Higher Algebra

✍ Scribed by Lurie J.

• Year: 2017
• Tongue: English
• Leaves: 1553
• Category: Library

✦ Table of Contents

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 3942"613A``4547"603AD-(3942"613A45`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`"613A4547"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 3942"613A``4547"603ALM Simplicial Models for Algebras and Modules Limits and Colimits of Modules Free Modules Bimodules The -Operad3942"613A45`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`"613A4547"603ARModA(3942"613A``4547"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 Patterns3942"613A45`47`"603AP-Anodyne Morphisms The Model Structure on (Set+)/`39`42`"613A4547"603AP
Flat Inner Fibrations
Functoriality
General Index
Notation Index