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Operadic tensor products and smash products
 Operads: Proceedings of Renaissance Conferences, volume 202 of Contemporary Math
, 1997
"... Abstract. Let k be a commutative ring. E ∞ kalgebras are associative and commutative kalgebras up to homotopy, as codified in the action of an E ∞ operad; A ∞ kalgebras are obtained by ignoring permutations. Using a particularly wellbehaved E ∞ algebra, we explain an associative and commutative ..."
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Abstract. Let k be a commutative ring. E ∞ kalgebras are associative and commutative kalgebras up to homotopy, as codified in the action of an E ∞ operad; A ∞ kalgebras are obtained by ignoring permutations. Using a particularly wellbehaved E ∞ algebra, we explain an associative and commutative operadic tensor product that effectively hides the operad: an A ∞ algebra or E ∞ algebra A is defined in terms of maps k − → A and A A − → A such that the obvious diagrams commute, and similarly for modules over A. This makes it little more difficult to study these algebraic objects than it is to study their classical counterparts. We also explain a topological analogue of the theory. This gives a symmetric monoidal category of modules