I will describe a map that associates to every deformation of an object in a higher category a collection of generalized symmetries of the object. Building on work by Lurie, we will see that the failure of this map to be an equivalence can be quantified. Under favorable circumstances, the map is an equivalence, and this leads to an explicit description of the space of deformations in terms of solutions to certain equations. I will discuss applications of these results to topological field theory and holomorphic symplectic geometry. This talk is based on joint work with Bhanu Kiran.