  The main goal of this paper is to introduce a framework for infinitesimal
deformation problems, using new methods coming from operadic calculus. We
construct an adjunction between infinitesimal deformation problems over some
type of algebras and their Koszul dual algebras, in any characteristic. This
adjunction is an equivalence if and only if some algebras are equivalent to
their completions. We give a concrete homological criterion for it. It gives us
a new proof of the celebrated Lurie--Pridham theorem, as well as of many other
generalizations of it. Our methods are effective, meaning they directly produce
point-set models for the algebras that encode infinitesimal deformation
problems.
