Derived \(A\)-infinity algebras and their homotopies. (English) Zbl 1439.16009
Summary: 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.
MSC:
| 16E45 | Differential graded algebras and applications (associative algebraic aspects) |
| 18G50 | Nonabelian homological algebra (category-theoretic aspects) |
| 16T15 | Coalgebras and comodules; corings |
| 18D20 | Enriched categories (over closed or monoidal categories) |
| 18M60 | Operads (general) |
Keywords:
derived \(A\)-infinity algebra; filtered \(A\)-infinity algebra; twisted complex; multicomplexReferences:
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