In this work we consider the question of realizing triangulated
dg-categories by derived categories of algebraic varieties. For this, we
introduce the notion of "system of points" in saturated dg-categories. We
show that given such a system on a dg-category T, we can construct an
algebraic space M, of finite type, smooth and separated, together with a
dg-functor from T to a certain twisted dg-category of sheaves on M. We
prove that this functor is furthermore an equivalence if and only if M is
proper. We also study t-strutcures on algebraic families of
objects in T, which might be of independent interest.