We use unimodular ribbon categories to construct quantum invariants of ribbon surfaces in 4-dimensional 2-handlebodies up to 1-isotopy. In the process, we recover invariants due to Bobtcheva-Messia, Broda-Petit, Gainutdinov-Geer-Patureau-Runkel (in collaboration with the second author), and Lee-Yetter. Our approach does not assume semisimplicity, and is based on a generalization of the Reshetikhin-Turaev functor to the category of labeled Kirby graphs which also yields invariants of framed links in the boundary of 4-dimensional 2-handlebodies up to 2-deformations. The setup is very flexible, and allows for several different constructions, using central elements satisfying equations introduced by Hennings and Bobtcheva-Messia, modified traces, and modules over Frobenius algebras satisfying conditions dictated by the diagrammatic calculus for embedded surfaces developed by Hughes, Kim, and Miller.
