Higher Steenrod squares for Khovanov homology (Q2182257)

\textit{R. Lipshitz} and \textit{S. Sarkar} [J. Am. Math. Soc. 27, No. 4, 983--1042 (2014; Zbl 1345.57014)] constructed a stable homotopy type which refines Khovanov homology and thus allowed them to obtain Steenrod Squares on Khovanov homology. Their original construction involved framed flow categories, but together with \textit{T. Lawson} they gave another construction in [Contemp. Math. 684, 63--85 (2017; Zbl 1380.57006)], using a functor from a cube poset to a Burnside category. The current paper aims to describe the action of the Steenrod Algebra on Khovanov homology in terms of the Burnside category. To do this, the author considers symmetric multiplications on cochain complexes coming from simplicial sets. In particular, for a cochain complex of an ordered augmented semi-simplicial object in the Burnside category with coefficients in the field with two elements he shows the existence of a natural stable symmetric multiplication. The author continues to define Steenrod squares on Khovanov homology using this construction. That these Steenrod squares agree with the ones from Lipshitz-Sarkar is announced in ongoing work of the author.