A-infinity-bimodules and Serre A-infinity-functors

TL;DR

This paper develops the theory of A-infinity-bimodules and Serre A-infinity-functors, establishing their equivalence to certain A-infinity-functors and characterizing when such functors exist for A-infinity-categories over a field.

Contribution

It introduces a new framework for A-infinity-bimodules and Serre A-infinity-functors, including a generalized Yoneda Lemma, and characterizes their existence in terms of homotopy categories.

Findings

A-infinity-bimodules are equivalent to A-infinity-functors with two arguments.

A unital A-infinity-category admits a Serre A-infinity-functor iff its homotopy category admits a Serre functor.

The paper generalizes the Yoneda Lemma to the A-infinity setting.

Abstract

We define A-infinity-bimodules similarly to Tradler and show that this notion is equivalent to an A-infinity-functor with two arguments which takes values in the differential graded category of complexes of k-modules, where k is a ground commutative ring. Serre A-infinity-functors are defined via A-infinity-bimodules likewise Kontsevich and Soibelman. We prove that a unital closed under shifts A-infinity-category A over a field k admits a Serre A-infinity-functor if and only if its homotopy category H^0(A) admits a Serre k-linear functor. The proof uses categories enriched in K, the homotopy category of complexes of k-modules, and Serre K-functors. Also we use a new A-infinity-version of the Yoneda Lemma generalizing the previously obtained result.