In this letter, we highlight the structure and main properties of an abstract approach to quantum cosmology and gravity, dubbed SU(∞)-QGR. Beginning from the concept of the Universe as an isolated quantum system, the main axiom of the model is the existence of an infinite number of mutually commuting observables. Consequently, the Hilbert space of the Universe represents SU(∞) symmetry. This Universe as a whole is static and topological. Nonetheless, quantum fluctuations induce local clustering in its quantum state and divide it into approximately isolated subsystems representing G×SU(∞), where G is a generic finite-rank internalsymmetry. Due to the global SU(∞) each subsystem is entangled to the rest of the Universe. In addition to parameters characterizing the representation of G, quantum states of subsystems depend on four continuous parameters: two of them characterize the representation of SU(∞), a dimensionful parameter arises from the possibility of comparing representations of SU(∞) by different subsystems, and the fourth parameter is a measurable used as time registered by an arbitrary subsystem chosen as a quantum clock. It introduces a relative dynamics for subsystems, formulated by a symmetry-invariant effective Lagrangian defined on the (3+1)D space of the continuous parameters. At lowest quantum order, the Lagrangian is a Yang–Mills field theory for both SU(∞) and internal symmetries. We identify the common SU(∞) symmetry and its interaction with gravity. Consequently, SU(∞)-QGR predicts a spin-1 mediator for quantum gravity (QGR). Apparently, this is in contradiction with classical gravity. Nonetheless, we show that an observer who is unable to detect the quantumness of gravity perceives its effect as curvature of the space of average values of the continuous parameters. We demonstrate Lorentzian geometry of this emergent classical spacetime.