We discuss the geometrization of entire evolution histories by means of a single, (d+1)-dimensional manifold furnished with a fixed (pseudo-) Riemannian structure. We propose a universal form of the higher-dimensional metric and discuss its properties. The non-degeneracy of the higher-dimensional metric is linked to a monotonicity requirement for the running of the cosmological constant, which we test in the case of Asymptotic Safety. Furthermore, we allow the higher-dimensional manifold to be an arbitrary Einstein space, admitting a Lorentzian signature, a prime example being a stack of de Sitter spaces. We solve the corresponding functional RG and the effective Einstein equations, and we embed the 4D metrics into a single 5D one. We show that if the scale invariance of the fixed points extends to full conformal invariance, the 5D picture of the resulting geometric and field theoretic structure displays a novel kind of AdS/CFT correspondence. While strongly reminiscent of the usual string theory-based correspondence, also differences are discussed. This seminar is based on 2103.15709 and 2205.12030.