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

TL;DR

This paper introduces a higher representation theoretic structure to Ozsvath-Szabo's bordered knot Floer homology, revealing new algebraic compatibilities and connecting it to advanced categorification frameworks.

Contribution

It categorifies the intertwining property of local crossing bimodules in bordered knot Floer homology using 2-representations, linking it to higher representation theory.

Findings

Establishes a 1-morphism structure for crossing bimodules

Reformulates compatibility properties algebraically

Connects bordered knot Floer homology with higher representation theory

Abstract

We equip the basic local crossing bimodules in Ozsv\'ath-Szab\'o'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 Ozsv\'ath-Szab\'o's bimodules that is important when building their theory up from local crossings to more global tangles and knots.