Abstract: Very often in homotopy theory, one passes from one infinity category to another by stabilizing with respect to some set of objects. For example one stabilizes with respect to the ordinary sphere in ordinary spaces, the representation spheres in equivariant spaces, and the Tate sphere in motivic spaces. Given an object S in a symmetric monoidal infinity category C, I will introduce the infinity category of S-symmetric spectrum objects in C and explain how this gives the universal category which is stable with respect to S. This description has many technical advantages. To illustrate this I will end with an application to proper equivariant spectra.