Braidings on skew monoidal categories

Steve Lack – 1 February 2017

I defined a notion of braiding for a skew monoidal category, generalizing the usual notion for monoidal categories, and involving isomorphisms (XA)B->(XB)A. In the case where the left unit maps IA->A are invertible, one can take X=I and so obtain an isomorphism AB-> BA, but in general this may not be possible.

One example is the category of categories with chosen finite limits, and functors which strictly preserve them. This has a skew closed structure for which the internal hom [A,B] consists of the functors which preserve finite limits in the usual sense. There is a corresponding tensor product, and this the resulting skew monoidal category has a braiding in our sense. There are many related examples where the structure of finite limits is replaced by some other "commutative" structure.

Another example involves bialgebras. Given a bialgebra B, there is an induced skew monoidal structure on Vect, with the tensor product of V and W given by the usual tensor product VBW. Braidings on this skew monoidal category correspond to quasitriangular structures on B.

This is joint work with John Bourke.