Operads, Algebras, Modules, and Motives, by Igor Kriz and Peter May

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:

  • Definitions and examples of operads and their actions
  • Partial algebraic structures and conversion theorems
  • Derived categories from a topological point of view
  • Rational derived categories and mixed Tate motives
  • Derived categories of modules over E_\infty 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, n-Lie algebras, n-braid 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. This monograph will appear in Asterisque.

    Here is a pointer to the paper at Purdue.

    Igor Kriz <ikriz@math.lsa.umich.edu>
    Peter May <may@math.uchicago.edu>