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Stream: theory: category theory

Topic: braiding as lax monoidal structure

Matteo Capucci (he/him) (Nov 28 2020 at 22:47):

Is anyone aware of a proof that a braiding for a monoidal category is equivalent to the lax monoidality of the 'identity' from $(M, I, \otimes) \to (M,I, \otimes^{op})$?
It seems so involve a lot of diagram chasing and I'm lazy

John Baez (Nov 28 2020 at 23:37):

Hmm, I know Joyal and Street's proof that a braided monoidal category is the same as a monoidal category in MonCat, but that's slightly different.

Matteo Capucci (he/him) (Nov 28 2020 at 23:43):

Is that an higher Eckmann-Hilton argument?

Matteo Capucci (he/him) (Nov 28 2020 at 23:47):

Matteo Capucci said:

Is anyone aware of a proof that a braiding for a monoidal category is equivalent to the lax monoidality of the 'identity' from $(M, I, \otimes) \to (M,I, \otimes^{op})$?
It seems so involve a lot of diagram chasing and I'm lazy

Btw, I don't know if this is actually true. My imagination got tickled by the fact that lax monoidal functors and braiding obey suspiciously similar hexagonal laws. However if you write down the two hexagons, they don't match on the nose.

Matteo Capucci (he/him) (Nov 28 2020 at 23:48):

It feels weird since how many ways could there be to arrange parenthesized triple products, associators and a natural transformation in two variables?

John Baez (Nov 29 2020 at 06:28):

Matteo Capucci said:

Is that a higher Eckmann-Hilton argument?

Yes, it's in here on page 12.