Ancestors

Written by Trebor on 2024-12-27 at 14:15

This periodic table thing is a scam. The correct thing to do is to take iterated (homotopy-coherent) monoids, so Mon(Mon(Cat)) yields the correct bicategory of braided categories. A 4-category with one object and one morphism happens to provide two associative products compatible with each other, but it makes no guarantee about not having any extra structure or property. And it's just lucky that for small cases no such things are introduced, just like how it's lucky that bicategories can all be made strict.

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Toot

Written by Trebor on 2024-12-27 at 14:16

To be fair, it's interesting in its own right to investigate exactly how much extra structure is there. For example, a tricategory with one object, one morphism and one 2-morphism is a commutative monoid with some extra elements being selected, and all functors will preserve those elements!

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Descendants

Written by Naïm Camille Favier on 2024-12-27 at 14:29

@trebor Where do the extra elements come from?

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Written by Trebor on 2024-12-27 at 14:35

@ncf after you prove the third monoidal product coincides with the first two, the associators and unitors etc are actually left over unused and uneliminated (unlike the situation in eg. half adjoint equivalences where you can add two levels of data which bundles up into a contractible type and disappears), so in the final form you just get an invertible element in that monoid

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