Twisted complexes and simplicial homotopies

Abstract

In this paper we consider the dg-category of twisted complexes over simplicial spaces. It is clear that a simplicial map f : U → V between simplicial spaces induces a dg-functor f * : Tw(V, R) → Tw(U, R).In this paper we prove that for simplicial homotopic maps f and g, there exists an A∞-natural transformation Φ : f * ⇒ g * between induced dg-functors. Moreover the 0th component of Φ is a weak equivalence. If we restrict ourselves to the full dg-subcategory of twisted perfect complexes, we prove that Φ admits an A∞-quasiinverse.

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

“…Remark 7. In [Wei19] the formula in this note is used to construct A ∞ -inverse of dg-natural transformations between twisted complexes. Moreover in an upcoming work [Wei], the formula in this note, together with the method in [Wei16], [Wei18], [BHW17], and [AØ18], can be used to obtain an injective dg-resolution of the equivariant derived category [BL94].…”