Compatibility in Ozsvath-Szabo's bordered HFK via higher representations

Abstract

We 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 Rouquier and the second author, 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.

arXiv comment25 pages; 10 figures

References in corpus (7)

• Bimodules in bordered Heegaard Floer homology
• Homological mirror symmetry for higher dimensional pairs of pants
• On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology
• On the algebra of cornered Floer homology
• Cornered Heegaard Floer homology
• Strands algebras and Ozsváth-Szabó's Kauffman-states functor
• Ozsvath-Szabo bordered algebras and subquotients of category O