Quantum geometry has emerged as a new framework that unites different disciplines and underpins novel transport and nonlinear phenomena in topological states of matter. In this colloquium, I will discuss an emerging system for topological studies: designable hybrid light-matter fluids. By integration of 2D material excitations — such as moiré excitons and Fermi polarons — with engineered photonic crystals, we create a platform where topology can be designed and dynamically controlled.