Lipshitz-Sarkar defined a stable homotopy type refining Khovanov homology, producing cohomology operations $\text{ Sq}^i$ on the Khovanov homology $Kh(L)$ of a link $L$. Later, Mor\'an proposed a sequence of cup-i products on the $\mathbb{ F}_2$-coefficient cochain complex of any augmented semi-simplicial object in the Burnside category. Applied to the Khovanov functor, he obtained another sequence of operations $\mathfrak{ sq}^n$ on $Kh(L)$, where $\mathfrak{ sq}^0$, $\mathfrak{ sq}^1$ agree with the usual Steenrod squares. We prove that Lipshitz-Sarkar's $\text{ Sq}^2$, the first Steenrod operation that cannot be computed from merely homological data, agrees with Mor\'an's $\mathfrak{ sq}^2$.
Categories: math.GT math.AT