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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.


EDIT:

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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1 Answer 1

Monoidal categories are a special kind of (coloured) operad. It is this viewpoint in which the unbiased version is most natural. In light of this, it would be most natural to consider the nerve of the associated operad. So, you're on to something in your suspecting that simplicial sets are not the most natural choice for capturing the nerve of a monoidal category. If your monoidal category is symmetric, then its nerve can be taken as a dendroidal set (otherwise, you it should be taken as a planar dendroidal set). Dendroidal sets were invented (discovered?) by Ieke Moerdijk and Ittay Weiss. Roughly speaking, dendroidal sets are to operads as simplicial sets are to categories. Like simplicial sets, they are presheaves on a certain test-category. Simplicial sets are $Set^{\Delta^{op}}$ where $\Delta$ consists of finite linear orders. Dendroidal sets are $Set^{\Omega^{op}}$, where $\Omega$ consists of finite rooted trees. One can even extend Joyal's model structure for quasicategories to dendroidal sets, and start talking about infinity-operads.

In fact, there is some very general machinery developed by Mark Weber which automatically produces the "correct" combinatorial objects to use to take a nerve of a certain type of algebraic object (given as an algebra for a monad). See: http://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html . Basically, given a monad $T$, this machinery produces a category $\Theta(T)$, consisting of certain "linear-like" free $T$-algebras, which has $Set^{\Theta(T)}$ some how the canonical choice for taking nerves of $T$-algebras. So, you COULD plug in the free-unbiased-monoidal category monad into this machinery and see what you get out, and use THIS to take the nerve. However, I think this is a bit over-kill. Using dendroidal sets should do just fine. However, it is worth mentioning that Cisinski has extended Joyal's model structure in this setting as well.

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David, could you give a reference for the paper of Cisinski? –  Harry Gindi Jul 21 '10 at 4:55
    
Thank you for the suggestion; I will definitely look into dendroidal sets. As I mention in my edit, trying to make use of more exotic structures instead of simplicial sets may be taking me further away from my ultimate goal, but it does seem that there is a very interesting "big picture" that this all fits into, as your link describing Weber's work indicates. –  Evan Jenkins Jul 21 '10 at 4:59
    
@Harry: I'm not sure. I saw him give a talk about this at MIT last summer. But, he's on MO. Try asking him directly :-) –  David Carchedi Jul 21 '10 at 5:32
    
@Evan, P.S., I will be at U Chicago from the 8th to the 13th of August if you want to discuss this some more. –  David Carchedi Jul 21 '10 at 17:16
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