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

Abstract

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