Bar category of modules and homotopy adjunction for tensor functors

Anno, Rina and Logvinenko, Timothy ORCID: https://orcid.org/0000-0001-5279-6977 2021. Bar category of modules and homotopy adjunction for tensor functors. International Mathematics Research Notices 2021 (2) , pp. 1353-1462. 10.1093/imrn/rnaa066

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Abstract

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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Oxford University Press
ISSN:	1073-7928
Date of First Compliant Deposit:	3 March 2020
Date of Acceptance:	3 March 2020
Last Modified:	07 Nov 2023 00:00
URI:	https://orca.cardiff.ac.uk/id/eprint/130090

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