  To every compact oriented surface that is composed entirely out of
2-dimensional 0- and 1-handles, we construct a dg category using structures
arising in Khovanov homology. These dg categories form part of the
2-dimensional layer (a.k.a. modular functor) of a categorified version of the
sl(2) Turaev--Viro topological field theory. As a byproduct, we obtain a
unified perspective on several hitherto disparate constructions in categorified
quantum topology, including the Rozansky--Willis invariants,
Asaeda--Przytycki--Sikora homologies for links in thickened surfaces,
categorified Jones--Wenzl projectors and associated spin networks, and dg
horizontal traces.
