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Stream: learning: questions

Topic: euler characteristic in braided monoidal categories

Dhruva Divate (Jul 07 2020 at 09:50):

Hello all, I write to ask if there is a well developed theory of dualisable objects and traces, like euler characteristic for symmetric monoidal categories, in braided monoidal categories.

Dhruva Divate (Jul 07 2020 at 09:50):

if not, are there "no-go" theorems for this

Oscar Cunningham (Jul 07 2020 at 09:59):

There's a thing called a ribbon category

Oscar Cunningham (Jul 07 2020 at 09:59):

https://ncatlab.org/nlab/show/ribbon+category

Oscar Cunningham (Jul 07 2020 at 10:05):

The idea is that in a braided monoidal category with duals you can draw string diagrams with the objects no longer being lines but rather being ribbons with some thickness. Then putting a loop in a ribbon is the same as introducing a twist

Oscar Cunningham (Jul 07 2020 at 10:05):

image.png

Oscar Cunningham (Jul 07 2020 at 10:06):

Image stolen from John Baez https://golem.ph.utexas.edu/category/2011/01/the_threefold_way_part_4_1.html

Simon Burton (Jul 07 2020 at 12:04):

Have you seen the paper by Joyal, Street and Verity called "Traced monoidal categories" ? I get lost in all the adjectives, but it does seem related to your question.

Dhruva Divate (Jul 08 2020 at 11:25):

Oscar Cunningham said:

There's a thing called a ribbon category

Thanks a lot for your response. I think if we have braided categories with duals, we can define such traces, so this should work. I will think about this!

Dhruva Divate (Jul 08 2020 at 11:28):

Simon Burton said:

Have you seen the paper by Joyal, Street and Verity called "Traced monoidal categories" ? I get lost in all the adjectives, but it does seem related to your question.

Thanks a lot for your response. I see that they're symmetric monoidal by definition, maybe as above, if we can work with braided monoidal categories admitting duals, we can ask of it to be traced as well.

John Baez (Jul 08 2020 at 18:53):

A braided monoidal category with duals is traced, I think.