  Let L be a null homologous link in $\mathbb{RP}^3$. We define Khovanov-type
homologies of L which depend on an extra input $\alpha = (V_0,V_1,f,g)$
consisting of two graded vectors spaces and two maps between them. With some
specific choice of $\alpha = \alpha_{APS}$, we recover the categorification of
the Kauffman bracket due to Asaeda-Przytycki-Sikora. With another choice of
$\alpha = \alpha_{HF}$, we construct a spectral sequence from our theory
converging to the Heegaard Floer homology of the even branched double cover of
$\mathbb{RP}^3$.
