Multifunctor from the signed Burnside multicategory to spectra

Develop a multifunctor from the multicategory version of the signed Burnside category \underline{\mathscr{B}_{\sigma}} to a symmetric monoidal category of spectra (such as symmetric spectra) with the Elmendorf–Mandell-type multiplicativity and multifunctorial properties required to spectrify Blanchet–Khovanov algebraic structures.

Background

The paper seeks to spectrify Blanchet–Khovanov algebras, which fix sign issues in the foam-based categorification related to Khovanov homology. Spectrification of Khovanov’s arc algebras and related tangle bimodules currently uses the Elmendorf–Mandell infinite loop space machine via a multifunctor from a shape multicategory into a Burnside multicategory. However, Blanchet–Khovanov structures involve signs and thus call for a signed Burnside framework.

The authors construct a multifunctor from shape multicategories into the signed Burnside multicategory \underline{\mathscr{B}{\sigma}} that recovers the Blanchet–Khovanov algebras after forgetting signs, but they note a key missing step: a suitable multifunctor from \underline{\mathscr{B}{\sigma}} into spectra with the necessary multiplicativity, analogous to Elmendorf–Mandell, to obtain spectral algebras. This is explicitly identified as unclear and left unaddressed in the paper.