  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.
