I will report on joint work with D. Jordan, I. Le and A. Shapiro in which we construct categorical invariants of decorated surfaces using the stratified factorization homology of Ayala, Francis and Tanaka, together with the representation theory of quantum groups. The categories we obtain can be regarded as `quantizations' of the categories of quasicoherent sheaves on the stacks of decorated local systems on surfaces, and satisfy strong functoriality and locality properties reminiscent of those of a TQFT. I will give an overview of their construction, and explain how to recover Fock-Goncharov-Shen's cluster quantizations of related moduli spaces within this framework.