Spectral version of the \mathfrak{gl}_2-foam 2-category \mathfrak{F}

Construct a spectral enhancement of the \mathfrak{gl}_2-foam 2-category \mathfrak{F} in which each morphism category is promoted to a spectral category, providing spectra of 2-morphisms and enabling fully homotopy-coherent 2-representations.

Background

The 2-category \mathfrak{F} of \mathfrak{gl}_2-foams encodes morphism categories that recover Blanchet–Khovanov structures and are related to categorified quantum groups via foamation. For spectral 2-representations beyond homotopy categories, one would like morphism categories to carry spectra of 2-morphisms.

While the authors give signed Burnside lifts for Blanchet–Khovanov algebras and extend these to general morphism categories in \mathfrak{F}, they explicitly state that a spectral version of \mathfrak{F} is not yet available, highlighting a concrete missing construction required for richer spectral representation theory.