ABSTRACT

This chapter reviews Quillen’s rational homotopy theory. It highlights the relation between the Lie algebra model LQ(X) and the coalgebra model CQ(X), which can be expressed in terms of Koszul duality. The chapter discusses spectral Lie algebras and their connection with the Goodwillie tower of the identity functor of S∗. The duality between Lie algebras and commutative coalgebras expressed by Theorem 16.2.8 is a special case of what is usually called Koszul duality or bar-cobar duality. The chapter describes an extension of the theory of Lie algebras to the ∞-category Sp of spectra. It examines some fundamental concepts of chromatic homotopy theory and their role in unstable homotopy theory. The chapter also describes the general philosophy of stable chromatic homotopy theory, in analogy with the theory of localizations in algebra, and its application to unstable homotopy theory through the Bousfield–Kuhn functors.