Higher arity monoidal products Suppose we have a monoidal category $\mathbb{C}$ with monoidal product $\otimes.$ The $\otimes$ operation gives us binary monoidal products, so we would have to build up an expression for a product of three objects using brackets, with expressions like $A\otimes (B \otimes C).$ However, in some places, like this nlab page, people seem to write expressions like $A\otimes B \otimes C,$ excluding the brackets. This suggests to me that there is a way to define "higher arity monoidal product functors", which (for example) send $(A,B,C)$ to $A\otimes B \otimes C.$ I am wondering if this is so, and how such things are defined. I also wonder if there is related literature.
 A: There is the notion of unbiased monoidal category, which is a category equipped with $n$-ary products $\otimes_n:\mathcal C^n\to \mathcal C$ for every $n$ except $n=1,$ together with naturality isomorphism between $\otimes_{mn}$ and $\otimes_n\circ \otimes_m^n,$ the functor that multiplies together each row of an $n\times m$ matrix of objects and then multiplies the result. These satisfy coherence conditions that are more obvious than those for traditional monoidal categories: basically, if you want to use these isomorphisms to multiply together a $3D$ array of objects, either of the two ways to do it is the same. Abstractly, they are pseudo-algebras for the strict monoidal category monad on the $2$-category of categories, or else strict algebras for a related monad, or similarly, algebras for a certain fattening of the associative operad. Thus by either general or special strictification results, one proves that any unbiased monoidal category is equivalent to a strict one. Strict unbiased monoidal categories being obviously equivalent to traditional strict monoidal categories, this is one level of a proof that the 2-categories of biased and unbiased monoidal categories are equivalent.
Such definitions are particularly important to higher-categorical applications, where the signature of the desired generalization of a monoidal category is no longer finite (since there are further coherence conditions between different ways of filling the pentagon in that axiom, and so on ad infinitum.) Thus trying to give a traditional definition no longer has the advantage of concision while having a completely unmanageable disadvantage in uniformity of the definition.
Tom Leinster has a book that starts precisely with this concept. It was one of the very first books on higher category theory, all the way back in 2003!
