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• CommentRowNumber1.
• CommentAuthorJohn Baez
• CommentTimeDec 20th 2009
The definition of braided monoidal category was wrong or at least nonstandard, because it left out one of the hexagon axioms and included a 'compatibility with the unit object' law which follows from the usual definition.

I changed it to the usual definition.

If the nonstandard definition is equivalent to the usual one, I'd love to know why! But I don't see how you get two hexagons from one, even given compatibility with the unit object.

(Of course for a symmetric monoidal category we just need one hexagon.)

I also beefed up the definition at symmetric monoidal category so the poor reader doesn't need to run back to braided monoidal, then monoidal.
• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeDec 20th 2009

I made a stub for pseudomonoid and referenced it from braided monoidal category.

• CommentRowNumber3.
• CommentAuthorTobyBartels
• CommentTimeDec 20th 2009

I put in a little stub at symmetric monoidal natural transformation. Really what we need now are braided monoidal functor and monoidal natural transformation. Luckily, the definitions are all in that little expository note of yours, John.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeDec 20th 2009
• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeDec 20th 2009
• (edited Dec 20th 2009)

Possibly. Sometimes it's nice to have something just explicitly say: there is nothing new here; Urs does that sort of thing sometimes. But if anybody puts in the definition explicitly, that should be moved to monoidal natural transformation.

I'd do all of this now, but I don't have time. Maybe after Christmas.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeDec 21st 2009
• (edited Dec 21st 2009)
This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> Sometimes it's nice to have something just explicitly say: there is nothing new here; </blockquote> <p>Yes, sometimes it's a difficult design decision. For instance whether to have <a href="http://ncatlab.org/nlab/show/domain+opfibration">domain opfibration</a> and <a href="http://ncatlab.org/nlab/show/codomain+fibration">codomain fibration</a> be separate entries.</p> <p>On the one hand it may be useful to have separate entries for separate terms. The danger is that we keep track of which information we keep where, and which is duplicated. We should in any case be sure to include a prominently featured remark along the lines "See also the closely relate entry xyz".</p> </div>
• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJun 1st 2016

I have edited the formatting of the Definition-section a little, such as to make it look more like a mathematical definition.

• CommentRowNumber8.
• CommentAuthorIngoBlechschmidt
• CommentTimeFeb 3rd 2018
• (edited Feb 3rd 2018)

Added to braided monoidal category an absolute minimum of two easy examples. We should mention more! The current list doesn’t give a well-rounded view. For anyone wanting to help, examples could, for instance, be lifted from the the first-mentioned reference (Joyal/Street).

1. Yes, certainly the ’canonical’ example (free braided monoidal category on one object) should be there (coproduct of the braid groups). No time at the moment unfortunately.

2. Added example of crossed G-sets. Will create a page which defines the category of these and the braided monoidal structure upon this category.