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I’m looking for references on the structure which can be roughtly described as follows: given a (braided? symmetric?) monoidal category $C$, I want to consider a simplicial set $N(\mathbf{B}C)$ with a single vertex, an edge for every object of $C$, a triangle with edges $X,Y,Z$ for every morphism $phi:Z\to X\otimes Y$, a tethraedron for every four triangles making up a commutative diagram involving the associator of $C$, higher coherences..
Any suggestion? thanks
There is a long description of this in http://www.tac.mta.ca/tac/volumes/9/n10/9-10abs.html. The point is to note that you are looking at $C$ as a bicategory, then the Duskin nerve in that case is very fully studied. There are other ways some of which are mentioned the lab entry: http://ncatlab.org/nlab/show/nerve#nerve_of_a_2category_19.
Generally nerves of omega cats are useful here.
If your $C$ is strict then it corresponds to a 2-cat and hence a S-cat (simplicially enriched cat) and you can use the homotopy coherent nerve construction. That can be useful as a means of thinking about interpretations of the maps. There is some discussion on various aspects of this somewhere in the Menagerie. For instance simplicial morphisms between these things corresponds to lax functors, etc. There are some references there, but if Duskin does not give you what you need get back to me and I will dig some more exact links out!
Thanks Tim.
I’m interested in this kind of structure since it seems to lie hidden in the definition of Turaev-Viro 3-manifold invariants.
In what way? In the stuff I did on TQFTs I was using crossed modules and those are (small) categorical groups so I have always felt that the same stuff should generalise to nice enough monoidal categories. I did not know enough about Tureav-Viro at the time and then got diverted by Turaev’s HQFTs. (I find that Turaev sometimes writes clearly and beautifully and on other occasions I do not see why he is emphasising some point that to me seems more or less trivial.)
Don’t know exactly, I just happened to meet the definition of Turaev-Viro inavriants yesterday.. :)
But there it seems one has a triangulation of a 3-manifold and decorates it with an irreducible representation of $SU(2)$ (or rather, a quantum version of this) on edges, in such a way that the 2-simplices one obtains are “admissible” (which seems to amount to saying that, if one calls $V_i,V_j$ and $V_k$ the three representations involved, then there is a nontrivial morphism $V_k\to V_i\otimes V_j$, i.e., $V_k$ appears in the decomposition of $V_i\otimes V_j$ in irreducible representations), and to 3-simplices are associated (quantum) 6-$j$ symbols. So I was trying to figure out whch could be a nice general setting for this construction.
Have a look at what I did for the TQFTs with a crossed module as ’coefficients’. That was based on Yetter’s earlier paper. A version of some of that was added to the Menagerie but is not in the version available on the nLab yet as I have not finished the TQFT section. In fact I have not worked on it for 4 months as my visit to Lyon pushed me into other equally interesting areas.
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