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I'm looking for references on the structure which can be roughtly described as follows: given a (braided or 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 $\varphi:Z\to X\otimes Y$, a tethraedron for every four triangles making up a commutative diagram involving the associator of $C$, higher coherences..

Any suggestion? thanks

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3 Answers 3

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If you want to capture the structure of the category together with its monoidal structure, you may need a $k$-fold simplicial set for $k>1$, i.e., a functor from $(\Delta^{op})^k$ to sets. One of the simplicial coordinates encodes the composition law in the category, another encodes the monoidal structure, and the rest decribe compatibility between monoidal structures (if the monoidal structure is braided or symmetric). See also Double nerve. You may want to look up work by Baez and Dolan on their periodic table that expresses monoidal categories of various types as higher categories with connectedness properties. In particular, there is an equivalence between monoidal categories and 2-categories with one object, and an equivalence between braided monoidal categories and 3-categories with one object and one 1-morphism.

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    $\begingroup$ Thanks. I'm currently interested in this since when the braided category is the category $Rep(SU(2))$ of finite dimensional representations of $SU(2)$ (or, rather, a quantum verion of it), such a nerve seems to lie behind Turaev-Viro invariants of 3-manifolds. $\endgroup$
    – domenico fiorenza
    Aug 10, 2010 at 15:38
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For plain old monoidal categories, you could regard them as a bicategory with a single object and use the Duskin nerve. For braided or symmetric categories there might be higher nerves you can take, but I'm not sure how well those work. You might be better off using the k-fold simplicial sets that Scott suggested.

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Generalising Omar's answer, braided monoidal categories are (a special case of) tricategories with one 1-morphism. Symmetric monoidal categories are tetracategories (whatever that is) with one 2-morphism. Writing these correspondences out explicitly, you should be able to work out the $\omega$-nerve of the underlying $\omega$-category (Ross Street's work with orientals). Actually, following Street's approach, an $\omega$-category already is a simplicial set and one would expect that you can expect a tricategory to be just an $\omega$-category where all the $n$-morphisms are identities above 3.

I believe the Duskin nerve is just taking a bicategory as an $\omega$-category following Street's approach and then considering its nerve.

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