–{: .query} Ben Webster: I would very much like to know: what structure on a (triangulated/dg-/stable infinity/whatever you like) monoidal category would make its 2-category of module categories (give that phrase any sensible construal you like) is braided monoidal.
If one decategorifies this question, one gets the question “what structure on a ring makes its category of representations braided monoidal” and the answer to this question is well-known: a quasi-triangular quasi-Hopf structure.
I asked a MathOverflow question on the same topic. No interesting answers yet. =–
Today I had had again the pleasure to chat with Larry Breen. In passing he reminded me that he had proposed the definition of braided monoidal (2-)category independently of Joyaj-Street in a letter to Deligne in 1988, as linked to from his webpage.
I have added this pointer to the nLab entries here and here.
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