Operadic Koszul Duality

1. OPERADS Our goal in this talk is to give a sort of categorified version of “Koszul duality”. One of the primary motivations for us to do so is to take the classical results about Koszul duality, which form a convoluted and complex body of literature, and stretch them apart to see which assumptions power which parts of the theory. Classically, Koszul duality results are concerned chiefly with two cases: derived categories over modules and various forms of homotopy theory (e.g., rational nilpotent spaces). Our first goal will be to give a definition of “algebra” and “action” which encompass both of these situations. The rough idea will be to take dreadfully seriously the notion of parse trees: the expression 1 · ((2+ 3+ 4) + 5) can be rendered graphically in a parse tree as ..