Colored Khovanov homology theories at prime roots of unity

We equip Khovanov’s colored $\mathfrak{g}\mathfrak{l}(2)$-homology groups with a degree-2 derivation. Over a field of characteristic $p>2$, the derivation specializes to a $p$-nilpotent differential, which leads to a categorification of colored $\mathfrak{g}\mathfrak{l}(2)$-homology at a $2p$th root of unity.