Higher Algebra II

Stable -Categories
	Foundations
		Stability
		The Homotopy Category of a Stable -Category
		Closure Properties of Stable -Categories
		Exact Functors
	Stable -Categories and Homological Algebra
		t-Structures on Stable -Categories
		Filtered Objects and Spectral Sequences
		The Dold-Kan Correspondence
		The -Categorical Dold-Kan Correspondence
	Homological Algebra and Derived Categories
		Nerves of Differential Graded Categories
		Derived -Categories
		The Universal Property of `39`42`\"613A``45`47`\"603AD-(`39`42`\"613A``45`47`\"603AA)
		Inverting Quasi-Isomorphisms
		Grothendieck Abelian Categories
	Spectra and Stabilization
		The Brown Representability Theorem
		Spectrum Objects
		The -Category of Spectra
		Presentable Stable -Categories
-Operads
	Foundations
		From Colored Operads to -Operads
		Maps of -Operads
		Algebra Objects
		-Preoperads
	Constructions of -Operads
		Subcategories of `39`42`\"613A``45`47`\"603AO-Monoidal -Categories
		Slicing -Operads
		Coproducts of -Operads
		Monoidal Envelopes
		Tensor Products of -Operads
		Day Convolution
	Disintegration and Assembly
		Unital -Operads
		Generalized -Operads
		Approximations to -Operads
		Disintegration of -Operads
	Products and Coproducts
		Cartesian Symmetric Monoidal Structures
		Monoid Objects
		CoCartesian Symmetric Monoidal Structures
		Wreath Products
Algebras and Modules over -Operads
	Free Algebras
		Operadic Colimit Diagrams
		Operadic Left Kan Extensions
		Construction of Free Algebras
		Transitivity of Operadic Left Kan Extensions
	Limits and Colimits of Algebras
		Unit Objects and Trivial Algebras
		Limits of Algebras
		Colimits of Algebras
		Tensor Products of Commutative Algebras
	Modules over -Operads
		Coherent -Operads
		A Coherence Criterion
		Module Objects
	General Features of Module -Categories
		Algebra Objects of -Categories of Modules
		Modules over Trivial Algebras
		Limits of Modules
		Colimits of Modules
Associative Algebras and Their Modules
	Associative Algebras
		The Associative -Operad
		Monoid Objects of -Categories
		Planar -Operads and A-Algebras
		Nonunital An-Algebras and Nonunital An-Monoids
		From An-Algebras to An+1-Algebras
		The Associahedron
		Monoidal Model Categories
		Rectification of Associative Algebras
	Left and Right Modules
		The -Operad `39`42`\"613A``45`47`\"603ALM
		Simplicial Models for Algebras and Modules
		Limits and Colimits of Modules
		Free Modules
	Bimodules
		The -Operad `39`42`\"613A``45`47`\"603AB M
		Bimodules, Left Modules, and Right Modules
		Limits, Colimits, and Free Bimodules
	The Relative Tensor Product
		Multilinear Maps
		Tensor Products and the Bar Construction
		Associativity of the Tensor Product
	Modules over Commutative Algebras
		Left and Right Modules over Commutative Algebras
		Tensor Products over Commutative Algebras
		Change of Algebra
		Rectification of Commutative Algebras
	Duality
		Duality in Monoidal -Categories
		Duality of Bimodules
		Exchanging Right and Left Actions
		Smooth and Proper Algebras
		Frobenius Algebras
	Monads and the Barr-Beck Theorem
		Endomorphism -Categories
		Split Simplicial Objects
		The Barr-Beck Theorem
		BiCartesian Fibrations
		Descent and the Beck-Chevalley Condition
	Tensor Products of -Categories
		Tensor Products of -Categories
		Smash Products of Spectra
		Algebras and their Module Categories
		Properties of `39`42`\"613A``45`47`\"603ARModA(`39`42`\"613A``45`47`\"603AC)
		Behavior of the Functor
Little Cubes and Factorizable Sheaves
	Definitions and Basic Properties
		Little Cubes and Configuration Spaces
		The Additivity Theorem
		Tensor Products of Ek-Modules
		Comparison of Tensor Products
	Bar Constructions and Koszul Duality
		Twisted Arrow -Categories
		The Bar Construction for Associative Algebras
		Iterated Bar Constructions
		Reduced Pairings
		Koszul Duality for Ek-Algebras
		Iterated Loop Spaces
	Centers and Centralizers
		Centers and Centralizers
		The Adjoint Representation
		Tensor Products of Free Algebras
	Little Cubes and Manifold Topology
		Embeddings of Topological Manifolds
		Variations on the Little Cubes Operads
		Digression: Nonunital Associative Algebras and their Modules
		Nonunital Ek-Algebras
		Little Cubes in a Manifold
	Topological Chiral Homology
		The Ran Space
		Topological Chiral Homology
		Properties of Topological Chiral Homology
		Factorizable Cosheaves and Ran Integration
		Verdier Duality
		Nonabelian Poincare Duality
The Calculus of Functors
	The Calculus of Functors
		n-Excisive Functors
		The Taylor Tower
		Functors of Many Variables
		Symmetric Functors
		Functors from Spaces to Spectra
		Norm Maps
	Differentiation
		Derivatives of Functors
		Stabilization of Differentiable Fibrations
		Differentials of Functors
		Generalized Smash Products
		Stabilization of -Operads
		Uniqueness of Stabilizations
	The Chain Rule
		Cartesian Structures
		Composition of Correspondences
		Derivatives of the Identity Functor
		Differentiation and Reduction
		Consequences of Theorem 6.3.3.14
		The Dual Chain Rule
Algebra in the Stable Homotopy Category
	Structured Ring Spectra
		E1-Rings and Their Modules
		Recognition Principles
		Change of Ring
		Algebras over Commutative Rings
	Properties of Rings and Modules
		Free Resolutions and Spectral Sequences
		Flat and Projective Modules
		Localizations and Ore Conditions
		Finiteness Properties of Rings and Modules
	The Cotangent Complex Formalism
		Stable Envelopes and Tangent Bundles
		Relative Adjunctions
		The Relative Cotangent Complex
		Tangent Bundles to -Categories of Algebras
		The Cotangent Complex of an Ek-Algebra
		The Tangent Correspondence
	Deformation Theory
		Square-Zero Extensions
		Deformation Theory of E-Algebras
		Connectivity and Finiteness of the Cotangent Complex
	ÉtaleMorphisms
		ÉtaleMorphisms of E1-Rings
		The Nonconnective Case
		Cocentric Morphisms
		ÉtaleMorphisms of Ek-Rings
Constructible Sheaves and Exit Paths
	Locally Constant Sheaves
	Homotopy Invariance
	The Seifert-van Kampen Theorem
	Singular Shape
	Constructible Sheaves
	-Categories of Exit Paths
	A Seifert-van Kampen Theorem for Exit Paths
	Digression: Recollement
	Exit Paths and Constructible Sheaves
Categorical Patterns
	`39`42`\"613A``45`47`\"603AP-Anodyne Morphisms
	The Model Structure on (Set+)/`39`42`\"613A``45`47`\"603AP
	Flat Inner Fibrations
	Functoriality
	General Index
	Notation Index