Lindbladians, one of the simplest extensions of Hamiltonian-based quantum mechanics, are used to describe “drainage” (i.e., decay) and decoherence of a quantum system induced by the system's environment. While traditionally viewed as detrimental to fragile quantum properties, a tunable environment offers the ability to drive the system toward exotic phases of matter, which may be difficult to stabilize in nature, or toward protected subspaces, which can be used to store and process quantum information. An important property of Lindbladians is their behavior in the limit of infinite time, and in this talk I will discuss a formula for the map corresponding to infinite-time Lindbladian evolution. This formula allows us to determine to what extent decay affects a system's linear or adiabatic response. It also allows us to determine geometrical structures (holonomy, curvature, and metric) associated with adiabatically deformed steady-state subspaces.