Since the birth of rational homotopy theory, the possibility of extending the Quillen approach – in terms of Lie algebras – to a more general category of spaces, including the non-simply connected case, has been a challenge for the algebraic topologist community. Despite the clear Eckmann-Hilton duality between Quillen and Sullivan treatments, the simplicity in the realization of algebraic structures in the latter contrasts with the complexity required by the Lie algebra version.
In this book, the authors develop new tools to address these problems. Working with complete Lie algebras, they construct, in a combinatorial way, a cosimplicial Lie model for the standard simplices. This is a key object, which allows the definition of a new model and realization functors that turn out to be homotopically equivalent to the classical Quillen functors in the simply connected case. With this, the authors open new avenues for solving old problems and posing new questions.
This monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.
Author(s): Urtzi Buijs, Yves Félix, Aniceto Murillo, Daniel Tanré
Series: Progress in Mathematics, 335
Publisher: Birkhäuser
Year: 2021
Language: English
Pages: 283
City: Cham
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
