Derived A-infinity algebras and their homotopies

TL;DR

This paper explores the homotopy theory of derived A-infinity algebras over commutative rings, introducing new notions of homotopy and establishing their relation to spectral sequences, with applications to twisted complexes.

Contribution

It introduces a hierarchy of homotopy notions for derived A-infinity algebras and provides new interpretations of these algebras in twisted complexes and split filtered complexes.

Findings

r-homotopy corresponds to E_r-quasi-isomorphisms

Established new interpretations of derived A-infinity algebras

Analyzed homotopy theory in the context of twisted complexes

Abstract

The notion of a derived A-infinity algebra, considered by Sagave, is a generalization of the classical notion of A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We initiate a study of the homotopy theory of these algebras, by introducing a hierarchy of notions of homotopy between the morphisms of such algebras. We define r-homotopy, for non-negative integers r, in such a way that r-homotopy equivalences underlie E_r-quasi-isomorphisms, defined via an associated spectral sequence. We study the special case of twisted complexes (also known as multicomplexes) first since it is of independent interest and this simpler case clearly exemplifies the structure we study. We also give two new interpretations of derived A-infinity algebras as A-infinity algebras in twisted complexes and as A-infinity algebras in split filtered cochain complexes.