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Stream: learning: questions

Topic: monoidal infinity-categories

Nathanael Arkor (Apr 05 2020 at 01:38):

I have a slightly tongue-in-cheek question: if a monoidal $n$-category is a one-object (weak) $(n + 1)$-category, then is it reasonable to define a "monoidal $\omega$-category" as a one-object $\omega$-category?

Fabrizio Genovese (Apr 05 2020 at 01:41):

That would actually be the most amazing example of "getting more with less"

John Baez (Apr 05 2020 at 01:46):

Yes, this is a fine definition of a monoidal $\omega$-category.

John Baez (Apr 05 2020 at 01:48):

No joke. Of course the details depend on what definition of $\omega$-category you use.

Nathanael Arkor (Apr 05 2020 at 01:53):

people seem to give infinity category theory a hard time for being very difficult, but this is far simpler than even the 1-categorical analogue — I don't see what all the complaints are about :wink:

Thibaut Benjamin (Apr 06 2020 at 11:08):

I have a preprint on my website, to propose a type theoretical definition of monoidal weak $\omega$-category. I was suppposed to make a running implementation of it, so that you could actually manipulate the terms and play around with them to see what actually happens there. The basic trick for this theory was to take the definition of weak $\omega$ category that I use, and impose it to have only one object, then shift every dimension by one, and describe the resulting theory. Rest of the paper is dedicated to showing correctness, assuming correctness for the theory weak $\omega$-category, and it probably isn't the most intesting regarding your question.
A few remarks on that though :

• I have a running implementation of a tool for weak $\omega$-category. Everything that you can define this that tool that sits above only one object is something you can define for monoidal weak $\omega$-categories (that's essentially what the proof of correctness of my paper says), and the proof that you define in any of these two theories would be essentially the same.
• This can be generalized to more coherent monoidal categories. For example, braided monoidal $\omega$-categories, really are monoidal $\omega$-categories with only one object, and you could derive the same way I did a theory for them. However, proving the correctness of this theory would be significantly more difficult, and it would require some strictification. In practice, this means that working with the theory for $\omega$-categories to prove anything about braided monoidal $\omega$-categories would be a big waste of time, as opposed to working directly in the theory of braided monoidal categories.
  My paper is available here : http://www.lix.polytechnique.fr/~tbenjamin/monoidal.pdf

Thibaut Benjamin (Apr 06 2020 at 11:09):

Also if you are interested in using my implementation for weak $omega$-categories, you can let me know, and I can show you some basics of what you can define and show with them