  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.
