  We generalize Illusie's definition of the Atiyah class to complexes with
quasi-coherent cohomology on arbitrary algebraic stacks. We show that this
gives a global obstruction theory for moduli stacks of complexes in algebraic
geometry without derived methods. We give a similar generalization of the
reduced Atiyah class, and show various useful properties for working with
Atiyah classes, such as compatibilities between the reduced and ordinary Atiyah
class, and compatibility with tensor products and determinants.
