Operads, Algebras, Modules, and Motives

With motivation from algebraic topology, algebraic geometry, and stringtheory, we study various topics in differential homological algebra. The work is dividedinto five largely independent Parts:I Definitions and examples of operads and their actionsII Partial algebraic structures and conversion theoremsIII Derived categories from a topological point of viewIV Rational derived categories and mixed Tate motivesV Derived categories of modules over E∞ algebrasIn differential algebra, operads are systems of parameter chain complexes for mul-tiplication on various types of differential graded algebras “up to homotopy”, forexample commutative algebras, n-Lie algebras, n-braid algebras, etc. Our primaryfocus is the development of the concomitant theory of modules up to homotopy andthe study of both classical derived categories of modules over DGA’s and derived cat-egories of modules up to homotopy over DGA’s up to homotopy. Examples of suchderived categories provide the appropriate setting for one approach to mixed Tatemotives in algebraic geometry, both rational and integral.