Rational Homotopical Models and Uniqueness.

Intro -- TABLE OF CONTENTS -- ABSTRACT -- KEYWORDS -- PREFACE -- INTRODUCTION -- 1. HOMOTOPY THEORY -- 1. HOMOTOPICAL CATEGORIES -- 1. The axioms -- 2. Left homotopical categories -- 3. Homotopical subcategories -- 2. FUNDAMENTAL RESULTS -- 1. Lifting and extension -- 2. The derived category -- 3. Homotopical functors and their derived functors -- 4. The Adjoint Functor Theorem -- 3. COMONOIDS UP TO HOMOTOPY -- 1. … as comonoids over the derived category -- 2. Derived tensor product -- 3. Generalizations -- A. EXAMPLES OF HOMOTOPICAL CATEGORIES -- 1. Cofibration categories -- 2. Model categories -- 3. Spaces -- 4. Simplicial objects -- 2. DIFFERENTIAL ALGEBRA -- 1. PRELIMINARIES -- 1. Chain complexes -- 2. DG (co)algebras -- 3. Tensor (co) algebras -- 2. TWISTING MAPS AND THE (CO) BAR CONSTRUCTION -- 1. Twisting maps and homotopies -- 2. The (co)bar construction -- 3. Compatibility with tensor product -- 4. Homological properties -- 3. ACYCLIC MODELS -- 1. Representable functors -- 2. The method of acyclic models -- 3. Duality -- 4. Acyclic model theorems for twisting maps -- 4. EZ-MORPHISMS -- 1. Extension of an EZ-morphism -- 2. A generalization -- 3. Properties of the extension -- B. CHAIN (CO) FUNCTORS -- 1. Monoidal categories -- 2. Normalization -- 3. Representable cofunctors for spaces -- 4. Cohomology theories -- 3. COMPLETE ALGEBRA -- 1. COMPLETE AUGMENTED ALGEBRAS -- 1. Ring systems -- 2. Complete modules -- 3. Complete augmented algebras and free groups -- 4. Rigidity -- 2. COMPLETE LIE ALGEBRAS AND COMPLETE HOPF ALGEBRAS -- 1. Complete Hopf algebras and the exponential mapping -- 2. The PBW-Theorem -- 3. Normal complete Hopf algebras -- 4. Rigidity -- 3. COMPLETE GROUPS -- 1. Nilpotent groups -- 2. Complete groups -- 3. The Lazard - Mal'cev correspondence -- 4. The Quillen functor -- C. FILTERED MODULES.

1. Filtered vs. cofiltered modules -- 2. Normal maps and exactness -- 3. Filtered tensor product -- 4. Complete Differential Algebra -- 4. THREE MODELS FOR SPACES -- 1. THE CELLULAR MODEL -- 1. The homotopical category of dg algebras -- 2. The homotopical category of dg Hopf algebras up to homotopy -- 3. The cobar - chain functor and the chain - loop functor -- 4. Compatibility with (tensor) products -- 5. The homotopy diagonals -- 2. THE SULLIVAN MODEL -- 1. The homotopical category of commutative dg* algebras -- 2. The Sullivan cofunctor and Stokes' map -- 3. Extension of Stokes' map -- 4. Compatibility with (tensor) products -- 5. Dualization -- 6. The homotopy diagonals -- 3. THE QUILLEN MODEL -- 1. The homotopical category of dg Lie algebras -- 2. The Quillen functor -- 3. Connection to the chain - loop functor -- 4. The group algebra of a free simplicial group -- 5. A proof of the Quillen equivalence -- 4. MAIN RESULTS -- 1. Summary -- 2. Anick's equivalence -- 3. Proof of the Baues - Lemaire conjecture -- 4. Rational equivalence -- D. THE CELLULAR LIE ALGEBRA MODEL -- 1. A natural diagonal for the cobar - chain functor -- 2. A natural Hopf diagonal -- 3. The category of dg Lie algebras (over any ring) -- 4. Anick's theorems and naturality -- NOTATIONS -- BIBLIOGRAPHY.

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