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Stream: learning: questions

Topic: braiding and symmetry

Asad Saeeduddin (Apr 18 2020 at 14:21):

Is it correct to say that a symmetric monoidal category is a braided monoidal category whose braiding is an involution?

James Wood (Apr 18 2020 at 14:23):

nLab seems to say exactly this: https://ncatlab.org/nlab/show/symmetric+monoidal+category#definition

Reid Barton (Apr 18 2020 at 14:23):

https://en.wikipedia.org/wiki/Braided_monoidal_category#Symmetric_monoidal_categories

Reid Barton (Apr 18 2020 at 14:24):

It's not literally an involution (except possibly in some parameterized sense) because the indices are different.

Reid Barton (Apr 18 2020 at 14:24):

Otherwise, yes.

John Baez (Apr 18 2020 at 16:56):

Due to what Reid points out, I wouldn't say the symmetry is an involution unless I clarified that right away. Might as well just say

$B_{y,x} B_{x,y} = 1_{x \otimes y}$

Joe Moeller (Apr 19 2020 at 01:46):

The components might not be involutions, but maybe the natural transformation itself is an involution, no?

sarahzrf (Apr 19 2020 at 01:47):

no, it can't be—it's not an endomorphism

sarahzrf (Apr 19 2020 at 01:48):

it's a transformation from ⊗ to ⊗ ∘ τ, where τ is the swapping functor C × C → C × C

sarahzrf (Apr 19 2020 at 01:49):

to be precise, symmetry says that braiding's inverse is not itself but its own whiskering by τ

sarahzrf (Apr 19 2020 at 01:50):

which is well-typed since τ is an involution

John Baez (Apr 19 2020 at 02:14):

So the symmetry is a kind of "quasi-involution" that's riding the involution $\tau$.

John Baez (Apr 19 2020 at 02:15):

Here "riding" is one of Jim Dolan's terms for a certain kind of microcosm principle thing...

John Baez (Apr 19 2020 at 02:15):

For example if you have 2 monoids in two different monoidal categories, you can have a morphism from one monoid to another riding a monoidal functor from one monoidal category to the other.

sarahzrf (Apr 19 2020 at 02:25):

oooh

sarahzrf (Apr 19 2020 at 02:25):

i like that

sarahzrf (Apr 19 2020 at 02:25):

ive definitely noticed that kind of thing before :eyes:

sarahzrf (Apr 19 2020 at 02:25):

nice to have a name for it

John Baez (Apr 19 2020 at 03:11):

Yes, I like it.

John Baez (Apr 19 2020 at 03:12):

It hasn't really caught on, but we should just start using it - it's pretty clear.