$(b, \nu)$-algebras and their twisted modules

TL;DR

This paper characterizes the closure under shifts of a strictly unital $A_ $-category, explores its arithmetical properties, and shows how the associated cohomology category forms a triangulated structure.

Contribution

It provides an intrinsic characterization of shift closures in $A_ $-categories and analyzes their higher operations and categorical structures.

Findings

The cohomology category ${ m H}(\\widehat{\ m A})$ is triangulated.

The precategory of cocycles ${\cal Z}({\cal A})$ has a Frobenius-like structure.

${\cal H}(\widehat{\cal A})$ is shown to be a stable category.

Abstract

We give an intrinsic characterization of the closure under shifts $\widehat{\cal A}$ of a given strictly unital $A_\infty$-category ${\cal A}$. We study some arithmetical properties of its higher operations and special conflations in the precategory of cocycles ${\cal Z}({\cal A})$ of its $A_\infty$-category of twisted modules. We exhibit a structure for ${\cal Z}(\widehat{\cal A})$ similar to a special Frobenius category. We derive that the cohomology category ${\cal H}(\widehat{\cal A})$ appears as the corresponding stable category and then we review how this implies that ${\cal H}(\widehat{\cal A})$ is a triangulated category.