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I'd like to clarify why I'm interested in nerves (and consequently, why I'd really prefer that my nerve be a simplicial set instead of something more exotic, unless I can be convinced that more exotic objects can be easily adapted to my needs).

My poking around in all of this was inspired by the preprint by Etingof, Nikshych, and Ostrik, "Fusion categories and homotopy theory." The main results of this paper are proved by formulating their questions in terms of classical obstruction theory on the nerves of certain 3-groupoids. The obstruction theory itself can be justified using elementary fiddling with simplicial sets, as the reference Gregory Arone provided to my earlier question on obstrucion theory reveals. However, I wanted to understand the category theory side of the equation better, which led me to try to formulate things in terms of unbiased monoidal 2-categories.

So ultimately, the goal is to have a definition of the nerve to which I can apply my favorite classical obstruction theory techniques. While some people appear to have studied obstruction theory in more general settings, it's not clear to me how to squeeze out the appropriate concrete computational gadgets (e.g., the cohomology groups $H^n(X; \pi_{n - 1}(Y))$) from the relevant abstract nonsense. Of course, if somebody could elucidate how that works, that would be wonderful, although perhaps that should be the subject of another question...

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# What is the right way to define the nerve of an unbiased monoidal category?

I've been toying around with unbiased composition in higher categorical structures on and off for a while now. In particular, I've been playing around with unbiased monoidal 2-categories. One motivation for this, as I discussed in my last question on the matter, is that unbiased tensor products and compositions often seem to be better descriptions of what goes on "in nature" than biased ones.

Another motivation was the hope that such gadgets would provide a cleaner notion of nerve than what one gets in the biased setting, where higher associators are floating around everywhere. However, directly transcribing the ordinary notion of nerve seems to work poorly, even for unbiased monoidal categories, for two reasons. First, in each dimension, one is forced to consider products of fixed numbers of objects, which is antithetical to the unbiased philosophy. Secondly, degeneracies are difficult to write down because one has, in place of a unit object, a zero-fold tensor product, which requires a bit of care to handle. A more natural "nerve" for an unbiased monoidal categories might involve having simplicies of dimension $n$ correspond to nested tensor products of depth $n$. I can't quite get such a definition to work, although I'm pretty sure that something like it should be possible.

Is there a construction of the nerve of an unbiased monoidal category that is natural to write down? (The definition of unbiased monoidal category can be found in section 3.1 of Leinster's Higher Operads, Higher Categories.) It strikes me that the problem might be simplicial sets themselves; are there some more exotic combinatorial objects that are better suited to capturing unbiased compositions? I'm aware of the existence of things like opetopes, but I have no idea if they're relevant to this particular issue.