Bordered invariants from Khovanov homology

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.