(joint with Kai Behrend) The invertible elements of a Banach algebra form a Lie group. There is a generalization of this observation to differential graded Banach algebras, in which the quasi-invertible elements form the morphisms of an n-groupoid in the category of analytic Banach varieties. I will explain how this observation forms the basis for a generalization of Kuranishi's work on deformation of holomorphic vector bundles to perfect complexes (bounded complexes of holomorphic vector bundles).