Weakly curved A-infinity algebras over a topological local ring

We define and study the derived categories of the first kind for curved DG and A-infinity algebras complete over a pro-Artinian local ring with the curvature elements divisible by the maximal ideal of the local ring. We develop the Koszul duality theory in this setting and deduce the generalizations of the conventional results about A-infinity modules to the weakly curved case. The formalism of contramodules and comodules over pro-Artinian topological rings is used throughout the paper. Our motivation comes from the Floer-Fukaya theory.