Compatibility in Ozsváth-Szabó's bordered HFK via higher representations (Q6105387)

Ozsváth-Szabó's theory of bordered knot Floer homology has proven to be highly efficient for computations. W. Chang and A. Manion equip the basic local crossing bimodules in Ozsváth-Szabó's theory of bordered knot Floer homology with the structure of 1-morphisms of 2-representations, categorifying the \(U_q(\mathfrak{gl}(1|1)^+)\)-intertwining property of the corresponding maps between ordinary representations. Besides yielding a new connection between bordered knot Floer homology and higher representation theory in line with work of \textit{R. Rouquier} and the second author [``Higher representations and cornered Heegaard Floer homology'', Preprint, \url{arXiv:2009.09627}], this structure gives an algebraic reformulation of a ``compatibility between summands'' property for Ozsváth-Szabó's bimodules that is important when building their theory up from local crossings to more global tangles and knots.