One of the initial motivating factors for learning category theory, besides needing it for my work, was the idea that almost all mathematical notions I would encounter could be understood using categories one way or another.
That’s largely been borne out at the $1$-categorical level, and (almost?) completely vindicated at the $\infty$-categorical level, but I keep encountering statements about multicategories that make me feel like I might still be missing out on some ‘big picture’ understanding of the sort typically furnished by categories.
For a specific example I recently came across an MO question about direct sums/tensor products of vector spaces, and the answer by Qiaochu Yuan seemed to essentially assert that although we can understand what’s happening in terms of categories the most natural view is furnished by multicategories, and further the notion of a monoidal category is strictly generalized by the notion of a multicategory in a satisfying way. Monoidal categories are also generalized by bicategories in a satisfying way though, so my first question is:
Do multicategories generalize categories in a way that bicategories don’t?
This is kind if vague, but returning to the example above I would ask if we can understand the situation involving vector spaces using bicategories to clarify things instead of multicategories.
If the answer to the first highlighted question is yes,
Is there a way to recapture the additional understanding imparted by multicategories using higher categories?
If not, then I would ask if a theory of higher multicategories exists and if the additional work of learning it over higher category theory is worth the understanding payoff.
If the answer to the first highlighted question is no, I am happy to stick with higher categories for now — I have a bonus question though:
For those familiar with it, does the theory of augmented virtual double categories have any significant ‘big picture understanding’ advantages over the theory of bicategories? What about compared to higher categories?
I am immediately attracted to the fact that the collection of all large categories (not even locally small), functors and natural transformations form an augmented virtual double category (what a mouthful), but is there any other nice conceptual payoff for the leap from categories to augmented virtual double categories?