  Using techniques due to Dwyer-Greenlees-Iyengar we construct weight
structures in triangulated categories generated by compact objects. We apply
our result to show that, for a dg category whose homology vanishes in negative
degrees and is semi-simple in degree 0, each simple module over the homology
lifts to a dg module which is unique up to isomorphism in the derived category.
This allows us, in certain situations, to deduce the existence of a canonical
t-structure on the perfect derived category of a dg algebra. From this, we can
obtain a bijection between hearts of t-structures and sets of so-called
simple-minded objects for some dg algebras (including Ginzburg algebras
associated to quivers with potentials). In three appendices, we elucidate the
relation between Milnor colimits and homotopy colimits and clarify the
construction of t-structures from sets of compact objects in triangulated
categories as well as the construction of a canonical weight structure on the
unbonded derived category of a non positive dg category.
