Spectrification of Khovanov’s arc algebra, introduced by Lawson, Lipshitz, and Sarkar, has provided a powerful framework for studying link homology at the spectrum level, offering refinements that are strictly stronger than classical invariants. In this talk, we present how this spectrification naturally leads to 2-representations of categorified quantum groups over F_2, which we call spectral 2-representations. These take values in the homotopy category of spectral bimodules, opening the door to a spectrum-level understanding of higher representation theory.
A technical innovation in our work is using a set of canonical cobordisms that we call frames to spectrify algebras. As a step towards extending these spectral 2-representations to integer coefficients, we also work in the gl_2 setting and lift the Blanchet–Khovanov algebra to a multifunctor into a multicategory version of Sarkar–Scaduto–Stoffregen’s signed Burnside category. We also discuss future directions on the integer lifting of these spectral 2-representations.
