Two ways to make a braiding into a transfor
Sjoerd Crans – 7 January 1998
Setting: the setting is of semistrict higher-dimensional algebra. In dimension 3 this means Gray-categories, in dimension 4 this means 4-dimensional teisi, defined as (Gray-Cat)-categories, for the monoidal structure on Gray-Cat as in "A tensor product for Gray-categories".
A q-transfor is a functor C tensor 2_q -> D.
A k-monoidal (n-dimensional) tas is a ((k+n)-dimensional) tas with one (k-1)-arrow (and hence with one i-arrow for every i Sigma (C). Fixing an object A of C gives a transfor R_{A,-}: Sigma (C) tensor 2_1 -> Sigma (C). I indicated how, for braided 2-categories, this induces a transfor A tensor - -> - tensor A: C -> C. Similarly, an arrow f of C gives a 2-transfor R_{f,-}: Sigma (C) tensor 2_2 -> Sigma (C), which again induces a 2-transfor C tensor 2_1 -> C. There are problems in generalizing this to braidings on higher-dimensional teisi, however.
Another way: by definition, a braiding is a functor R: Sigma (C) tensor Sigma (C) -> Sigma (C). Does it induce a transfor C tensor C -> C? It does for braided 2-categories, but not trivially. In higher dimensions, the question becomes whether there is a canonical relation between Omega (2_p tensor 2_q) (where Omega should be some extension of the above Omega to diagrams) and 2_{p-1} tensor 2_{q-1} tensor 2_1, which seems to me an interesting question in its own, topological, right.
Conclusion: the usual definition of a braiding (e.g. as in Generalized centers of braided and sylleptic monoidal 2-categories) indeed corresponds to the definition from the teisi point of view. However, the latter definition should be regarded as more fundamental, and is indicated in higher dimensions.
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