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

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.