Lie Models in Topology

Contents
Introduction
	Acknowledgement
Chapter 1 Background
	1.1 Simplicial categories
		1.1.1 Simplicial sets
		1.1.2 Simplicial complexes
		1.1.3 Simplicial chains
	1.2 Differential categories
		1.2.1 Commutative differential graded algebras and the Sullivan model of a space
		1.2.2 Differential graded Lie algebras and the Quillen model of a space
		1.2.3 Differential graded coalgebras
		1.2.4 Differential graded Lie coalgebras
		1.2.5 A∞-algebras
	1.3 Model categories
		1.3.1 Differential model categories
		1.3.2 Cofibrantly generated model categories
Chapter 2 The Quillen Functors L, C and their Duals A , E
	2.1 The functors L and C
	2.2 The functors A and E
Chapter 3 Complete Differential Graded Lie Algebras
	3.1 Complete differential graded Lie algebras
	3.2 The completion of free Lie algebras
	3.3 Completion vs profinite completion
Chapter 4 Maurer–Cartan Elements and the Deligne Groupoid
	4.1 Maurer–Cartan elements
	4.2 Exponential automorphisms and the Baker–Campbell–Hausdorff product
	4.3 The gauge action and the Deligne groupoid
	4.4 Applications to deformation theory
	4.5 The Goldman–Millson Theorem
Chapter 5 The Lawrence–Sullivan Interval
	5.1 Introducing the Lawrence–Sullivan interval
	5.2 The LS interval as a cylinder
	5.3 The flow of a differential equation, the gauge action and the LS interval
	5.4 Subdivision of the LS interval and a model of the triangle
	5.5 Paths in a cdgl
	Bibliographical notes
Chapter 6 The Cosimplicial cdgl
	6.1 The main result
	6.2 Inductive sequences of models of the standard simplices
	6.3 Sequences of equivariant models of the standard simplices
	6.4 The cosimplicial cdgl
	6.5 An explicit model for the tetrahedron
	6.6 Symmetric MC elements of simplicial complexes
Chapter 7 The Model and Realization Functors
	7.1 Introducing the global model and realization functors. Adjointness
	7.2 First features of the global model and realization functors
	7.3 The path components and homotopy groups of
	7.4 Homological behaviour of
	7.5 The Deligne groupoid of the global model
Chapter 8 A Model Category for cdgl
	8.1 The model category
	8.2 Weak equivalences and free extensions
	8.3 A path object, a cylinder object and homotopy of morphisms
	8.4 Minimal models of simplicial sets
	Bibliographical notes
Chapter 9 The Global Model Functor via Homotopy Transfer
	9.1 The Dupont calculus on APL(Δ•)
	9.2 Obtaining L• and LX by transfer
	Bibliographical notes
Chapter 10 Extracting the Sullivan, Quillen and Neisendorfer Models from the Global Model
	10.1 Connecting the global model with the Sullivan, Quillen and Neisendorfer models
	10.2 From the Lie minimal model to the Sullivan model and vice versa
	10.3 Coformal spaces
Chapter 11 The Deligne–Getzler–Hinich Functor MC• and Equivalence of Realizations
	11.1 The set of Maurer–Cartan elements as a set of morphisms
	11.2 Simplicial contractions of APL(Δ•)
	11.3 The Deligne–Getzler–Hinich ∞-groupoid
	11.4 Equivalence of realizations and Bousfield–Kan completion
	Bibliographical notes
Chapter 12 Examples
	12.1 Lie models of 2-dimensional complexes. Surfaces
	12.2 Lie models of tori and classifying spaces of right-angled Artin groups
	12.3 Lie model of a product
	12.4 Mapping spaces
		12.4.1 Lie models of mapping spaces
		12.4.2 Lie models of pointed mapping spaces
		12.4.3 Lie models of free loop spaces
		12.4.4 Simplicial enrichment of cdgl and cdga
		12.4.5 Complexes of derivations and homotopy groups of mapping spaces
	12.5 Homotopy invariants of the realization functor
		12.5.1 Action of π1 on π∗
		12.5.2 The rational homotopy Lie algebra of 
		12.5.3 Postnikov decomposition of 
	Bibliographical notes
Notation Index
	General notation
	Categories
Bibliography
Index