“This research monograph presents a major breakthrough by the authors in rational homotopy theory. Solving a hard classical problem, to be explained next, the book provides new tools and methods, and poses intriguing new questions. The major contribution is a new approach to the rational homotopy theory of arbitrary topological spaces and simplicial sets (not necessarily 1-connected, nilpotent, or even connected) by using complete differential graded Lie Q-algebras whose underlying complexes are potentially unbounded (cDGLs, henceforth).” (José M. Moreno-Fernández, Mathematical Reviews, May, 2022)“The book is a timely and authoritative treatment of exciting breakthroughs and powerful techniques in homotopytheory. It will surely become a standard reference in the field.” (Samuel Smith, zbMATH 1469.55001, 2021)“It is likely that modern topology graduate students have the background needed to appreciate the efficacy and sheer beauty of this whole approach to Lie models, so it would certainly be appropriate (and recommended!) for advanced courses and seminars.” (John Oprea, MAA Reviews, July 18, 2021)

• Background.- The Quillen functors L and C , and their duals.- Complete differential graded Lie algebras.- Maurer-Cartan elements and the Deligne groupoid.- The Lawrence-Sullivan Interval.- The cosimplicial cdgl L.- The model and realization functors.- A model category for cdgl.- The cosimplicial cdgl L via transfer.- The Deligne-Getzler-Hinich functor MC and equivalence of realizations.- Examples.- Index of notation.

“This research monograph presents a major breakthrough by the authors in rational homotopy theory. Solving a hard classical problem, to be explained next, the book provides new tools and methods, and poses intriguing new questions. The major contribution is a new approach to the rational homotopy theory of arbitrary topological spaces and simplicial sets (not necessarily 1-connected, nilpotent, or even connected) by using complete differential graded Lie Q-algebras whose underlying complexes are potentially unbounded (cDGLs, henceforth).” (José M. Moreno-Fernández, Mathematical Reviews, May, 2022)“The book is a timely and authoritative treatment of exciting breakthroughs and powerful techniques in homotopytheory. It will surely become a standard reference in the field.” (Samuel Smith, zbMATH 1469.55001, 2021)“It is likely that modern topology graduate students have the background needed to appreciate the efficacy and sheer beauty of this whole approach to Lie models, so it would certainly be appropriate (and recommended!) for advanced courses and seminars.” (John Oprea, MAA Reviews, July 18, 2021)

• Background.- The Quillen functors L and C , and their duals.- Complete differential graded Lie algebras.- Maurer-Cartan elements and the Deligne groupoid.- The Lawrence-Sullivan Interval.- The cosimplicial cdgl L.- The model and realization functors.- A model category for cdgl.- The cosimplicial cdgl L via transfer.- The Deligne-Getzler-Hinich functor MC and equivalence of realizations.- Examples.- Index of notation.