Simplicial Methods for Higher Categories: Segal-type Models of Weak n-Categories (Algebra and Applications, 26)

Preface
Acknowledgements
Contents
List of Symbols
List of Figures
Part I Higher Categories: Introduction and Background
	1 An Introduction to Higher Categories
		1.1 Motivation and Context
		1.2 Different Types of Higher Structures
			1.2.1 ω-Categories
			1.2.2 Truncated Higher Categories
			1.2.3 Strict Versus Weak n-Categories
			1.2.4 n-Fold Categories
			1.2.5 n-Fold Structures Versus Strict and Weakn-Categories
		1.3 The Homotopy Hypothesis
			1.3.1 Homotopy Types and Their Algebraic Models
			1.3.2 Modelling Homotopy Types with n-Fold Structures
	2 Multi-Simplicial Techniques
		2.1 Multi-Simplicial Objects and Segal Maps
			2.1.1 Simplicial Objects and Their Segal Maps
			2.1.2 Multi-Simplicial Objects
		2.2 Multi-Simplicial Sets
			2.2.1 The Functors p(r) and q(r)
			2.2.2 Closure Properties
		2.3 n-Fold Internal Categories
		2.4 Multi-Nerve Functors
		2.5 n-Fold Categories
		2.6 A Multi-Simplicial Description of Strict n-Categories
		2.7 The Functor Décalage
	3 An Introduction to the Three Segal-Type Models
		3.1 Geometric Versus Higher Categorical Equivalences
		3.2 Multi-Simplicial Structures as an Environment for Higher Categories
		3.3 The Idea of Weak Globularity
		3.4 The Three Segal-Type Models
			3.4.1 Notational Conventions
			3.4.2 Common Features of the Three Segal-Type Models
			3.4.3 Main Results
			3.4.4 Organization of This Work
			3.4.5 Informal Discussions
	4 Techniques from 2-Category Theory
		4.1 Some Functors on Cat
		4.2 Pseudo-Functors and Their Strictification
			4.2.1 Adjunctions and Equivalences in 2-Categories
			4.2.2 The Notion of Pseudo-Functor
			4.2.3 Pseudo T-Algebras
			4.2.4 Strictification of Pseudo-Functors
		4.3 Transport of Structure
Part II The Three Segal-Type Models and Segalic Pseudo-Functors
	5 Homotopically Discrete n-Fold Categories
		5.1 The Definition of Homotopically Discrete n-Fold Categories
			5.1.1 The Idea of a Homotopically Discrete n-FoldCategory
			5.1.2 The Formal Definition of Cathdn
			5.1.3 Homotopically Discrete n-Fold Categories As Internal Equivalence Relations
		5.2 Properties of Homotopically Discrete n-Fold Categories
			5.2.1 Closure Properties of Cathdn
			5.2.2 n-Equivalences in Cathdn
		5.3 Homotopically Discrete n-Fold Categories and 0-Types
	6 The Definition of the Three Segal-Type Models
		6.1 Weakly Globular Tamsamani n-Categories
			6.1.1 The Idea of Weakly Globular Tamsamanin-Categories
			6.1.2 Closure Properties
			6.1.3 The Formal Definition of the Category Tawgn
		6.2 Tamsamani n-Categories
		6.3 Weakly Globular n-Fold Categories
			6.3.1 The Idea of Weakly Globular n-Fold Categories
			6.3.2 The Formal Definition of the Category Catwgn
	7 Properties of the Segal-Type Models
		7.1 Properties of Weakly Globular Tamsamani n-Categories
			7.1.1 Properties of n-Equivalences
			7.1.2 The Functor q(n-1)
			7.1.3 Pullback Constructions Using q(n-1)
		7.2 Properties of Weakly Globular n-Fold Categories
			7.2.1 Weakly Globular n-Fold Categoriesand n-Equivalences
			7.2.2 A Criterion for an n-Fold Category to Be Weakly Globular
			7.2.3 A Geometric Interpretation
	8 Pseudo-Functors Modelling Higher Structures
		8.1 The Definition of a Segalic Pseudo-Functor
			8.1.1 Notational Conventions for Segalic Pseudo-Functors
			8.1.2 The Idea of a Segalic Pseudo-Functor
			8.1.3 The Formal Definition of a Segalic Pseudo-Functor
		8.2 Strictification of Segalic Pseudo-Functors
Part III Rigidification of Weakly Globular Tamsamani n-Categories
	9 Approximating Weakly Globular Tamsamani n-Categories by Simpler Ones
		9.1 The Category LTawgn
			9.1.1 The Idea of the Category LTawgn
			9.1.2 The Formal Definition of the Category LTawgn
			9.1.3 Properties of the Category LTawgn
			9.1.4 Catwgn and the Category LTawgn
		9.2 Approximating Tawgn by LTawgn
			9.2.1 The Main Steps in Approximating Tawgn by LTawgn
			9.2.2 Approximating Tawgn by LTawgn: The Formal Proofs
	10 Rigidifying Weakly Globular Tamsamani n-Categories
		10.1 From LTawgn to Pseudo-Functors
			10.1.1 The Idea of the Functor Trn
			10.1.2 The Formal Construction of the Functor Trn
		10.2 Rigidifying Weakly Globular Tamsamani n-Categories
			10.2.1 The Rigidification Functor Qn: Main Steps
			10.2.2 The Rigidification Functor: The Formal Proof
Part IV Weakly Globular n-Fold Categories as a Model of Weak n-Categories
	11 Functoriality of Homotopically Discrete Objects
		11.1 A Construction on Catwgn
			11.1.1 The Idea of the Construction X(f0)
		11.2 Weakly Globular n-Fold Categories and Functoriality of Homotopically Discrete Objects
			11.2.1 The Idea of the Functors Vn and Fn
			11.2.2 The Functors Vn and Fn
		11.3 The Category FCatwgn
			11.3.1 The Idea of the Category FCatwgn
			11.3.2 The Formal Definition of the Category FCatwgn
			11.3.3 The Idea of the Functor Gn
			11.3.4 The Functor Gn: The Formal Proof
	12 Weakly Globular n-Fold Categories as a Model of Weak n-Categories
		12.1 From FCatwgn to Tamsamani n-Categories
			12.1.1 The Idea of the Functor Dn
			12.1.2 The Functor Dn: Definition and Properties
		12.2 The Discretization Functor and the Comparison Result
			12.2.1 The Idea of the Functor Discn
			12.2.2 The Comparison Result
		12.3 Groupoidal Weakly Globular n-Fold Categories
		12.4 An Alternative Fundamental Functor
			12.4.1 The Functor Hn
			12.4.2 Some Examples
	13 Conclusions and Further Directions
		13.1 Algebraic Description of Postnikov Systems
		13.2 Model Comparisons
		13.3 Intermediate Levels of Weakness
		13.4 Model Structures
		13.5 A Weakly Globular Approach to (∞,n)-Categories
A Proof of Lemma 10.1.4
References
Index