$(\infty,n)$-operads?

Question

I wonder whether there is (or should be) a theory of colored $(\infty,n)$-operads or multicategories?

We know that multicategories are generalizations of categories, and nonsymmetric colored $\infty$-operads are generalizations of $\infty$-categories. Given that we have theories of $(\infty,n)$-categories, it seems natural to expect analogues for nonsymmetric operads. I also wonder the same for (symmetric) colored operads.

This grew out of discussions under https://mathoverflow.net/a/407227, which leads me to suspicion of existence of "delooping" of colored $\infty$-operads, which should be $(\infty,2)$ if exists.

Answer 1

Higher operads are discussed in one form in Leinster's Higher Operads, Higher Categories.

Likewise, there is a notion of 2-operads as an analogue of $2$-categories. See for instance $(A_\infty,2)$-categories and relative $2$-operads. Along the same path, I've seen 2-dendroidal sets appear in this paper of Stefan Forcey.

Of course, these are not necessarily the kind of higher operads you are after.

At this stage, my understanding is that we still have a hell of a lot to learn about $(\infty,1)$-operads, and examples thereof take care of most applications. I'm not aware of any application for a theory $(\infty,n)$-operads in current research, but I would very much like to see one.