Bar category of modules and homotopy adjunction for tensor functors
  Given a DG-category A we introduce the bar category of modules Modbar(A). It is a DG-enhancement of the derived category D(A) of A which is isomorphic to the category of DG A-modules with A-infinity morphisms between them. However, it is defined intrinsically in the language of DG-categories and requires no complex machinery or sign conventions of A-infinity categories. We define for these bar categories Tensor and Hom bifunctors, dualisation functors, and a convolution of twisted complexes. The intended application is to working with DG-bimodules as enhancements of exact functors between triangulated categories. As a demonstration we develop homotopy adjunction theory for tensor functors between derived categories of DG-categories. It allows us to show in an enhanced setting that given a functor F with left and right adjoints L and R the functorial complex $FR \rightarrow FRFR \rightarrow FR \rightarrow Id$ lifts to a canonical twisted complex whose convolution is the square of the spherical twist of F. We then write down four induced functorial Postnikov towers computing this convolution. 
Rina Anno and Timothy Logvinenko
math.CT math.AG