Abstract: In this talk I will compare several approaches to obtaining a deformation-obstruction calculus for diagrams of associative algebras. As main application, I will explain how this can be used to give an explicit description of deformations of the Abelian category of coherent sheaves on any separated Noetherian scheme, combining classical (commutative) deformations of schemes with quantizations of algebraic Poisson structures. I will also mention sufficient criteria for passing from formal to algebraic deformations. This talk is based on https://arxiv.org/abs/1806.05142 joint with Yaël Frégier and https://arxiv.org/abs/2002.10001 and https://arxiv.org/abs/2107.07490 joint with Zhengfang Wang.