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Deo/nitions: operads, algebras and modules
 Contemporary Mathematics 202
, 1997
"... There are many different types of algebra: associative, associative and commutative, Lie, Poisson, etc., etc. Each comes with an appropriate notion of a module. As is becoming more and more important in a variety of fields, it is often necessary to deal with algebras and modules of these sorts “up t ..."
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There are many different types of algebra: associative, associative and commutative, Lie, Poisson, etc., etc. Each comes with an appropriate notion of a module. As is becoming more and more important in a variety of fields, it is often necessary to deal with algebras and modules of these sorts “up to homotopy”. I shall give a very partial overview, concentrating on algebra, but saying a little about the original use of operads in topology. The development of abstract frameworks in which to study such algebras has a long history. As this conference attests, it now seems to be widely accepted that, for many purposes, the most convenient setting is that given by operads and their actions. While the notion was first written up in a purely topological framework [19], it was thoroughly understood by 1971 [12] that the basic definitions apply equally well in any underlying symmetric monoidal ( = tensor) category. The definitions and ideas had many precursors. I will indicate those that I was aware of at the time. • Algebraists such as Kaplansky, Herstein, and Jacobson systematically studied
OPERADS, ALGEBRAS, MODULES, AND MOTIVES
"... Abstract. With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent parts: I Definitions and examples of operads and their actions II Partial algebraic structures and ..."
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Abstract. With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent parts: I Definitions and examples of operads and their actions II Partial algebraic structures and conversion theorems III Derived categories from a topological point of view IV Rational derived categories and mixed Tate motives V Derived categories of modules over E ∞ algebras In differential algebra, operads are systems of parameter chain complexes for multiplication on various types of differential graded algebras “up to homotopy”, for example commutative algebras, nLie algebras, nbraid algebras, etc. Our primary focus is the development of the concomitant theory of modules up to homotopy and the study of both classical derived categories of modules over DGA’s and derived categories of modules up to homotopy over DGA’s up to homotopy. Examples of such derived categories provide the appropriate setting for one approach to mixed Tate motives in algebraic geometry, both rational and integral.
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