Homotopical finiteness of smooth and proper dg-algebras
  We show that any smooth and proper dg-algebra (over some base ring k) is determined, up to quasi-isomorphism, by its underlying A_n-algebra, for a certain integer n. Similarly, any morphism between two smooth and proper dg-algebras is determined, up to homotopy, by the morphism induced on the underlying A_n-algebras, for a certain integer n. When the base ring k is local, we show that the integer n can be chosen uniformally for all smooth and proper dg-algebras for which two numerical invariants (the "type" and the "cohomogical dimension") are bounded. 
B. Toen