Derived A-infinity algebras and their homotopies

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 Er-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.

Model category structures and spectral sequences

“…This amounts to saying that the collection ( h m ) m is an r-homotopy (of twisted complexes) from f to g as proven in [2, proposition 3.18]. Hence E r+1 (f ) = E r+1 (g) follows from proposition 3.24 of [2].…”

“…Note that the categories of filtered complexes and bicomplexes we will consider satisfy the assumptions of this theorem as well as conditions (1), (2) and 3. Indeed, the category bC R of bicomplexes is abelian and thus is complete and cocomplete.…”

Homotopy theory of bicomplexes

“…We think it is simpler to directly construct our Cartan-Eilenberg model structure, at the very least because we obtain easier generating (trivial) cofibrations which allow for a straightforward identification of (trivial) fibrations. For 0 ≤ r < ∞, E r -equivalences have been studied by Cirici, Santander, Livernet, and Whitehouse in [CESLW17], not only for maps of bicomplexes but for twisted maps of twisted complexes.…”

“…Furthermore, the homological perturbation lemma [Bro67] tells us that the vertical homology of every Cartan-Eilenberg resolution can be equipped with the structure of a twisted complex. Therefore, in order to understand the homotopy theory of derived A ∞ -algebras, specifically in an operadic context [LRW13,CESLW17], it is necessary to understand the homotopy theory of the underlying twisted complexes.…”

Derived universal Massey products

“…They are filtered homotopy algebras which allow the construction of minimal models over rings, even for algebras with torsion homology. This theory was initiated by Sagave [22] in the associative case, which attracted some attention [13,4], and then continued in [15,16] over other operads.…”