Talk:Braided monoidal category

Braided category

Anybody willing to fill in this redlink, either by an appropriate redirect, or a new article? Oleg Alexandrov (talk) 07:11, 23 April 2006 (UTC)

I retract my comments I made May 9, 2007 on Wikipedia's article on Braided Monoidal Categories. D.S.

Question about usage of opposite category here

I think the usage of ${\mathcal {C}}^{op}$ in the first sentence is confusing. In particular, it seems like ${\mathcal {C}}^{op}$ is being described as the same category as ${\mathcal {C}}$ but with a different (opposite) monoidal structure, rather than a monoidal structure on the opposite category of ${\mathcal {C}}$ as described in Opposite (category theory), which is the much more customary definition of ${\mathcal {C}}^{op}$.

So, given a monoidal category, $({\mathcal {C}},\otimes )$, are we saying that $({\mathcal {C}},\otimes )^{op}$ is of the form $({\mathcal {C}}^{op},\otimes ^{op})$ or is it of the form $({\mathcal {C}},\otimes ^{op})$?

A "natural isomorphism" would mean between two functors between the same categories, so it seems like this would mean that you'd want $\otimes ^{op}:{\mathcal {C}}\times {\mathcal {C}}\rightarrow {\mathcal {C}}$ and a natural isomorphism between $\otimes$ and $\otimes ^{op}$.

Thomaso (talk) 18:03, 10 February 2012 (UTC)

I agree. I'm trying to work out if a few different characterisations are equal and the description given in the opening line is not exactly correct. The differences are subtle but it's best to be as exact as possible. I believe the natural isomorphism is from $\otimes$ to $\otimes ^{op}$ as you say.

MitchB (talk) 04:04, 12 June 2013 (UTC)