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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 16th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexat _[[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 category|triangulated]]/[[differential graded category|dg-]]/[[stable (infinity,1)-category|stable infinity]]/whatever you like) [[monoidal category]] would make its 2-category of [[module category|module categories]] (give that phrase any sensible construal you like) is braided monoidal. If one [[decategorification|decategorifies]] this question, one gets the question "what structure on a [[ring]] makes its category of representations [[braided monoidal category|braided monoidal]]" and the answer to this question is well-known: a [[quasi-triangular quasi-Hopf algebra|quasi-triangular]] [[quasi-Hopf algebra|quasi-Hopf]] structure. I asked a [MathOverflow question](http://mathoverflow.net/questions/4790/what-structure-on-a-monoidal-category-would-make-its-2-category-of-module-catego) on the same topic. No interesting answers yet. =--

   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. =–

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 19th 2014
   • (edited Jun 19th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexToday 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](http://ncatlab.org/nlab/show/braided+monoidal+2-category#Breen88) and [here](http://ncatlab.org/nlab/show/braided+monoidal+category#Breen88).

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 21st 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[André Joyal]], [[Ross Street]], Section 5 of: _Braided tensor categories_, Adv. Math. **102** (1993), 20--78 ([doi:10.1006/aima.1993.1055](https://doi.org/10.1006/aima.1993.1055)) <a href="https://ncatlab.org/nlab/revision/diff/braided+monoidal+2-category/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/braided+monoidal+2-category/11">v11</a>, <a href="https://ncatlab.org/nlab/show/braided+monoidal+2-category">current</a>

   added pointer to:

   • André Joyal, Ross Street, Section 5 of: Braided tensor categories, Adv. Math. 102 (1993), 20–78 (doi:10.1006/aima.1993.1055)

   diff, v11, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 7th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * Thibault D. Décoppet, Hao Xu, *Local Modules in Braided Monoidal 2-Categories* &lbrack;[arXiv:2307.02843](https://arxiv.org/abs/2307.02843)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/braided+monoidal+2-category/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/braided+monoidal+2-category/13">v13</a>, <a href="https://ncatlab.org/nlab/show/braided+monoidal+2-category">current</a>

   added pointer to today’s

   • Thibault D. Décoppet, Hao Xu, Local Modules in Braided Monoidal 2-Categories [arXiv:2307.02843]

   diff, v13, current