Abstract: To each compact, orientable surface S whose connected components have non-empty boundary, we define a dg category whose objects are neatly embedded 1-manifolds in S. In the case when S is a disk, this recovers the so-called “Bar-Natan category”: the natural setting for Khovanov’s celebrated categorification of the Jones polynomial. We expect that these dg categories form part of the 2-dimensional layer of a categorified version of the Turaev–Viro TQFT associated to the quantum group U_q(sl(2)). In particular, the (twisted) Hom-pairing on our dg categories recovers the canonical bilinear form on the state spaces of the Turaev–Viro theory. (This is joint work with M. Hogancamp and P. Wedrich.)