Table Of Content

Higher Algebra May 16, 2016 2 Contents 1 Stable ∞-Categories 13 1.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.2 The Homotopy Category of a Stable ∞-Category . . . . . . . . . . . . . . . . . . . . . 17 1.1.3 Closure Properties of Stable ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.1.4 Exact Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.2 Stable ∞-Categories and Homological Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.2.1 t-Structures on Stable ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.2.2 Filtered Objects and Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.2.3 The Dold-Kan Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.2.4 The ∞-Categorical Dold-Kan Correspondence. . . . . . . . . . . . . . . . . . . . . . . 53 1.3 Homological Algebra and Derived Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 1.3.1 Nerves of Differential Graded Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 63 1.3.2 Derived ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 1.3.3 The Universal Property of D−(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 1.3.4 Inverting Quasi-Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 1.3.5 Grothendieck Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 1.4 Spectra and Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 1.4.1 The Brown Representability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 1.4.2 Spectrum Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 1.4.3 The ∞-Category of Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 1.4.4 Presentable Stable ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 2 ∞-Operads 127 2.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 2.1.1 From Colored Operads to ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 2.1.2 Maps of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 2.1.3 Algebra Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 2.1.4 ∞-Preoperads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 2.2 Constructions of ∞-Operads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 2.2.1 Subcategories of O-Monoidal ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . 148 2.2.2 Slicing ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 2.2.3 Coproducts of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 2.2.4 Monoidal Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 2.2.5 Tensor Products of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 2.3 Disintegration and Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 2.3.1 Unital ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 2.3.2 Generalized ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 2.3.3 Approximations to ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 3 4 CONTENTS 2.3.4 Disintegration of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 2.4 Products and Coproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 2.4.1 Cartesian Symmetric Monoidal Structures . . . . . . . . . . . . . . . . . . . . . . . . . 198 2.4.2 Monoid Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 2.4.3 CoCartesian Symmetric Monoidal Structures . . . . . . . . . . . . . . . . . . . . . . . 207 2.4.4 Wreath Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 3 Algebras and Modules over ∞-Operads 221 3.1 Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 3.1.1 Operadic Colimit Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 3.1.2 Operadic Left Kan Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 3.1.3 Construction of Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 3.1.4 Transitivity of Operadic Left Kan Extensions . . . . . . . . . . . . . . . . . . . . . . . 247 3.2 Limits and Colimits of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 3.2.1 Unit Objects and Trivial Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 3.2.2 Limits of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.2.3 Colimits of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 3.2.4 Tensor Products of Commutative Algebras . . . . . . . . . . . . . . . . . . . . . . . . 262 3.3 Modules over ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 3.3.1 Coherent ∞-Operads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 3.3.2 A Coherence Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 3.3.3 Module Objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 3.4 General Features of Module ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 3.4.1 Algebra Objects of ∞-Categories of Modules . . . . . . . . . . . . . . . . . . . . . . . 286 3.4.2 Modules over Trivial Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 3.4.3 Limits of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 3.4.4 Colimits of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 4 Associative Algebras and Their Modules 325 4.1 Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 4.1.1 The ∞-Operad Ass⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 4.1.2 Simplicial Models for Associative Algebras. . . . . . . . . . . . . . . . . . . . . . . . . 331 4.1.3 Monoidal Model Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 4.1.4 Rectification of Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 4.2 Left and Right Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 4.2.1 The ∞-Operad LM⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 4.2.2 Simplicial Models for Algebras and Modules . . . . . . . . . . . . . . . . . . . . . . . . 356 4.2.3 Limits and Colimits of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 4.2.4 Free Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 4.3 Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 4.3.1 The ∞-Operad BM⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 4.3.2 Bimodules, Left Modules, and Right Modules . . . . . . . . . . . . . . . . . . . . . . . 376 4.3.3 Limits, Colimits, and Free Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 4.4 The Relative Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 4.4.1 Multilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 4.4.2 Tensor Products and the Bar Construction . . . . . . . . . . . . . . . . . . . . . . . . 401 4.4.3 Associativity of the Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 4.5 Modules over Commutative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 4.5.1 Left and Right Modules over Commutative Algebras . . . . . . . . . . . . . . . . . . . 416 4.5.2 Tensor Products over Commutative Algebras . . . . . . . . . . . . . . . . . . . . . . . 420 4.5.3 Change of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 CONTENTS 5 4.5.4 Rectification of Commutative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 4.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 4.6.1 Duality in Monoidal ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 4.6.2 Duality of Bimodules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 4.6.3 Exchanging Right and Left Actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 4.6.4 Smooth and Proper Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 4.6.5 Frobenius Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 4.7 Monads and the Barr-Beck Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 4.7.1 Technical Digression: Simplicial Models for Planar ∞-Operads . . . . . . . . . . . . . 465 4.7.2 Endomorphism ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 4.7.3 Split Simplicial Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 4.7.4 The Barr-Beck Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 4.7.5 BiCartesian Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 4.7.6 Descent and the Beck-Chevalley Condition . . . . . . . . . . . . . . . . . . . . . . . . 504 4.8 Tensor Products of ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 4.8.1 Tensor Products of ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 4.8.2 Smash Products of Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 4.8.3 Algebras and their Module Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 4.8.4 Properties of RMod (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 A 4.8.5 Behavior of the Functor Θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 5 Little Cubes and Factorizable Sheaves 545 5.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 5.1.1 Little Cubes and Configuration Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 5.1.2 The Additivity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 5.1.3 Tensor Products of E -Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 k 5.1.4 Comparison of Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 5.2 Bar Constructions and Koszul Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 5.2.1 Twisted Arrow ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 5.2.2 The Bar Construction for Associative Algebras . . . . . . . . . . . . . . . . . . . . . . 595 5.2.3 Iterated Bar Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 5.2.4 Reduced Pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 5.2.5 Koszul Duality for E -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 k 5.2.6 Iterated Loop Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 5.3 Centers and Centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 5.3.1 Centers and Centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 5.3.2 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 5.3.3 Tensor Products of Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 5.4 Little Cubes and Manifold Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 5.4.1 Embeddings of Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 5.4.2 Variations on the Little Cubes Operads . . . . . . . . . . . . . . . . . . . . . . . . . . 676 5.4.3 Digression: Nonunital Associative Algebras and their Modules . . . . . . . . . . . . . 679 5.4.4 Nonunital E -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 k 5.4.5 Little Cubes in a Manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 5.5 Topological Chiral Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 5.5.1 The Ran Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 5.5.2 Topological Chiral Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 5.5.3 Properties of Topological Chiral Homology . . . . . . . . . . . . . . . . . . . . . . . . 713 5.5.4 Factorizable Cosheaves and Ran Integration . . . . . . . . . . . . . . . . . . . . . . . . 717 5.5.5 Verdier Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 5.5.6 Nonabelian Poincare Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 6 CONTENTS 6 The Calculus of Functors 739 6.1 The Calculus of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 6.1.1 n-Excisive Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 6.1.2 The Taylor Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748 6.1.3 Functors of Many Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757 6.1.4 Symmetric Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765 6.1.5 Functors from Spaces to Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772 6.1.6 Norm Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776 6.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 6.2.1 Derivatives of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785 6.2.2 Stabilization of Differentiable Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . 793 6.2.3 Differentials of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802 6.2.4 Generalized Smash Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812 6.2.5 Stabilization of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815 6.2.6 Uniqueness of Stabilizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 6.3 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 6.3.1 Cartesian Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 6.3.2 Composition of Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 6.3.3 Derivatives of the Identity Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849 6.3.4 Differentiation and Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854 6.3.5 Consequences of Theorem 6.3.3.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862 6.3.6 The Dual Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869 7 Algebra in the Stable Homotopy Category 881 7.1 Structured Ring Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882 7.1.1 E -Rings and Their Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883 1 7.1.2 Recognition Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888 7.1.3 Change of Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893 7.1.4 Algebras over Commutative Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898 7.2 Properties of Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904 7.2.1 Free Resolutions and Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 904 7.2.2 Flat and Projective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911 7.2.3 Localizations and Ore Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919 7.2.4 Finiteness Properties of Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . 928 7.3 The Cotangent Complex Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941 7.3.1 Stable Envelopes and Tangent Bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . 943 7.3.2 Relative Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948 7.3.3 The Relative Cotangent Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956 7.3.4 Tangent Bundles to ∞-Categories of Algebras . . . . . . . . . . . . . . . . . . . . . . . 966 7.3.5 The Cotangent Complex of an E -Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 974 k 7.3.6 The Tangent Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 7.4 Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988 7.4.1 Square-Zero Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988 7.4.2 Deformation Theory of E -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999 ∞ 7.4.3 Connectivity and Finiteness of the Cotangent Complex . . . . . . . . . . . . . . . . . 1007 7.5 E´taleMorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018 7.5.1 E´taleMorphisms of E -Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019 1 7.5.2 The Nonconnective Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 7.5.3 Cocentric Morphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1030 7.5.4 E´taleMorphisms of E -Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035 k CONTENTS 7 A Constructible Sheaves and Exit Paths 1041 A.1 Locally Constant Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042 A.2 Homotopy Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045 A.3 The Seifert-van Kampen Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 A.4 Singular Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053 A.5 Constructible Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 A.6 ∞-Categories of Exit Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060 A.7 A Seifert-van Kampen Theorem for Exit Paths . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 A.8 Digression: Recollement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 A.9 Exit Paths and Constructible Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 B Categorical Patterns 1091 B.1 P-Anodyne Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 B.2 The Model Structure on (Set+) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103 ∆ /P B.3 Flat Inner Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112 B.4 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122 General Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138 Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146 8 CONTENTS LetK denotethefunctorofcomplex K-theory,whichassociatestoeverycompactHausdorffspaceX the GrothendieckgroupK(X)ofisomorphismclassesofcomplexvectorbundlesonX. ThefunctorX (cid:55)→K(X) is an example of a cohomology theory: that is, one can define more generally a sequence of abelian groups {Kn(X,Y)} for every inclusion of topological spaces Y ⊆X, in such a way that the Eilenberg-Steenrod n∈Z axioms are satisfied (see [49]). However, the functor K is endowed with even more structure: for every topological space X, the abelian group K(X) has the structure of a commutative ring (when X is compact, the multiplication on K(X) is induced by the operation of tensor product of complex vector bundles). One would like that the ring structure on K(X) is a reflection of the fact that K itself has a ring structure, in a suitable setting. To analyze the problem in greater detail, we observe that the functor X (cid:55)→K(X) is representable. That is,thereexistsatopologicalspaceZ =Z×BU andauniversalclassη ∈K(Z),suchthatforeverysufficiently nice topological space X, the pullback of η induces a bijection [X,Z] → K(X); here [X,Z] denotes the set of homotopy classes of maps from X into Z. According to Yoneda’s lemma, this property determines the space Z up to homotopy equivalence. Moreover, since the functor X (cid:55)→ K(X) takes values in the category of commutative rings, the topological space Z is automatically a commutative ring object in the homotopy category H of topological spaces. That is, there exist addition and multiplication maps Z ×Z → Z, such that all of the usual ring axioms are satisfied up to homotopy. Unfortunately, this observation is not very useful. We would like to have a robust generalization of classical algebra which includes a good theory of modules, constructions like localization and completion, and so forth. The homotopy category H is too poorly behaved to support such a theory. An alternate possibility is to work with commutative ring objects in the category of topological spaces itself: that is, to require the ring axioms to hold “on the nose” and not just up to homotopy. Although this does lead to a reasonable generalization of classical commutative algebra, it not sufficiently general for manypurposes. Forexample,ifZ isatopologicalcommutativering,thenonecanalwaysextendthefunctor X (cid:55)→ [X,Z] to a cohomology theory. However, this cohomology theory is not very interesting: in degree zero, it simply gives the following variant of classical cohomology: (cid:89) Hn(X;π Z). n n≥0 In particular, complex K-theory cannot be obtained in this way. In other words, the Z =Z×BU for stable vector bundles cannot be equipped with the structure of a topological commutative ring. This reflects the fact that complex vector bundles on a space X form a category, rather than just a set. The direct sum and tensor product operation on complex vector bundles satisfy the ring axioms, such as the distributive law E⊗(F⊕F(cid:48))(cid:39)(E⊗F)⊕(E⊗F(cid:48)), but only up to isomorphism. However, although Z×BU has less structure than a commutative ring, it has morestructurethansimplyacommutativeringobjectinthehomotopycategoryH,becausetheisomorphism displayed above is actually canonical and satisfies certain coherence conditions (see [91] for a discussion). To describe the kind of structure which exists on the topological space Z × BU, it is convenient to introduce the language of commutative ring spectra, or, as we will call them, E -rings. Roughly speaking, ∞ an E -ring can be thought of as a space Z which is equipped with an addition and a multiplication for ∞ which the axioms for a commutative ring hold not only up to homotopy, but up to coherent homotopy. The E -rings play a role in stable homotopy theory analogous to the role played by commutative rings in ∞ ordinary algebra. As such, they are the fundamental building blocks of derived algebraic geometry. One of our ultimate goals in this book is to give an exposition of the theory of E -rings. Recall that ∞ ordinary commutative ring R can be viewed as a commutative algebra object in the category of abelian groups, which we view as endowed with a symmetric monoidal structure given by tensor product of abelian groups. ToobtainthetheoryofE -ringswewillusethesamedefinition,replacingabeliangroupsbyspectra ∞ (certain algebro-topological objects which represent cohomology theories). To carry this out in detail, we need to say exactly what a spectrum is. There are many different definitions in the literature, having a CONTENTS 9 variety of technical advantages and disadvantages. Some modern approaches to stable homotopy theory have the feature that the collection of spectra is realized as a symmetric monoidal category (and one can define an E -ring to be a commutative algebra object of this category): see, for example, [73]. ∞ We will take a different approach, using the framework of ∞-categories developed in [97]. The collection of all spectra can be organized into an ∞-category, which we will denote by Sp: it is an ∞-categorical counterpart of the ordinary category of abelian groups. The tensor product of abelian groups also has a counterpart: the smash product functor on spectra. In order to describe the situation systematically, we introduce the notion of a symmetric monoidal ∞-category: that is, an ∞-category C equipped with a tensor product functor ⊗ : C×C → C which is commutative and associative up to coherent homotopy. For any symmetric monoidal ∞-category C, there is an associated theory of commutative algebra objects, which are themselves organized into an ∞-category CAlg(C). We can then define an E -ring to be a commutative ∞ algebraobjectofthe∞-categoryofspectra,endowedwiththesymmetricmonoidalstructuregivenbysmash products. We now briefly outline the contents of this book (more detailed outlines can be found at the beginning of individual sections and chapters). Much of this book is devoted to developing an adequate language to make sense of the preceding paragraph. We will begin in Chapter 1 by introducing the notion of a stable ∞-category. Roughly speaking, the notion of stable ∞-category is obtained by axiomatizing the essential feature of stable homotopy theory: fiber sequences are the same as cofiber sequences. The ∞-category Sp of spectra is an example of a stable ∞-category. In fact, it is universal among stable ∞-categories: we will show that Sp is freely generated (as a stable ∞-category which admits small colimits) by a single object (see Corollary 1.4.4.6). However, there are a number of stable ∞-categories that are of interest in other contexts. Forexample,thederivedcategoryofanabeliancategorycanberealizedasthehomotopycategory of a stable ∞-category. We may therefore regard the theory of stable ∞-categories as a generalization of homological algebra, which has many applications in pure algebra and algebraic geometry. We can think of an ∞-category C as comprised of a collection of objects X,Y,Z,... ∈ C, together with a mapping space Map (X,Y) for every pair of objects X,Y ∈ C (which are equipped with coherently C associativecompositionlaws). InChapter2,wewillstudyavariationonthenotionof∞-category,whichwe callan∞-operad. Roughlyspeaking, an∞-operadOconsistsofacollectionofobjectstogetherwithaspace of operations MulO({Xi}1≤i≤n,Y)} for every finite collection of objects X1,...,Xn,Y ∈O (again equipped with coherently associative multiplication laws). As a special case, we will obtain a theory of symmetric monoidal ∞-categories. Given a pair of ∞-operads O and C, the collection of maps from O to C is naturally organized into an ∞-category which we will denote by Alg (C), and refer to as the ∞-category of O-algebra objects of C. An O important special case is when O is the commutative ∞-operad and C is a symmetric monoidal ∞-category: in this case, we will refer to Alg (C) as the ∞-category of commutative algebra objects of C and denote it by O CAlg(C). We will make a thorough study of algebra objects (commutative and otherwise) in Chapter 3. In Chapter 4, we will specialize our general theory of algebras to the case where O is the associative ∞- operad. In this case, we will denote Alg (C) by Alg(C) and refer to it the ∞-category of associative algebra O objects of C. The ∞-categorical theory of associative algebra objects is an excellent formal parallel of the usualtheoryofassociativealgebras. Forexample, onecanstudyleftmodules, rightmodules, andbimodules over associative algebras. This theory of modules has some nontrivial applications; for example, in §4.7 we will use it to prove an ∞-categorical analogue of the Barr-Beck theorem, which has many applications in higher category theory. In ordinary algebra, there is a thin line dividing the theory of commutative rings from the theory of associative rings: a commutative ring R is just an associative ring whose elements satisfy the additional identity xy = yx. In the ∞-categorical setting, the situation is rather different. Between the theory of associative and commutative algebras is a whole hierarchy of intermediate notions of commutativity, which are described by the “little cubes” operads of Boardman and Vogt. In Chapter 5, we will introduce the notion of an E -algebra for each 0≤k ≤∞. This definition reduces to the notion of an associative algebra k in the case k =1, and to the notion of a commutative algebra when k =∞. The theory of E -algebras has k many applications in intermediate cases 1<k <∞, and is closely related to the topology of k-dimensional 10 CONTENTS manifolds. The theory of differential calculus provides techniques for analyizing a general (smooth) function f : R → R by studying linear functions which approximate f. A fundamental insight of Goodwillie is that the same ideas can be fruitfully applied to problems in homotopy theory. More precisely, we can sometimes reduce questions about general ∞-categories and general functors to questions about stable ∞-categories and exact functors, which are more amenable to attack by algebraic methods. In Chapter 6 we will develop Goodwillie’s calculus of functors in the ∞-categorical setting. Moreover, we will apply our theory of ∞- operads to formulate and prove a Koszul dual version of the chain rule of Arone-Ching. In Chapter 7, we will study E -algebra objects in the symmetric monoidal ∞-category of spectra, which k werefertoasE -rings. Thiscanberegardedasarobustgeneralizationofordinarynoncommutativealgebra k (when k =1) or commutative algebra (when k ≥2). In particular, we will see that a great deal of classical commutative algebra can be extended to the setting of E -rings. ∞ We close the book with two appendices. Appendix A develops the theory of constructible sheaves on stratified topological spaces. Aside from its intrinsic interest, this theory has a close connection with some of the geometric ideas of Chapter 5 and should prove useful in facilitating the application of those ideas. Appendix B is devoted to some rather technical existence results for model category structures on (and Quillen functors between) certain categories of simplicial sets. We recommend that the reader refer to this material only as necessary. Prerequisites The following definition will play a central role in this book: Definition 0.0.0.1. An ∞-category is a simplicial set C which satisfies the following extension condition: (∗) Every map of simplicial sets f :Λn →C can be extended to an n-simplex f :∆n →C, provided that 0 i 0<i<n. Remark 0.0.0.2. The notion of ∞-category was introduced by Boardman and Vogt under the name weak Kancomplexin[19]. TheyhavebeenstudiedextensivelybyJoyal,andareoftenreferredtoasquasicategories in the literature. If E is a category, then the nerve N(E) of E is an ∞-category. Consequently, we can think of the theory of ∞-categories as a generalization of category theory. It turns out to be a robust generalization: most of the important concepts from classical category theory (limits and colimits, adjoint functors, sheaves and presheaves, etcetera) can be generalized to the setting of ∞-categories. For a detailed exposition, we refer the reader to our book [97]. Remark 0.0.0.3. For a different treatment of the theory of ∞-categories, we refer the reader to Joyal’s notes [78]. Other references include [19], [82], [79], [80], [115], [39], [40], [121], and [63]. Apartfrom[97],theformalprerequisitesforreadingthisbookarefew. Wewillassumethatthereaderis familiarwiththehomotopytheoryofsimplicialsets(goodreferencesonthisinclude[105]and[57])andwith abitofhomologicalalgebra(forwhichwerecommend[160]). Familiaritywithotherconceptsfromalgebraic topology (spectra, cohomology theories, operads, etcetera) will be helpful, but not strictly necessary: one of the main goals of this book is to give a self-contained exposition of these topics from an ∞-categorical perspective. Notation and Terminology We now call the reader’s attention to some of the terminology used in this book: • We will make extensive use of definitions and notations from the book [97]. If the reader encounters something confusing or unfamiliar, we recommend looking there first. We adopt the convention that