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Homological algebra of homotopy algebras. (English) Zbl 0894.18008

If \(k\) is a ring and \(C(k)\) the category of unbounded complexes of \(k\)-modules, then there is a well structured homological (in fact homotopical) algebra defined on \(C(k)\). Replacing \(k\) by a d.g. algebra, the same result holds. This possibility of working with unbounded complexes is important if one wants to go beyond d.g. algebras to weak algebras in which associativity, etc. hold only up to higher homotopies. A convenient setting then is that of operads and operad algebras and in this paper the author shows that the usual theory extends with great simplicity to that very much more general setting. This enables the definition of cohomology of operad algebras and a corresponding cotangent complex to be defined.
Reviewer: T.Porter (Bangor)

MSC:

18G55 Nonabelian homotopical algebra (MSC2010)
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
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