Homotopical Quantum Field Theory

Preface
Contents
1. Introduction
	1.1 Algebraic Quantum Field Theory
	1.2 Homotopy Algebraic Quantum Field Theory
	1.3 Homotopy Prefactorization Algebra
	1.4 Comparison
	1.5 Organization
2. Category Theory
	2.1 Basics of Categories Definition
	2.2 Examples of Categories
	2.3 Limits and Colimits Definition 2.3.1.
	2.4 Adjoint Functors
	2.5 Symmetric Monoidal Categories
	2.6 Monoids
	2.7 Monads Definition
	2.8 Localization
	Theorem 2.8.3.
3. Trees
	3.1 Graphs
	3.2 Tree Substitution
	3.3 Grafting
4. Colored Operads
	4.1 Operads as Monoids
	4.2 Operads in Terms of Generating Operations
	4.3 Operads in Terms of Partial Compositions
	4.4 Operads in Terms of Trees
	4.5 Algebras over Operads
5. Constructions on Operads
	5.1 Change-of-Operad Adjunctions
	5.2 Model Category Structures
	5.3 Changing the Base Categories
	5.4 Localizations of Operads
	5.5 Algebras over Localized Operads
6. Boardman-Vogt Construction of Operads
	6.1 Overview
	6.2 Commutative Segments
	6.3 Coend Definition of the BV Construction
	6.4 Augmentation
	6.5 Homotopy Morita Equivalence
	6.6 Filtration
7. Algebras over the Boardman-Vogt Construction
	7.1 Overview
	7.2 Coherence Theorem
	7.3 Homotopy Coherent Diagrams
	7.4 Homotopy Inverses
	7.5 Aª-Algebras
	7.6 Eª-Algebras
	7.7 Homotopy Coherent Diagrams of Aª-Algebras
	7.8 Homotopy Coherent Diagrams of Eª-Algebras
8. Algebraic Quantum Field Theories
	8.1 From Haag-Kastler Axioms to Operads
	8.2 AQFT as Functors
	8.3 AQFT as Operad Algebras
	8.4 Examples of AQFT
	8.5 Homotopical Properties
9. Homotopy Algebraic Quantum Field Theories
	9.1 Overview
	9.2 Homotopy AQFT as Operad Algebras
	9.3 Examples of Homotopy AQFT
	9.4 Coherence Theorem
	9.5 Homotopy Causality Axiom
	9.6 Homotopy Coherent Diagrams
	9.7 Homotopy Time-Slice Axiom
	9.8 Objectwise
	9.9 Homotopy Coherent Diagrams of
10. Prefactorization Algebras
	10.1 Costello-Gwilliam Prefactorization Algebras
	10.2 Configured Categories
	10.3 Prefactorization Algebras as Operad Algebras
	10.4 Pointed Diagram Structure
	10.5 Commutative Monoid Structure
	10.6 Diagrams of Modules over a Commutative Monoid
	10.7 Diagrams of Commutative Monoids
	10.8 Configured and Homotopy Morita Equivalences
11. Homotopy Prefactorization Algebras
	11.1 Overview
	11.2 Homotopy Prefactorization Algebras as Operad Algebras
	11.3 Examples
	11.4 Coherence Theorem
	11.5 Homotopy Coherent Pointed Diagrams
	11.6 Homotopy Time-Slice Axiom
	11.7 Eª-Algebra Structure
	11.8 Objectwise
	11.9 Homotopy Coherent Diagrams of
	11.10 Homotopy Coherent Diagrams of Eª-Algebras
12. Comparing Prefactorization Algebras and AQFT
	12.1 Orthogonal Categories as Configured Categories
	12.2 Configured Categories to Orthogonal Categories
	12.3 Comparison Adjunctions
	12.4 Examples of Comparison
	12.5 Prefactorization Algebras from AQFT
List of Notations
	Notation Page Description Chapter 2
	Chapter 3
	Chapter 4
	Chapter 5
	Chapter 6
	Chapter 7
	Chapter 8
	Chapter 9
	Chapter 10
	Chapter 11
	Chapter 12
Bibliography
Index