The Bar Construction of an E-infinite Algebra
نویسنده
چکیده
defined by the shuffle of tensors. If the product of A is commutative, then the bar differential is a derivation with respect to the shuffle product so that B(A) is still an associative and commutative differential graded algebra. Unfortunately, in algebraic topology, algebras are usually commutative only up to homotopy: a motivating example is provided by the cochain algebra of a topological space C(X). In this context, the shuffle product is no longer compatible with the differential. Thus, the problem is to use commutativity homotopies in order to add perturbations to the shuffle product so that B(A) can still be equipped with the structure of a differential graded algebra. In order to state precise results, we introduce E∞-algebra structures (strongly homotopy associative and commutative
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