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.

中文翻译：

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张量函子的模和同伦附属的条形类别

给定 DG 类别 A，我们引入模块 Modbar(A) 的条形类别。它是 A 的派生范畴 D(A) 的 DG 增强，它同构于在它们之间具有 A 无穷态射的 DG A 模的范畴。然而，它本质上是用 DG 类别的语言定义的，不需要复杂的机器或 A 无穷大类别的符号约定。我们为这些条形定义了 Tensor 和 Hom 双函子、二元化函子和扭曲复合物的卷积。预期的应用是使用 DG-bimodules 作为三角化类别之间精确函子的增强。作为演示，我们为 DG 类别的派生类别之间的张量函子开发了同伦附属理论。