From Categories to Homotopy Theory

Contents
Introduction page
Part I Category Theory
	1 Basic Notions in Category Theory
		1.1 Definition of a Category and Examples
		1.2 EI Categories and Groupoids
		1.3 Epi- and Monomorphisms
		1.4 Subcategories and Functors
		1.5 Terminal and Initial Objects
	2 Natural Transformations and the Yoneda Lemma
		2.1 Natural Transformations
		2.2 The Yoneda Lemma
		2.3 Equivalences of Categories
		2.4 Adjoint Pairs of Functors
		2.5 Equivalences of Categories via Adjoint Functors
		2.6 Skeleta of Categories
	3 Colimits and Limits
		3.1 Diagrams and Their Colimits
		3.2 Existence of Colimits and Limits
		3.3 Colimits and Limits in Functor Categories
		3.4 Adjoint Functors and Colimits and Limits
		3.5 Exchange Rules for Colimits and Limits
	4 Kan Extensions
		4.1 Left Kan Extensions
		4.2 Right Kan Extensions
		4.3 Functors Preserving Kan Extensions
		4.4 Ends
		4.5 Coends as Colimits and Ends as Limits
		4.6 Calculus Notation
		4.7 “All Concepts are Kan Extensions”
	5 Comma Categories and the Grothendieck Construction
		5.1 Comma Categories: Definition and Special Cases
		5.2 Changing Diagrams for Colimits
		5.3 Sifted Colimits
		5.4 Density Results
		5.5 The Grothendieck Construction
	6 Monads and Comonads
		6.1 Monads
		6.2 Algebras over Monads
		6.3 Kleisli Category
		6.4 Lifting Left Adjoints
		6.5 Colimits and Limits of Algebras over a Monad
		6.6 Monadicity
		6.7 Comonads
	7 Abelian Categories
		7.1 Preadditive Categories
		7.2 Additive Categories
		7.3 Abelian Categories
	8 Symmetric Monoidal Categories
		8.1 Monoidal Categories
		8.2 Symmetric Monoidal Categories
		8.3 Monoidal Functors
		8.4 Closed Symmetric Monoidal Categories
		8.5 Compactly Generated Spaces
		8.6 Braided Monoidal Categories
	9 Enriched Categories
		9.1 Basic Notions
		9.2 Underlying Category of an Enriched Category
		9.3 Enriched Yoneda Lemma
		9.4 Cotensored and Tensored Categories
		9.5 Categories Enriched in Categories
		9.6 Bicategories
		9.7 Functor Categories
		9.8 Day Convolution Product
Part II From Categories to Homotopy Theory
	10 Simplicial Objects
		10.1 The Simplicial Category
		10.2 Simplicial and Cosimplicial Objects
		10.3 Interlude: Joyal’s Category of Intervals
		10.4 Bar and Cobar Constructions
		10.5 Simplicial Homotopies
		10.6 Geometric Realization of a Simplicial Set
		10.7 Skeleta of Simplicial Sets
		10.8 Geometric Realization of Bisimplicial Sets
		10.9 The Fat Realization of a (Semi)Simplicial Set or Space
		10.10 The Totalization of a Cosimplicial Space
		10.11 Dold–Kan Correspondence
		10.12 Kan Condition
		10.13 Quasi-Categories and Joins of Simplicial Sets
		10.14 Segal Sets
		10.15 Symmetric Spectra
	11 The Nerve and the Classifying Space of a Small Category
		11.1 The Nerve of a Small Category
		11.2 The Classifying Space and Some of Its Properties
		11.3 π0 and π1 of Small Categories
		11.4 The Bousfield Kan Homotopy Colimit
		11.5 Coverings of Classifying Spaces
		11.6 Fibers and Homotopy Fibers
		11.7 Theorems A and B
		11.8 Monoidal and Symmetric Monoidal Categories, Revisited
	12 A Brief Introduction to Operads
		12.1 Definition and Examples
		12.2 Algebras Over Operads
		12.3 Examples
		12.4 E∞-monoidal Functors
	13 Classifying Spaces of Symmetric Monoidal Categories
		13.1 Commutative H-Space Structure on BC for C Symmetric Monoidal
		13.2 Group Completion of Discrete Monoids
		13.3 Grayson–Quillen Construction
		13.4 Group Completion of H-Spaces
	14 Approaches to Iterated Loop Spaces via Diagram Categories
		14.1 Diagram Categories Determine Algebraic Structure
		14.2 Reduced Simplicial Spaces and Loop Spaces
		14.3 Gamma-Spaces
		14.4 Segal K-Theory of a Permutative Category
		14.5 Injections and Infinite Loop Spaces
		14.6 Braided Injections and Double Loop Spaces
		14.7 Iterated Monoidal Categories as Models for Iterated Loop Spaces
		14.8 The Category n
	15 Functor Homology
		15.1 Tensor Products
		15.2 Tor and Ext
		15.3 How Does One Obtain a Functor Homology Description?
		15.4 Cyclic Homology as Functor Homology
		15.5 The Case of Gamma Homology
		15.6 Adjoint Base-Change
	16 Homology and Cohomology of Small Categories
		16.1 Thomason Cohomology and Homology of Categories
		16.2 Quillen’s Definition
		16.3 Spectral Sequence for Homotopy Colimits in Chain Complexes
		16.4 Baues–Wirsching Cohomology and Homology
		16.5 Comparison of Functor Homology and Homology of Small Categories
References
Index