Multicategories vs Categories

Question

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?

Answer 1

Multicategories and bicategories, to me, are first of all completely orthogonal generalisations of monoidal categories, with virtual double categories as a common generalisation of multicategories and (strict) $2$-categories (they are to multicategories as categories are to monoids, or $2$-categories to monoidal categories). As for generalising simply categories, they are even more different, as multicategories are still a strictly associative structure while coherence issues appear for bicategories.

So, to your second highlighted question, I would say that the answer is no. A way to more precisely understand the difference, and a positive answer to your first question, lies in your final (bonus) question.

In addition to the aforementioned algebraic aspect, double categories have a large conceptual advantage over bicategories for formal category theory; in short, if you try to treat the objects of an arbitrary $2$-category as abstract categories, you will be lacking a lot of elements (the Yoneda structure coming from profunctors a.k.a. bimodules) to speak about limits in them, while double categories (at least the ones equipping their vertical category with proarrows) will give you enough. Virtual double categories are just the relevant generalisation for when bimodules do not compose, and the augmented version deals with the case lacking identity bimodules (e.g. for non-locally small categories). This is, in my opinion, the main conceptual payoff for (augmented) virtual double categories.

To finish, two technical generalisations, the latter of which was your second-and-a-half question:

• Virtual double categories (though not the augmented ones, as far as I know), are an example of "generalised multicategories", something that can be defined relative to any monad acting on a virtual proarrow equipment. For this, see Leinster's book suggested in varkor's comment (chapters 4 and 5) or the more general and recent A unified framework for generalized multicategories by Cruttwell and Shulman.

• There does exist a higher-categorical version of multicategories and even virtual double categories, defined (as "generalised non-symmetric $\infty$-operads") and put to great use (precisely to unify $\infty$-multicategories and double $\infty$-categories) in Gepner–Haugseng's Enriched $\infty$-categories via non-symmetric $\infty$-operads and follow-ups. A more general form, close to the generalised multicategories, is offered by Chu–Haugseng's formalism of algebraic patterns developed in Homotopy-coherent algebra via Segal conditions (see in particular section 9 with ex. 9.8). For the moment this is only used for algebraic aspects rather than formal (higher) category theory, but I would argue it is definitely worth learning (not over higher category theory, but as an extension of it).

Answer 2

Is there a way to recapture the additional understanding imparted by multicategories using higher categories?

Well I would say so. Multicategories are basically categories whose morphisms have multiple sources instead of only one. Bicategories generalize categories by adding 2-morphisms, i.e. relations among morphisms, this is completely different from adding multiple sources to the domains of the morphisms.

Is there a way to recapture the additional understanding imparted by multicategories using higher categories?

None that I'm aware of, but that should be expected since they provide different additions to categories (multiple sources vs morphisms among morphisms).

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.

Again not that I'm aware of.

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'm not really familiar with that, but far I haven't seen lots of work on the subject. Probably time will tell.

Hope this helps.