2016/12/30 by Rina Anno, Anno, Rina, Timothy Logvinenko +1
Mathematics · #14F05 #18D20 #18E30 #18G99 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1612.09530
openalex publication_date 2016/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 → FRFR → FR → 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.