  For every oriented surface of finite type, we construct a functorial Khovanov
homology for links in a thickening of the surface, which takes values in a
categorification of the corresponding gl(2) skein module. The latter is a mild
refinement of the Kauffman bracket skein algebra, and its categorification is
constructed using a category of gl(2) foams that admits an interesting
non-negative grading. We expect that the natural algebra structure on the gl(2)
skein module can be categorified by a tensor product that makes the surface
link homology functor monoidal. We construct a candidate bifunctor on the
target category and conjecture that it extends to a monoidal structure. This
would give rise to a canonical basis of the associated gl(2) skein algebra and
verify an analogue of a positivity conjecture of Fock--Goncharov and Thurston.
We provide evidence towards the monoidality conjecture by checking several
instances of a categorified Frohman-Gelca formula for the skein algebra of the
torus. Finally, we recover a variant of the Asaeda--Przytycki--Sikora surface
link homologies and prove that surface embeddings give rise to spectral
sequences between them.
