ScalingStacks

Remark 6.3. An analogous trigraded semistrict braided monoidal 2-category $\boldsymbol{\mathrm{KhR}}_{\infty}$ can be constructed from the triply-graded Khovanov–Rozansky homology, which categorifies the HOMFLY-PT polynomial. This uses the functoriality of Rouquier complexes in the homotopy categories of type A Soergel bimodules under braid cobordisms, which has been proven by [EK10b]. The 2-category $\boldsymbol{\mathrm{KhR}}_{\infty}$ admits vertical duals $*$, but it has no duality $\#$ with respect to its monoidal structure. It is an open problem to find a categorification of the HOMFLY-PT polynomial that allows the construction of a version of $\boldsymbol{\mathrm{KhR}}_{\infty}$ that admit duals, and beyond that an $SO(4)$-pivotal structure.