Topological equivalences for differential graded algebras

We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an Eilenberg–Mac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasi-isomorphic dgas are topologically equivalent, but we produce explicit counterexamples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting.