We introduce the notion of a braiding on a skew monoidal category, whose curious feature is that the defining isomorphisms involve three objects rather than two. Examples are shown to arise from 2-category theory and from bialgebras. In order to describe the 2-categorical examples, we take a multicategorical approach. We explain how certain braided skew monoidal structures in the 2-categorical setting give rise to braided monoidal bicategories. For the bialgebraic examples, we show that, for a skew monoidal category arising from a bialgebra, braidings on the skew monoidal category are in bijection with cobraidings (also known as coquasitriangular structures) on the bialgebra.
Keywords: Braiding, skew monoidal category, bialgebra, quasitriangular, 2-category
2010 MSC: 18M50, 18M15, 18N10, 18N40, 16T10
Theory and Applications of Categories, Vol. 35, 2020, No. 2, pp 19-63.