A recurrent formula of $A_{\infty}$-quasi inverses of dg-natural transformations between dg-lifts of derived functors

Abstract

A dg-natural transformation between dg-functors is called an objectwise homotopy equivalence if its induced morphism on each object admits a homotopy inverse. In general an objectwise homotopy equivalence does not have a dg-inverse but has an A ∞ quasi-inverse. In this note we give a recurrent formula of the A ∞ quasi-inverse. This result is useful in studying the compositions of dg-lifts of derived functors of schemes.

Twisted complexes and simplicial homotopies

“…In this section we review A ∞ -natural transformations between dg-functors. For more details see [Wei19]. See for example [Lyu03] or [AØ18] for an introduction of more general A ∞ -categories, A ∞ -functors and A ∞ -natural transformations.…”