Higher Algebra

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