Transfers of 𝐴_{∞}- and other homotopy structures as Grothendieck bifibrations

We study the functor that assigns to an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{A_\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -morphism between isotopy classes of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{A_\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras, whose linear part is a chain homotopy equivalence, its underlying chain map. We show that this functor is a discrete Grothendieck bifibration. We then generalize our results to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">P</mml:mi> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathscr {P}_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -structures over a field of characteristic zero, for an arbitrary quadratic Koszul operad <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {P}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . An immediate application is a categorical framework in which the transfers of, for example, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">L_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> - and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">C_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -structures are strictly functorial.