Deformation theory of objects in homotopy and derived categories I: General theory

This is the first paper in a series. We develop a general deformation theory of objects in homotopy and derived categories of DG categories. Namely, for a DG module E over a DG category we define four deformation functors Defh(E), coDefh(E), Def(E), coDef(E). The first two functors describe the deformations (and co-deformations) of E in the homotopy category, and the last two – in the derived category. We study their properties and relations. These functors are defined on the category of artinian (not necessarily commutative) DG algebras.