Homotopical resolutions associated to deformable adjunctions
  Given an adjunction connecting reasonable categories with weak equivalences, we define a new derived bar and cobar construction associated to the adjunction. This yields homotopical models of the completion and cocompletion associated to the monad and comonad of the adjunction. We discuss applications of these resolutions to spectral sequences for derived completions and Goodwillie calculus in general model categories. 
Andrew J. Blumberg, Emily Riehl
math.AT math.CT