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

Abstract We give an intrinsic characterization of the closure under shifts $$\widehat{{\mathcal {A}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> of a given strictly unital $$A_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -category $${{\mathcal {A}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> . We study some arithmetical properties of its higher operations and special conflations in the precategory of cocycles $${{\mathcal {Z}}}({{\mathcal {A}}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of its $$A_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -category of twisted modules. We exhibit a structure for $${{\mathcal {Z}}}(\widehat{{\mathcal {A}}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> similar to a special Frobenius category. We derive that the cohomology category $$\mathcal{H}(\widehat{{\mathcal {A}}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> appears as the corresponding stable category and then we review how this implies that $${{\mathcal {H}}}(\widehat{\mathcal{A}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a triangulated category.