higher algebra - people.math.harvard.edupeople.math.harvard.edu/~lurie/papers/ha.pdf · contents 5...

Higher Algebra

September 18, 2017

Contents

1 Stable ∞-Categories 151.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.1.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.1.2 The Homotopy Category of a Stable ∞-Category . . . . . . . . . . . . . . . . 201.1.3 Closure Properties of Stable ∞-Categories . . . . . . . . . . . . . . . . . . . . 301.1.4 Exact Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.2 Stable ∞-Categories and Homological Algebra . . . . . . . . . . . . . . . . . . . . . 361.2.1 t-Structures on Stable ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . 371.2.2 Filtered Objects and Spectral Sequences . . . . . . . . . . . . . . . . . . . . . 47

1.2.3 The Dold-Kan Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1.2.4 The ∞-Categorical Dold-Kan Correspondence . . . . . . . . . . . . . . . . . . 661.3 Homological Algebra and Derived Categories . . . . . . . . . . . . . . . . . . . . . . 78

1.3.1 Nerves of Differential Graded Categories . . . . . . . . . . . . . . . . . . . . . 78

1.3.2 Derived ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871.3.3 The Universal Property of D−(A) . . . . . . . . . . . . . . . . . . . . . . . . 97

1.3.4 Inverting Quasi-Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

1.3.5 Grothendieck Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . . 121

1.4 Spectra and Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

1.4.1 The Brown Representability Theorem . . . . . . . . . . . . . . . . . . . . . . 134

1.4.2 Spectrum Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

1.4.3 The ∞-Category of Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1501.4.4 Presentable Stable ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . 154

2 ∞-Operads 1642.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

2.1.1 From Colored Operads to ∞-Operads . . . . . . . . . . . . . . . . . . . . . . 1712.1.2 Maps of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1782.1.3 Algebra Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

2.1.4 ∞-Preoperads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

2

CONTENTS 3

2.2 Constructions of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1922.2.1 Subcategories of O-Monoidal ∞-Categories . . . . . . . . . . . . . . . . . . . 1932.2.2 Slicing ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2002.2.3 Coproducts of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2062.2.4 Monoidal Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

2.2.5 Tensor Products of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . 2192.2.6 Day Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

2.3 Disintegration and Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

2.3.1 Unital ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2432.3.2 Generalized ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2492.3.3 Approximations to ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . 2582.3.4 Disintegration of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

2.4 Products and Coproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

2.4.1 Cartesian Symmetric Monoidal Structures . . . . . . . . . . . . . . . . . . . . 283

2.4.2 Monoid Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

2.4.3 CoCartesian Symmetric Monoidal Structures . . . . . . . . . . . . . . . . . . 297

2.4.4 Wreath Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

3 Algebras and Modules over ∞-Operads 3153.1 Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

3.1.1 Operadic Colimit Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

3.1.2 Operadic Left Kan Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 329

3.1.3 Construction of Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

3.1.4 Transitivity of Operadic Left Kan Extensions . . . . . . . . . . . . . . . . . . 349

3.2 Limits and Colimits of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

3.2.1 Unit Objects and Trivial Algebras . . . . . . . . . . . . . . . . . . . . . . . . 354

3.2.2 Limits of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

3.2.3 Colimits of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

3.2.4 Tensor Products of Commutative Algebras . . . . . . . . . . . . . . . . . . . 369

3.3 Modules over ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3733.3.1 Coherent ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3753.3.2 A Coherence Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

3.3.3 Module Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

3.4 General Features of Module ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . 3993.4.1 Algebra Objects of ∞-Categories of Modules . . . . . . . . . . . . . . . . . . 4003.4.2 Modules over Trivial Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

3.4.3 Limits of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

3.4.4 Colimits of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

4 CONTENTS

4 Associative Algebras and Their Modules 449

4.1 Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

4.1.1 The Associative ∞-Operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4544.1.2 Monoid Objects of ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . 4604.1.3 Planar ∞-Operads and A∞-Algebras . . . . . . . . . . . . . . . . . . . . . . . 4654.1.4 Nonunital An-Algebras and Nonunital An-Monoids . . . . . . . . . . . . . . . 4694.1.5 From An-Algebras to An+1-Algebras . . . . . . . . . . . . . . . . . . . . . . . 4744.1.6 The Associahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

4.1.7 Monoidal Model Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

4.1.8 Rectification of Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . 491

4.2 Left and Right Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

4.2.1 The ∞-Operad LM⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5044.2.2 Simplicial Models for Algebras and Modules . . . . . . . . . . . . . . . . . . . 512

4.2.3 Limits and Colimits of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 524

4.2.4 Free Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

4.3 Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

4.3.1 The ∞-Operad BM⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5354.3.2 Bimodules, Left Modules, and Right Modules . . . . . . . . . . . . . . . . . . 539

4.3.3 Limits, Colimits, and Free Bimodules . . . . . . . . . . . . . . . . . . . . . . 549

4.4 The Relative Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556

4.4.1 Multilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

4.4.2 Tensor Products and the Bar Construction . . . . . . . . . . . . . . . . . . . 573

4.4.3 Associativity of the Tensor Product . . . . . . . . . . . . . . . . . . . . . . . 580

4.5 Modules over Commutative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

4.5.1 Left and Right Modules over Commutative Algebras . . . . . . . . . . . . . . 592

4.5.2 Tensor Products over Commutative Algebras . . . . . . . . . . . . . . . . . . 598

4.5.3 Change of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

4.5.4 Rectification of Commutative Algebras . . . . . . . . . . . . . . . . . . . . . . 606

4.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615

4.6.1 Duality in Monoidal ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . 6164.6.2 Duality of Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

4.6.3 Exchanging Right and Left Actions . . . . . . . . . . . . . . . . . . . . . . . . 630

4.6.4 Smooth and Proper Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

4.6.5 Frobenius Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

4.7 Monads and the Barr-Beck Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 656

4.7.1 Endomorphism ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 6584.7.2 Split Simplicial Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677

4.7.3 The Barr-Beck Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

CONTENTS 5

4.7.4 BiCartesian Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

4.7.5 Descent and the Beck-Chevalley Condition . . . . . . . . . . . . . . . . . . . 702

4.8 Tensor Products of ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7064.8.1 Tensor Products of ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . 7074.8.2 Smash Products of Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716

4.8.3 Algebras and their Module Categories . . . . . . . . . . . . . . . . . . . . . . 723

4.8.4 Properties of RModA(C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730

4.8.5 Behavior of the Functor Θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743

5 Little Cubes and Factorizable Sheaves 756

5.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758

5.1.1 Little Cubes and Configuration Spaces . . . . . . . . . . . . . . . . . . . . . . 761

5.1.2 The Additivity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

5.1.3 Tensor Products of Ek-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.1.4 Comparison of Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . 792

5.2 Bar Constructions and Koszul Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 801

5.2.1 Twisted Arrow ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 8035.2.2 The Bar Construction for Associative Algebras . . . . . . . . . . . . . . . . . 821

5.2.3 Iterated Bar Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839

5.2.4 Reduced Pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848

5.2.5 Koszul Duality for Ek-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 8585.2.6 Iterated Loop Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870

5.3 Centers and Centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880

5.3.1 Centers and Centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882

5.3.2 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896

5.3.3 Tensor Products of Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 905

5.4 Little Cubes and Manifold Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 922

5.4.1 Embeddings of Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . 922

5.4.2 Variations on the Little Cubes Operads . . . . . . . . . . . . . . . . . . . . . 929

5.4.3 Digression: Nonunital Associative Algebras and their Modules . . . . . . . . 933

5.4.4 Nonunital Ek-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9455.4.5 Little Cubes in a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955

5.5 Topological Chiral Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963

5.5.1 The Ran Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964

5.5.2 Topological Chiral Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 971

5.5.3 Properties of Topological Chiral Homology . . . . . . . . . . . . . . . . . . . 977

5.5.4 Factorizable Cosheaves and Ran Integration . . . . . . . . . . . . . . . . . . . 983

5.5.5 Verdier Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991

5.5.6 Nonabelian Poincare Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 998

6 CONTENTS

6 The Calculus of Functors 1011

6.1 The Calculus of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012

6.1.1 n-Excisive Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015

6.1.2 The Taylor Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023

6.1.3 Functors of Many Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035

6.1.4 Symmetric Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045

6.1.5 Functors from Spaces to Spectra . . . . . . . . . . . . . . . . . . . . . . . . . 1055

6.1.6 Norm Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

6.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070

6.2.1 Derivatives of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072

6.2.2 Stabilization of Differentiable Fibrations . . . . . . . . . . . . . . . . . . . . . 1082

6.2.3 Differentials of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094

6.2.4 Generalized Smash Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107

6.2.5 Stabilization of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11116.2.6 Uniqueness of Stabilizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121

6.3 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1130

6.3.1 Cartesian Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135

6.3.2 Composition of Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . 1149

6.3.3 Derivatives of the Identity Functor . . . . . . . . . . . . . . . . . . . . . . . . 1157

6.3.4 Differentiation and Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163

6.3.5 Consequences of Theorem 6.3.3.14 . . . . . . . . . . . . . . . . . . . . . . . . 1173

6.3.6 The Dual Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182

7 Algebra in the Stable Homotopy Category 1196

7.1 Structured Ring Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197

7.1.1 E1-Rings and Their Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 11997.1.2 Recognition Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206

7.1.3 Change of Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213

7.1.4 Algebras over Commutative Rings . . . . . . . . . . . . . . . . . . . . . . . . 1220

7.2 Properties of Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227

7.2.1 Free Resolutions and Spectral Sequences . . . . . . . . . . . . . . . . . . . . . 1228

7.2.2 Flat and Projective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237

7.2.3 Localizations and Ore Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 1247

7.2.4 Finiteness Properties of Rings and Modules . . . . . . . . . . . . . . . . . . . 1260

7.3 The Cotangent Complex Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276

7.3.1 Stable Envelopes and Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . 1280

7.3.2 Relative Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287

7.3.3 The Relative Cotangent Complex . . . . . . . . . . . . . . . . . . . . . . . . . 1297

7.3.4 Tangent Bundles to ∞-Categories of Algebras . . . . . . . . . . . . . . . . . . 1309

CONTENTS 7

7.3.5 The Cotangent Complex of an Ek-Algebra . . . . . . . . . . . . . . . . . . . . 13207.3.6 The Tangent Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328

7.4 Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334

7.4.1 Square-Zero Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335

7.4.2 Deformation Theory of E∞-Algebras . . . . . . . . . . . . . . . . . . . . . . . 13497.4.3 Connectivity and Finiteness of the Cotangent Complex . . . . . . . . . . . . 1359

7.5 ÉtaleMorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373

7.5.1 ÉtaleMorphisms of E1-Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 13757.5.2 The Nonconnective Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1381

7.5.3 Cocentric Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1388

7.5.4 ÉtaleMorphisms of Ek-Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395

A Constructible Sheaves and Exit Paths 1401

A.1 Locally Constant Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1402

A.2 Homotopy Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407

A.3 The Seifert-van Kampen Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1412

A.4 Singular Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418

A.5 Constructible Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1421

A.6 ∞-Categories of Exit Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427A.7 A Seifert-van Kampen Theorem for Exit Paths . . . . . . . . . . . . . . . . . . . . . 1436

A.8 Digression: Recollement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443

A.9 Exit Paths and Constructible Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 1456

B Categorical Patterns 1466

B.1 P-Anodyne Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1469

B.2 The Model Structure on (Set+∆)/P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1482

B.3 Flat Inner Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493

B.4 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507

General Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1527

Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538

8 CONTENTS

Let K denote the functor of complex K-theory, which associates to every compact Hausdorff

space X the Grothendieck group K(X) of isomorphism classes of complex vector bundles on X. The

functor X 7→ K(X) is an example of a cohomology theory: that is, one can define more generallya sequence of abelian groups {Kn(X,Y )}n∈Z for every inclusion of topological spaces Y ⊆ X, insuch a way that the Eilenberg-Steenrod 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 7→ K(X) is repre-sentable. That is, there exists a topological space Z = Z×BU and a universal class η ∈ K(Z),such that for every sufficiently 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. Accordingto Yoneda’s lemma, this property determines the space Z up to homotopy equivalence. Moreover,

since the functor X 7→ K(X) takes values in the category of commutative rings, the topologicalspace 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 theusual 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 many purposes. For example, if Z is a topological commutative ring, then

one can always extend the functor X 7→ [X,Z] to a cohomology theory. However, this cohomologytheory is not very interesting: in degree zero, it simply gives the following variant of classical

cohomology: ∏n≥0

Hn(X;πnZ).

In particular, complex K-theory cannot be obtained in this way. In other words, the Z = Z×BUfor 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′) ' (E⊗F)⊕ (E⊗F′),

but only up to isomorphism. However, although Z×BU has less structure than a commutative ring,it has more structure than simply a commutative ring object in the homotopy category H, because

CONTENTS 9

the isomorphism displayed above is actually canonical and satisfies certain coherence conditions

(see [92] for a discussion).

To describe the kind of structure which exists on the topological space Z×BU, it is convenientto introduce the language of commutative ring spectra, or, as we will call them, E∞-rings. Roughlyspeaking, an E∞-ring can be thought of as a space Z which is equipped with an addition and amultiplication 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 therole 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. To obtain the theory of E∞-rings we will use the samedefinition, replacing abelian groups by spectra (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 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∞-ringto be a commutative algebra object of this category): see, for example, [74].

We will take a different approach, using the framework of ∞-categories developed in [98]. Thecollection of all spectra can be organized into an ∞-category, which we will denote by Sp: it isan ∞-categorical counterpart of the ordinary category of abelian groups. The tensor product ofabelian 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: thatis, an ∞-category C equipped with a tensor product functor ⊗ : C×C → C which is commutativeand associative up to coherent homotopy. For any symmetric monoidal ∞-category C, there isan 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 algebra object of the∞-category of spectra, endowed with the symmetric monoidal structure given by smash 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 ∞-categoryis 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 astable∞-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. For example,

10 CONTENTS

the derived category of an abelian category can be realized as the homotopy category of a stable

∞-category. We may therefore regard the theory of stable ∞-categories as a generalization ofhomological 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, togetherwith a mapping space MapC(X,Y ) for every pair of objects X,Y ∈ C (which are equipped withcoherently associative composition laws). In Chapter 2, we will study a variation on the notion of

∞-category, which we call an ∞-operad. Roughly speaking, an ∞-operad O consists of a collectionof objects together with a space of operations MulO({Xi}1≤i≤n, Y )} for every finite collection ofobjects X1, . . . , Xn, Y ∈ O (again equipped with coherently associative multiplication laws). As aspecial 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 intoan∞-category which we will denote by AlgO(C), and refer to as the∞-category of O-algebra objectsof C. An important special case is when O is the commutative ∞-operad and C is a symmetricmonoidal ∞-category: in this case, we will refer to AlgO(C) as the ∞-category of commutativealgebra objects of C and denote it by 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 AlgO(C) by Alg(C) and refer to it the∞-categoryof associative algebra objects of C. The ∞-categorical theory of associative algebra objects is anexcellent formal parallel of the usual theory of associative algebras. For example, one can study

left modules, right modules, and bimodules over associative algebras. This theory of modules has

some nontrivial applications; for example, in §4.7 we will use it to prove an∞-categorical analogueof 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. Betweenthe 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 Ek-algebra for each 0 ≤ k ≤ ∞. This definition reducesto the notion of an associative algebra in the case k = 1, and to the notion of a commutative algebra

when k = ∞. The theory of Ek-algebras has many applications in intermediate cases 1 < k < ∞,and is closely related to the topology of k-dimensional 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 Goodwillieis 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 questionsabout stable ∞-categories and exact functors, which are more amenable to attack by algebraicmethods. In Chapter 6 we will develop Goodwillie’s calculus of functors in the ∞-categorical

CONTENTS 11

setting. Moreover, we will apply our theory of ∞-operads to formulate and prove a Koszul dualversion of the chain rule of Arone-Ching.

In Chapter 7, we will study Ek-algebra objects in the symmetric monoidal ∞-category of spec-tra, which we refer to as Ek-rings. This can be regarded as a robust generalization of ordinarynoncommutative algebra (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 extensioncondition:

(∗) Every map of simplicial sets f0 : Λni → C can be extended to an n-simplex f : ∆n → C,provided that 0 < i < n.

Remark 0.0.0.2. The notion of ∞-category was introduced by Boardman and Vogt under thename weak Kan complex in [19]. They have been studied extensively by Joyal, and are often

referred to as quasicategories in the literature.

If E is a category, then the nerve N(E) of E is an ∞-category. Consequently, we can thinkof the theory of ∞-categories as a generalization of category theory. It turns out to be a robustgeneralization: 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 [98].

Remark 0.0.0.3. For a different treatment of the theory of ∞-categories, we refer the reader toJoyal’s notes [79]. Other references include [19], [83], [80], [81], [116], [39], [40], [122], and [64].

Apart from [98], the formal prerequisites for reading this book are few. We will assume that

the reader is familiar with the homotopy theory of simplicial sets (good references on this include

[106] and [58]) and with a bit of homological algebra (for which we recommend [162]). Familiarity

with other concepts from algebraic 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.

12 CONTENTS

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 [98]. If the readerencounters something confusing or unfamiliar, we recommend looking there first. We adopt

the convention that references to [98] will be indicated by use of the letter T. For example,

Theorem HTT.6.1.0.6 refers to Theorem 6.1.0.6 of [98].

• We say that a category (or ∞-category) C is presentable if C admits small colimits andis generated under small colimits by a set of κ-compact objects, for some regular cardinal

number κ. This is departure from the standard category-theoretic terminology, in which such

categories are called locally presentable (see [1]).

• We let Set∆ denote the category of simplicial sets. If J is a linearly ordered set, we let ∆J

denote the simplicial set given by the nerve of J , so that the collection of n-simplices of

∆J can be identified with the collection of all nondecreasing maps {0, . . . , n} → J . We willfrequently apply this notation when J is a subset of {0, . . . , n}; in this case, we can identify∆J with a subsimplex of the standard n-simplex ∆n (at least if J 6= ∅; if J = ∅, then ∆J isempty).

• We will often use the term space to refer to a Kan complex (that is, a simplicial set satisfyingthe Kan extension condition).

• Let n ≥ 0. We will say that a space X is n-connective if it is nonempty and the homotopysets πi(X,x) are trivial for i < n and every vertex x of X (spaces with this property are more

commonly referred to as (n − 1)-connected in the literature). We say that X is connected ifit is 1-connective. By convention, we say that every space X is (−1)-connective. We will saythat a map of spaces f : X → Y is n-connective if the homotopy fibers of f are n-connective.

• Let n ≥ −1. We say that a space X is n-truncated if the homotopy sets πi(X,x) are trivialfor every i > n and every vertex x ∈ X. We say that X is discrete if it is 0-truncated. Byconvention, we say that X is (−2)-truncated if and only if X is contractible. We will say thata map of spaces f : X → Y is n-truncated if the homotopy fibers of f are n-truncated.

• Throughout this book, we will use homological indexing conventions whenever we discusshomological algebra. For example, chain complexes of abelian groups will be denoted by

· · · → A2 → A1 → A0 → A−1 → A−2 → · · · ,

with the differential lowering the degree by 1.

CONTENTS 13

• In Chapter 1, we will construct an ∞-category Sp, whose homotopy category hSp can beidentified with the classical stable homotopy category. In Chapter 7, we will construct a

symmetric monoidal structure on Sp, which gives (in particular) a tensor product functor

Sp×Sp → Sp. At the level of the homotopy category hSp, this functor is given by theclassical smash product of spectra, which is usually denoted by (X,Y ) 7→ X ∧ Y . We willadopt a different convention, and denote the smash product functor by (X,Y ) 7→ X ⊗ Y .

• If A is a model category, we let Ao denote the full subcategory of A spanned by the fibrant-cofibrant objects.

• Let C be an ∞-category. We let C' denote the largest Kan complex contained in C: that is,the ∞-category obtained from C by discarding all noninvertible morphisms.

• Let C be an ∞-category containing objects X and Y . We let CX/ and C/Y denote theundercategory and overcategory defined in §HTT.1.2.9 . We will generally abuse notation byidentifying objects of these ∞-categories with their images in C. If we are given a morphismf : X → Y , we can identify X with an object of C/Y and Y with an object of CX/, so thatthe ∞-categories

(CX/)/Y (C/Y )X/

are defined (and canonically isomorphic as simplicial sets). We will denote these∞-categoriesby CX//Y (beware that this notation is slightly abusive: the definition of CX//Y depends not

only on C, X, and Y , but also on the morphism f).

• Let C and D be ∞-categories. We let LFun(C,D) denote the full subcategory of Fun(C,D)spanned by those functors which admit right adjoints, and RFun(C,D) the full subcategory

of Fun(C,D) spanned by those functors which admit left adjoints. If C and D are presentable,

then these subcategories admit a simpler characterization: a functor F : C → D belongs toLFun(C,D) if and only if it preserves small colimits, and belongs to RFun(C,D) if and only

if it preserves small limits and small κ-filtered colimits for a sufficiently large regular cardinal

κ (see Corollary HTT.5.5.2.9 ).

• We will say that a map of simplicial sets f : S → T is left cofinal if, for every right fibrationX → T , the induced map of simplicial sets FunT (T,X)→ FunT (S,X) is a homotopy equiva-lence of Kan complexes (in [98], we referred to a map with this property as cofinal). We will

say that f is right cofinal if the induced map Sop → T op is left cofinal: that is, if f induces ahomotopy equivalence FunT (T,X)→ FunT (S,X) for every left fibration X → T . If S and Tare ∞-categories, then f is left cofinal if and only if for every object t ∈ T , the fiber productS ×T Tt/ is weakly contractible (Theorem HTT.4.1.3.1 ).

14 CONTENTS

Acknowledgements

In writing this book, I have benefited from the advice and assistance of many people. I would

like to thank Ben Antieau, Tobias Barthel, Clark Barwick, Dario Beraldo, Lukas Brantner, Daniel

Brügmann, Lee Cohn, Avirup Dutt, Saul Glassman, Moritz Groth, Rune Haugseng, Justin Hilburn,

Vladimir Hinich, Allen Knutson, Lev Livnev, Joseph Lipman, Sergey Lysenko, Akhil Mathew, Yo-

gesh More, Dmitri Pavlov, Anatoly Preygel, Steffen Sagave, Christian Schlichtkrull, Timo Schürg,

Elena Sendroiu, Markus Spitzweck, Hiro Tanaka, Arnav Tripathy, James Wallbridge, and Allen

Yuan for locating many mistakes in earlier versions of this book (though I am sure that there are

many left to find). I would also like to thank Matt Ando, Clark Barwick, David Ben-Zvi, Alexan-

der Beilinson, Julie Bergner, Andrew Blumberg, Dustin Clausen, Dan Dugger, Vladimir Drinfeld,

Matt Emerton, John Francis, Dennis Gaitsgory, Andre Henriques, Gijs Heuts, Mike Hopkins, Andre

Joyal, Tyler Lawson, Ieke Moerdijk, David Nadler, Anatoly Preygel, Charles Rezk, David Spivak,

Bertrand Toën, and Gabriele Vezzosi for useful conversations related to the subject matter of this

book. Finally, I would like to thank the National Science Foundation for supporting this project

under grant number 0906194.

Chapter 1

Stable ∞-Categories

There is a very useful analogy between topological spaces and chain complexes with values in

an abelian category. For example, it is customary to speak of homotopies between chain maps,

contractible complexes, and so forth. The analogue of the homotopy category of topological spaces

is the derived category of an abelian category A, a triangulated category which provides a good

setting for many constructions in homological algebra. However, it has long been recognized that

for many purposes the derived category is too crude: it identifies homotopic morphisms of chain

complexes without remembering why they are homotopic. It is possible to correct this defect by

viewing the derived category as the homotopy category of an underlying ∞-category D(A). The∞-categories which arise in this way have special features that reflect their “additive” origins: theyare stable.

We will begin in §1.1 by giving the definition of stability and exploring some of its conse-quences. For example, we will show that if C is a stable ∞-category, then its homotopy categoryhC is triangulated (Theorem 1.1.2.14), and that stable ∞-categories admit finite limits and col-imits (Proposition 1.1.3.4). The appropriate notion of functor between stable ∞-categories is anexact functor: that is, a functor which preserves finite colimits (or equivalently, finite limits: see

Proposition 1.1.4.1). The collection of stable ∞-categories and exact functors between them canbe organized into an ∞-category, which we will denote by CatEx∞ . In §1.1.4, we will establish somebasic closure properties of the ∞-category CatEx∞ ; in particular, we will show that it is closed underthe formation of limits and filtered colimits in Cat∞. The formation of limits in Cat

Ex∞ provides a

tool for addressing the classical problem of “gluing in the derived category”.

In §1.2, we recall the definition of a t-structure on a triangulated category. If C is a stable∞-category, we define a t-structure on C to be a t-structure on its homotopy category hC. If C isequipped with a t-structure, we show that every filtered object of C gives rise to a spectral sequence

taking values in the heart C♥ (Proposition 1.2.2.7). In particular, we show that every simplicial

object of C determines a spectral sequence, using an ∞-categorical analogue of the Dold-Kancorrespondence.

15

16 CHAPTER 1. STABLE ∞-CATEGORIES

We will return to the setting of homological algebra in §1.3. To any abelian category A withenough projective objects, one can associate a stable ∞-category D−(A), whose objects are (right-bounded) chain complexes of projective objects of A. This ∞-category provides useful tools fororganizing information in homological algebra. Our main result (Theorem 1.3.3.8) is a characteri-

zation of D−(A) by a universal mapping property.

In §1.4, we will focus our attention on a particular stable ∞-category: the ∞-category Sp ofspectra. The homotopy category of Sp can be identified with the classical stable homotopy category,

which is the natural setting for a large portion of modern algebraic topology. Roughly speaking, a

spectrum is a sequence of pointed spaces {X(n)}n∈Z equipped with homotopy equivalences X(n) 'ΩX(n + 1), where Ω denotes the functor given by passage to the loop space. More generally, one

can obtain a stable∞-category by considering sequences as above which take values in an arbitrary∞-category C which admits finite limits; we denote this ∞-category by Sp(C) and refer to it as the∞-category of spectrum objects of C.

1.1 Foundations

Our goal in this section is to introduce our main object of study for this chapter: the notion of

a stable ∞-category. The theory of stable ∞-categories can be regarded as an axiomatization ofthe essential features of stable homotopy theory: most notably, that fiber sequences and cofiber

sequences are the same. We will begin in §1.1.1 by reviewing some of the relevant notions (pointed∞-categories, zero objects, fiber and cofiber sequences) and using them to define the class of stable∞-categories.

In §1.1.2, we will review Verdier’s definition of a triangulated category. We will show that if Cis a stable ∞-category, then its homotopy category hC has the structure of a triangulated category(Theorem 1.1.2.14). The theory of triangulated categories can be regarded as an attempt to capture

those features of stable ∞-categories which are easily visible at the level of homotopy categories.Triangulated categories which arise naturally in mathematics are usually given as the homotopy

categories of stable ∞-categories, though it is possible to construct triangulated categories whichare not of this form (see [114]).

Our next goal is to study the properties of stable ∞-categories in greater depth. In §1.1.3, wewill show that a stable∞-category C admits all finite limits and colimits, and that pullback squaresand pushout squares in C are the same (Proposition 1.1.3.4). We will also show that the class of

stable ∞-categories is closed under various natural operations. For example, we will show that if Cis a stable ∞-category, then the ∞-category of Ind-objects Ind(C) is stable (Proposition 1.1.3.6),and that the ∞-category of diagrams Fun(K,C) is stable for any simplicial set K (Proposition1.1.3.1).

In §1.1.4, we shift our focus somewhat. Rather than concerning ourselves with the propertiesof an individual stable ∞-category C, we will study the collection of all stable ∞-categories. To

1.1. FOUNDATIONS 17

this end, we introduce the notion of an exact functor between stable ∞-categories. We will showthat the collection of all (small) stable ∞-categories and exact functors between them can itself beorganized into an ∞-category CatEx∞ , and study some of the properties of CatEx∞ .

Remark 1.1.0.1. The theory of stable∞-categories is not really new: most of the results presentedhere are well-known to experts. There exists a growing literature on the subject in the setting of

stable model categories: see, for example, [37], [127], [129], and [73]. For a brief account in the more

flexible setting of Segal categories, we refer the reader to [155].

Remark 1.1.0.2. Let k be a field. Recall that a differential graded category over k is a category

enriched over the category of chain complexes of k-vector spaces. The theory of differential graded

categories is closely related to the theory of stable ∞-categories. More precisely, one can showthat the data of a (pretriangulated) differential graded category over k is equivalent to the data

of a stable ∞-category C equipped with an enrichment over the monoidal ∞-category of k-modulespectra. The theory of differential graded categories provides a convenient language for working

with stable ∞-categories of algebraic origin (for example, those which arise from chain complexesof coherent sheaves on algebraic varieties), but is inadequate for treating examples which arise in

stable homotopy theory. There is a voluminous literature on the subject; see, for example, [85],

[102], [142], [35], and [149].

1.1.1 Stability

In this section, we introduce the definition of a stable ∞-category. We begin by reviewing somedefinitions from [98].

Definition 1.1.1.1. Let C be an ∞-category. A zero object of C is an object which is both initialand final. We will say that C is pointed if it contains a zero object.

In other words, an object 0 ∈ C is zero if the spaces MapC(X, 0) and MapC(0, X) are bothcontractible for every object X ∈ C. Note that if C contains a zero object, then that objectis determined up to equivalence. More precisely, the full subcategory of C spanned by the zero

objects is a contractible Kan complex (Proposition HTT.1.2.12.9 ).

Remark 1.1.1.2. Let C be an∞-category. Then C is pointed if and only if the following conditionsare satisfied:

(1) The ∞-category C has an initial object ∅.

(2) The ∞-category C has a final object 1.

(3) There exists a morphism f : 1→ ∅ in C.


The “only if” direction is obvious. For the converse, let us suppose that (1), (2), and (3) are satisfied.

We invoke the assumption that ∅ is initial to deduce the existence of a morphism g : ∅ → 1. Because∅ is initial, f ◦ g ' id∅, and because 1 is final, g ◦ f ' id1. Thus g is a homotopy inverse to f , sothat f is an equivalence. It follows that ∅ is also a final object of C, so that C is pointed.

Remark 1.1.1.3. Let C be an ∞-category with a zero object 0. For any X,Y ∈ C, the naturalmap

MapC(X, 0)×MapC(0, Y )→ MapC(X,Y )

has contractible domain. We therefore obtain a well defined morphism X → Y in the homotopycategory hC, which we will refer to as the zero morphism and also denote by 0.

Definition 1.1.1.4. Let C be a pointed ∞-category. A triangle in C is a diagram ∆1 ×∆1 → C,depicted as

Xf //

��

Y

g

��0 // Z

where 0 is a zero object of C. We will say that a triangle in C is a fiber sequence if it is a pullback

square, and a cofiber sequence if it is a pushout square.

Remark 1.1.1.5. Let C be a pointed ∞-category. A triangle in C consists of the following data:

(1) A pair of morphisms f : X → Y and g : Y → Z in C.

(2) A 2-simplex in C corresponding to a diagram

Yg

��X

f>>

h // Z

in C, which identifies h with the composition g ◦ f .

(3) A 2-simplex

0

��X

??

h // Z

in C, which we may view as a nullhomotopy of h.

We will generally indicate a triangle by specifying only the pair of maps

Xf→ Y g→ Z,

with the data of (2) and (3) being implicitly assumed.

1.1. FOUNDATIONS 19

Definition 1.1.1.6. Let C be a pointed ∞-category containing a morphism g : X → Y . A fiber ofg is a fiber sequence

W //

��

X

g

��0 // Y.

Dually, a cofiber of g is a cofiber sequence

Xg //

��

Y

��0 // Z.

We will generally abuse terminology by simply referring to W and Z as the fiber and cofiber of g.

We will also write W = fib(g) and Z = cofib(g).

Remark 1.1.1.7. Let C be a pointed ∞-category containing a morphism f : X → Y . A cofiber off , if it exists, is uniquely determined up to equivalence. More precisely, consider the full subcategory

E ⊆ Fun(∆1 × ∆1,C) spanned by the cofiber sequences. Let θ : E → Fun(∆1,C) be the forgetfulfunctor, which associates to a diagram

Xg //

��

Y

��0 // Z

the morphism g : X → Y . Applying Proposition HTT.4.3.2.15 twice, we deduce that θ is a Kanfibration, whose fibers are either empty or contractible (depending on whether or not a morphism

g : X → Y in C admits a cofiber). In particular, if every morphism in C admits a cofiber, then θ isa trivial Kan fibration, and therefore admits a section cofib : Fun(∆1,C)→ Fun(∆1×∆1,C), whichis well defined up to a contractible space of choices. We will often abuse notation by also letting

cofib : Fun(∆1,C)→ C denote the composition

Fun(∆1,C)→ Fun(∆1 ×∆1,C)→ C,

where the second map is given by evaluation at the final object of ∆1 ×∆1.

Remark 1.1.1.8. The functor cofib : Fun(∆1,C) → C can be identified with a left adjoint to theleft Kan extension functor C ' Fun({1},C) → Fun(∆1,C), which associates to each object X ∈ Ca zero morphism 0 → X. It follows that cofib preserves all colimits which exist in Fun(∆1,C)(Proposition HTT.5.2.3.5 ).

Definition 1.1.1.9. An ∞-category C is stable if it satisfies the following conditions:

(1) There exists a zero object 0 ∈ C.


(2) Every morphism in C admits a fiber and a cofiber.

(3) A triangle in C is a fiber sequence if and only if it a cofiber sequence.

Remark 1.1.1.10. Condition (3) of Definition 1.1.1.9 is analogous to the axiom for abelian cate-

gories which requires that the image of a morphism be isomorphic to its coimage.

Example 1.1.1.11. Recall that a spectrum consists of an infinite sequence of pointed topological

spaces {Xi}i≥0, together with homeomorphisms Xi ' ΩXi+1, where Ω denotes the loop spacefunctor. The collection of spectra can be organized into a stable ∞-category Sp. Moreover, Spis in some sense the universal example of a stable ∞-category. This motivates the terminology ofDefinition 1.1.1.9: an ∞-category C is stable if it resembles the ∞-category Sp, whose homotopycategory hSp can be identified with the classical stable homotopy category. We will return to the

theory of spectra (using a slightly different definition) in §1.4.3.

Example 1.1.1.12. Let A be an abelian category. Under mild hypotheses, we can construct

a stable ∞-category D(A) whose homotopy category hD(A) can be identified with the derivedcategory of A, in the sense of classical homological algebra. We will outline the construction of

D(A) in §1.3.2.

Remark 1.1.1.13. If C is a stable ∞-category, then the opposite ∞-category Cop is also stable.

Remark 1.1.1.14. One attractive feature of the theory of stable ∞-categories is that stability isa property of ∞-categories, rather than additional data. The situation for additive categories issimilar. Although additive categories are often presented as categories equipped with additional

structure (an abelian group structure on all Hom-sets), this additional structure is in fact deter-

mined by the underlying category: see Definition 1.1.2.1. The situation for stable ∞-categories issimilar: we will see later that every stable ∞-category is canonically enriched over the ∞-categoryof spectra.

1.1.2 The Homotopy Category of a Stable ∞-Category

Let M be a module over a commutative ring R. Then M admits a resolution

· · · → P2 → P1 → P0 →M → 0

by projective R-modules. In fact, there are generally many choices for such a resolution. Two

projective resolutions of M need not be isomorphic to one another. However, they are always

quasi-isomorphic: that is, if we are given two projective resolutions P• and P′• of M , then there

is a map of chain complexes P• → P ′• which induces an isomorphism on homology groups. Thisphenomenon is ubiquitous in homological algebra: many constructions produce chain complexes

which are not really well-defined up to isomorphism, but only up to quasi-isomorphism. In studying

1.1. FOUNDATIONS 21

these constructions, it is often convenient to work in the derived category D(R) of the ring R: that

is, the category obtained from the category of chain complexes of R-modules by formally inverting

all quasi-isomorphisms.

The derived category D(R) of a commutative ring R is usually not an abelian category. For

example, a morphism f : X ′ → X in D(R) usually does not have a cokernel in D(R). Instead,one can associate to f its cofiber (or mapping cone) X ′′, which is well-defined up to noncanonical

isomorphism. In [157], Verdier introduced the notion of a triangulated category in order to ax-

iomatize the structure on D(R) given by the formation of mapping cones. In this section, we will

review Verdier’s theory of triangulated categories (Definition 1.1.2.5) and show that the homotopy

category of a stable ∞-category C is triangulated (Theorem 1.1.2.14).We begin with some basic definitions.

Definition 1.1.2.1. Let A be a category. We will say that A is additive if it satisfies the following

four conditions:

(1) The category A admits finite products and coproducts.

(2) The category A has a zero object, which we will denote by 0.

For any pair of objects X,Y ∈ A, a zero morphism from X to Y is a map f : X → Y which factorsas a composition X → 0 → Y . It follows from (2) that for every pair X,Y ∈ A, there is a uniquezero morphism from X to Y , which we will denote by 0.

(3) For every pair of objects X,Y , the map X∐Y → X × Y described by the matrix[

idX 0

0 idY

]is an isomorphism; let φX,Y denote its inverse.

Assuming (3), we can define the sum of two morphisms f, g : X → Y to be the morphism f + ggiven by the composition

X → X ×X f,g→ Y × YφY,Y→ Y

∐Y → Y.

It is easy to see that this construction endows HomA(X,Y ) with the structure of a commutative

monoid, whose identity is the unique zero morphism from X to Y .

(4) For every pair of objects X,Y ∈ A, the addition defined above determines a group structureon HomA(X,Y ). In other words, for every morphism f : X → Y , there exists anothermorphism −f : X → Y such that f + (−f) is a zero morphism from X to Y .

Remark 1.1.2.2. An additive category A is said to be abelian if every morphism f : X → Y in Aadmits a kernel and a cokernel, and the canonical map coker(ker(f) → X) → ker(Y → coker(f))is an isomorphism.


Remark 1.1.2.3. Let A be an additive category. Then the composition law on A is bilinear: for

pairs of morphisms f, f ′ ∈ HomA(X,Y ) and g, g′ ∈ HomA(Y,Z), we have

g ◦ (f + f ′) = (g ◦ f) + (g ◦ f ′) (g + g′) ◦ f = (g ◦ f) + (g′ ◦ f).

In other words, the composition law on A determines abelian group homomorphisms

HomA(X,Y )⊗HomA(Y, Z)→ HomA(X,Z).

We can summarize the situation by saying that the category A is enriched over the category of

abelian groups.

Remark 1.1.2.4. Let A be an additive category. It follows from condition (3) of Definition 1.1.2.1

that for every pair of objects X,Y ∈ A, the product X × Y is canonically isomorphic to thecoproduct X

∐Y . It is customary to emphasize this identification by denoting both the product

and the coproduct by X ⊕ Y ; we will refer to X ⊕ Y as the direct sum of X and Y .

Definition 1.1.2.5 (Verdier). A triangulated category consists of the following data:

(1) An additive category D.

(2) A translation functor D → D which is an equivalence of categories. We denote this functorby X 7→ X[1].

(3) A collection of distinguished triangles

Xf→ Y g→ Z h→ X[1].

These data are required to satisfy the following axioms:

(TR1) (a) Every morphism f : X → Y in D can be extended to a distinguished triangle in D.(b) The collection of distinguished triangles is stable under isomorphism.

(c) Given an object X ∈ D, the diagram

XidX→ X → 0→ X[1]

is a distinguished triangle.

(TR2) A diagram

Xf→ Y g→ Z h→ X[1]

is a distinguished triangle if and only if the rotated diagram

Yg→ Z h→ X[1] −f [1]→ Y [1]

is a distinguished triangle.

1.1. FOUNDATIONS 23

(TR3) Given a commutative diagram

X //

f��

Y //

��

Z

��

// X[1]

f [1]��

X ′ // Y ′ // Z ′ // X ′[1]

in which both horizontal rows are distinguished triangles, there exists a dotted arrow rendering

the entire diagram commutative.

(TR4) Suppose given three distinguished triangles

Xf→ Y u→ Y/X d→ X[1]

Yg→ Z v→ Z/Y d

′→ Y [1]

Xg◦f→ Z w→ Z/X d

′′→ X[1]

in D. There exists a fourth distinguished triangle

Y/Xφ→ Z/X ψ→ Z/Y θ→ Y/X[1]

such that the diagram

Xg◦f //

f

��

Z

w

##

v // Z/Y

d′

""

θ // Y/X[1]

Y

u

!!

g

==

Z/X

ψ;;

d′′

##

Y [1]

u[1]::

Y/X

φ;;

d // X[1]

f [1]


induces a trivial fibration MΣ → C. Let s : C → MΣ be a section of this trivial fibration, andlet e : MΣ → C be the functor given by evaluation at the final vertex. The composition e ◦ s is afunctor from C to itself, which we will denote by Σ : C→ C and refer to as the suspension functoron C. Dually, we define MΩ to be the full subcategory of Fun(∆1 × ∆1,C) spanned by diagramsas above which are pullback squares with 0 and 0′ zero objects of C. If C admits fibers, then the

same argument shows that evaluation at the final vertex induces a trivial fibration MΩ → C. Ifwe let s′ denote a section to this trivial fibration, then the composition of s′ with evaluation at

the initial vertex induces a functor from C to itself, which we will refer to as the loop functor and

denote by Ω : C→ C. If C is stable, then MΩ = MΣ. It follows that Σ and Ω are mutually inverseequivalences from C to itself.

Remark 1.1.2.6. If the∞-category C is not clear from context, then we will denote the suspensionand loop functors Σ,Ω : C→ C by ΣC and ΩC, respectively.

Notation 1.1.2.7. If C is a stable ∞-category and n ≥ 0, we let

X 7→ X[n]

denote the nth power of the suspension functor Σ : C→ C constructed above (this functor is well-defined up to canonical equivalence). If n ≤ 0, we let X 7→ X[n] denote the (−n)th power of theloop functor Ω. We will use the same notation to indicate the induced functors on the homotopy

category hC.

Remark 1.1.2.8. If the ∞-category C is pointed but not necessarily stable, the suspension andloop space functors need not be homotopy inverses but are nevertheless adjoint to one another

(provided that both functors are defined).

If C is a pointed ∞-category containing a pair of objects X and Y , then the space MapC(X,Y )has a natural base point, given by the zero map. Moreover, if C admits cofibers, then the suspension

functor ΣC : C→ C is essentially characterized by the existence of natural homotopy equivalences

MapC(Σ(X), Y )→ Ω MapC(X,Y ).

In particular, we conclude that π0 MapC(Σ(X), Y ) ' π1 MapC(X,Y ), so that π0 MapC(Σ(X), Y ) hasthe structure of a group (here the fundamental group of MapC(X,Y ) is taken with base point given

by the zero map). Similarly, π0 MapC(Σ2(X), Y ) ' π2 MapC(X,Y ) has the structure of an abelian

group. If the suspension functor X 7→ Σ(X) is an equivalence of∞-categories, then for every Z ∈ Cwe can choose X such that Σ2(X) ' Z to deduce the existence of an abelian group structure onMapC(Z, Y ). It is easy to see that this group structure depends functorially on Z, Y ∈ hC. We aretherefore most of the way to proving the following result:

Lemma 1.1.2.9. Let C be a pointed ∞-category which admits cofibers, and suppose that the sus-pension functor Σ : C→ C is an equivalence. Then hC is an additive category.

1.1. FOUNDATIONS 25

Proof. The argument sketched above shows that hC is (canonically) enriched over the category

of abelian groups. It will therefore suffice to prove that hC admits finite coproducts. We will

prove a slightly stronger statement: the ∞-category C itself admits finite coproducts. Since C hasan initial object, it will suffice to treat the case of pairwise coproducts. Let X,Y ∈ C, and letcofib : Fun(∆1,C) → C denote the functor which assign to each morphism its cofiber, so that wehave equivalences X ' cofib(X[−1] u→ 0) and Y ' cofib(0 v→ Y ). Proposition HTT.5.1.2.2 impliesthat u and v admit a coproduct in Fun(∆1,C) (namely, the zero map X[−1] 0→ Y ). Since thefunctor cofib preserves coproducts (Remark 1.1.1.8), we conclude that X and Y admit a coproduct

(which can be constructed as the cofiber of the zero map from X[−1] to Y ).

Let C be a pointed ∞-category which admits cofibers. By construction, any diagram

X //

��

0

��0′ // Y

which belongs to MΣ determines a canonical isomorphism X[1]→ Y in the homotopy category hC.We will need the following observation:

Lemma 1.1.2.10. Let C be a pointed ∞-category which admits cofibers, and let

Xf //

f ′

��

0

��0′ // Y

be a diagram in C, classifying a morphism θ ∈ HomhC(X[1], Y ). (Here 0 and 0′ are zero objects ofC.) Then the transposed diagram

Xf ′ //

f��

0′

��0 // Y

classifies the morphism −θ ∈ HomhC(X[1], Y ). Here −θ denotes the inverse of θ with respect to thegroup structure on HomhC(X[1], Y ) ' π1 MapC(X,Y ).

Proof. Without loss of generality, we may suppose that 0 = 0′ and f = f ′. Let σ : Λ20 → C be thediagram

0f← X f→ 0.

For every diagram p : K → C, let D(p) denote the Kan complex Cp/×C{Y }. Then HomhC(X[1], Y ) 'π0 D(σ). We note that

D(σ) ' D(f)×D(X) D(f).


Since 0 is an initial object of C, D(f) is contractible. In particular, there exists a point q ∈ D(f).Let

D′ = D(f)×Fun({0},D(X)) Fun(∆1,D(X))×Fun({1},D(X)) D(f)

D′′ = {q} ×Fun({0},D(X)) Fun(∆1,D(X))×Fun({1},D(X)) {q}

so that we have canonical inclusions

D′′ ↪→ D′ ←↩ D(σ).

The left map is a homotopy equivalence because D(f) is contractible, and the right map is a

homotopy equivalence because the projection D(f) → D(X) is a Kan fibration. We observe thatD′′ can be identified with the simplicial loop space of HomLC(X,Y ) (taken with the base point

determined by q, which we can identify with the zero map from X to Y ). Each of the Kan complexes

D(σ), D′, D′′ is equipped with a canonical involution. On D(σ), this involution corresponds to the

transposition of diagrams as in the statement of the lemma. On D′′, this involution corresponds to

reversal of loops. The desired conclusion now follows from the observation that these involutions

are compatible with the inclusions D′′,D(σ) ⊆ D′.

Definition 1.1.2.11. Let C be a pointed ∞-category which admits cofibers. Suppose given adiagram

Xf→ Y g→ Z h→ X[1]

in the homotopy category hC. We will say that this diagram is a distinguished triangle if there

exists a diagram ∆1 ×∆2 → C as shown

Xf̃ //

��

Y

g̃��

// 0

��0′ // Z

h̃ //W,

satisfying the following conditions:

(i) The objects 0, 0′ ∈ C are zero.

(ii) Both squares are pushout diagrams in C.

(iii) The morphisms f̃ and g̃ represent f and g, respectively.

(iv) The map h : Z → X[1] is the composition of (the homotopy class of) h̃ with the equivalenceW ' X[1] determined by the outer rectangle.

Remark 1.1.2.12. We will generally only use Definition 1.1.2.11 in the case where C is a stable

∞-category. However, it will be convenient to have the terminology available in the case where Cis not yet known to be stable.

1.1. FOUNDATIONS 27

The following result is an immediate consequence of Lemma 1.1.2.10:

Lemma 1.1.2.13. Let C be a stable ∞-category. Suppose given a diagram ∆2 ×∆1 → C, depictedas

X

f��

// 0

��Y

��

g // Z

h��

0′ //W,

where both squares are pushouts and the objects 0, 0′ ∈ C are zero. Then the diagram

Xf→ Y g→ Z −h

′→ X[1]

is a distinguished triangle in hC, where h′ denotes the composition of h with the isomorphism

W ' X[1] determined by the outer square, and −h′ denotes the composition of h′ with the map− id ∈ HomhC(X[1], X[1]) ' π1 MapC(X,X[1]).

We can now state the main result of this section:

Theorem 1.1.2.14. Let C be a pointed ∞-category which admits cofibers, and suppose that thesuspension functor Σ is an equivalence. Then the translation functor of Notation 1.1.2.7 and the

class of distinguished triangles of Definition 1.1.2.11 endow hC with the structure of a triangulated

category.

Remark 1.1.2.15. The hypotheses of Theorem 1.1.2.14 hold whenever C is stable. In fact, the

hypotheses of Theorem 1.1.2.14 are equivalent to the stability of C: see Corollary 1.4.2.27.

Proof. We must verify that Verdier’s axioms (TR1) through (TR4) are satisfied.

(TR1) Let E ⊆ Fun(∆1 ×∆2,C) be the full subcategory spanned by those diagrams

Xf //

��

Y

��

// 0

��0′ // Z //W

of the form considered in Definition 1.1.2.11, and let e : E → Fun(∆1,C) be the restrictionto the upper left horizontal arrow. Repeated use of Proposition HTT.4.3.2.15 implies e is a

trivial fibration. In particular, every morphism f : X → Y can be completed to a diagrambelonging to E. This proves (a). Part (b) is obvious, and (c) follows from the observation

that if f = idX , then the object Z in the above diagram is a zero object of C.


(TR2) Suppose that

Xf→ Y g→ Z h→ X[1]

is a distinguished triangle in hC, corresponding to a diagram σ ∈ E as depicted above. Extendσ to a diagram

X //

��

Y

��

// 0

��0′ // Z //

��

W

u��

0′′ // V

where the lower right square is a pushout and 0′′ is a zero object of C. We have a map between

the squares

X //

��

0

��

Y //

��

0

��0′ //W 0′′ // V

which induces a commutative diagram in the homotopy category hC

W

u

��

// X[1]

f [1]

��V // Y [1]

where the horizontal arrows are isomorphisms. Applying Lemma 1.1.2.13 to the rectangle on

the right of the large diagram, we conclude that

Yg→ Z h→ X[1] −f [1]→ Y [1]

is a distinguished triangle in hC.

Conversely, suppose that

Yg→ Z h→ X[1] −f [1]→ Y [1]

is a distinguished triangle in hC. Since the functor Σ : C→ C is an equivalence, we concludethat the triangle

Y [−2] g[−2]→ Z[−2] h[−2]→ X[−1] −f [−1]→ Y [−1]

is distinguished. Applying the preceding argument five times, we conclude that the triangle

Xf→ Y g→ Z h→ X[1]

is distinguished, as desired.

1.1. FOUNDATIONS 29

(TR3) Suppose we are given distinguished triangles

Xf→ Y → Z → X[1]

X ′f ′→ Y ′ → Z ′ → X ′[1]

in hC. Without loss of generality, we may suppose that these triangles are induced by diagrams

σ, σ′ ∈ E. Any commutative diagram

Xf //

��

Y

��X ′

f ′ // Y ′

in the homotopy category hC can be lifted (nonuniquely) to a square in C, which we may

identify with a morphism φ : e(σ)→ e(σ′) in the ∞-category Fun(∆1,C). Since e is a trivialfibration of simplicial sets, φ can be lifted to a morphism σ → σ′ in E, which determines anatural transformation of distinguished triangles

X

��

// Y

��

// Z //

��

X[1]

��X ′ // Y ′ // Z ′ // X ′[1].

(TR4) Let f : X → Y and g : Y → Z be morphisms in C. In view of the fact that e : E→ Fun(∆1,C)is a trivial fibration, any distinguished triangle in hC beginning with f , g, or g ◦ f is uniquelydetermined up to (nonunique) isomorphism. Consequently, it will suffice to prove that there

exist some triple of distinguished triangles which satisfies the conclusions of (TR4). To prove

this, we construct a diagram in C

Xf //

��

Yg //

��

Z //

��

0

��0 // Y/X

��

// Z/X

��

// X ′ //

��

0

��0 // Z/Y // Y ′ // (Y/X)′

where 0 is a zero object of C, and each square in the diagram is a pushout (more precisely,

we apply Proposition HTT.4.3.2.15 repeatedly to construct a map from the nerve of the

appropriate partially ordered set into C). Restricting to appropriate rectangles contained in


the diagram, we obtain isomorphisms X ′ ' X[1], Y ′ ' Y [1], (Y/X)′ ' Y/X[1], and fourdistinguished triangles

Xf→ Y → Y/X → X[1]

Yg→ Z → Z/Y → Y [1]

Xg◦f→ Z → Z/X → X[1]

Y/X → Z/X → Z/Y → Y/X[1].

The commutativity in the homotopy category hC required by (TR4) follows from the

(stronger) commutativity of the above diagram in C itself.

Remark 1.1.2.16. The definition of a stable ∞-category is quite a bit simpler than that of atriangulated category. In particular, the octahedral axiom (TR4) is a consequence of∞-categoricalprinciples which are basic and easily motivated.

Notation 1.1.2.17. Let C be a stable ∞-category containing a pair of objects X and Y . We letExtnC(X,Y ) denote the abelian group HomhC(X[−n], Y ). If n is negative, this can be identifiedwith the homotopy group π−n MapC(X,Y ). More generally, Ext

nC(X,Y ) can be identified with the

(−n)th homotopy group of an appropriate spectrum of maps from X to Y .

1.1.3 Closure Properties of Stable ∞-Categories

According to Definition 1.1.1.9, a pointed∞-category C is stable if it admits certain pushout squaresand certain pullback squares, which are required to coincide with one another. Our goal in this

section is to prove that a stable ∞-category C admits all finite limits and colimits, and that thepushout squares in C coincide with the pullback squares in general (Proposition 1.1.3.4). To prove

this, we will need the following easy observation (which is quite useful in its own right):

Proposition 1.1.3.1. Let C be a stable ∞-category, and let K be a simplicial set. Then the∞-category Fun(K,C) is stable.

Proof. This follows immediately from the fact that fibers and cofibers in Fun(K,C) can be computed

pointwise (Proposition HTT.5.1.2.2 ).

Definition 1.1.3.2. If C is stable∞-category, and C0 is a full subcategory containing a zero objectand stable under the formation of fibers and cofibers, then C0 is itself stable. In this case, we will

say that C0 is a stable subcategory of C.

Lemma 1.1.3.3. Let C be a stable ∞-category, and let C′ ⊆ C be a full subcategory which is stableunder cofibers and under translations. Then C′ is a stable subcategory of C.

1.1. FOUNDATIONS 31

Proof. It will suffice to show that C′ is stable under fibers. Let f : X → Y be a morphism in C.Theorem 1.1.2.14 shows that there is a canonical equivalence fib(f) ' cofib(f)[−1].

Proposition 1.1.3.4. Let C be a pointed ∞-category. Then C is stable if and only if the followingconditions are satisfied:

(1) The ∞-category C admits finite limits and colimits.

(2) A square

X //

��

Y

��X ′ // Y ′

in C is a pushout if and only if it is a pullback.

Proof. Condition (1) implies the existence of fibers and cofibers in C, and condition (2) implies

that a triangle in C is a fiber sequence if and only if it is a cofiber sequence. This proves the “if”

direction.

Suppose now that C is stable. We begin by proving (1). It will suffice to show that C admits

finite colimits; the dual argument will show that C admits finite limits as well. According to

Proposition HTT.4.4.3.2 , it will suffice to show that C admits coequalizers and finite coproducts.

The existence of finite coproducts was established in Lemma 1.1.2.9. We now conclude by observing

that a coequalizer for a diagram

Xf //f ′// Y

can be identified with cofib(f − f ′).We now show that every pushout square in C is a pullback; the converse will follow by a dual

argument. Let D ⊆ Fun(∆1 × ∆1,C) be the full subcategory spanned by the pullback squares.Then D is stable under finite limits and under translations. It follows from Lemma 1.1.3.3 that D

is a stable subcategory of Fun(∆1 ×∆1,C).Let i : Λ20 ↪→ ∆1 × ∆1 be the inclusion, and let i! : Fun(Λ20,C) → Fun(∆1 × ∆1,C) be a

functor of left Kan extension. Then i! preserves finite colimits, and is therefore exact (Proposition

1.1.4.1). Let D′ = i−1! D. Then D′ is a stable subcategory of Fun(Λ20,C); we wish to show that

D′ = Fun(Λ20,C). To prove this, we observe that any diagram

X ′ ← X → X ′′

can be obtained as a (finite) colimit

e′X′∐e′X

eX∐e′′X

e′′X′′


where eX ∈ Fun(Λ20,C) denotes the diagram X ← X → X, e′Z ∈ Fun(Λ20,C) denotes the diagramZ ← 0→ 0, and e′′Z ∈ Fun(Λ20,C) denotes the diagram 0← 0→ Z. It will therefore suffice to provethat a pushout of any of these five diagrams is also a pullback. This follows immediately from the

following more general observation: any pushout square

A //

f��

A′

��B // B′

in an (arbitrary) ∞-category C is also a pullback square, provided that f is an equivalence.

Remark 1.1.3.5. Let C be a stable ∞-category. Then C admits finite products and finite co-products (Proposition 1.1.3.4). Moreover, for any pair of objects X,Y ∈ C, there is a canonicalequivalence

X q Y → X × Y,

given by the matrix [idX 0

0 idY

].

Theorem 1.1.2.14 implies that this map is an equivalence. We will sometimes use the notation

X ⊕ Y to denote a product or coproduct of X and Y in C.

We conclude this section by establishing a few closure properties for the class of stable ∞-categories.

Proposition 1.1.3.6. Let C be a (small) stable ∞-category and let κ be a regular cardinal. Thenthe ∞-category Indκ(C) is stable.

Proof. The functor j preserves finite limits and colimits (Propositions HTT.5.1.3.2 and HTT.5.3.5.14 ).

It follows that j(0) is a zero object of Indκ(C), so that Indκ(C) is pointed.

We next show that every morphism f : X → Y in Indκ(C) admits a fiber and a cofiber.According to Proposition HTT.5.3.5.15 , we may assume that f is a κ-filtered colimit of morphisms

fα : Xα → Yα which belong to the essential image C′ of j. Since j preserves fibers and cofibers,each of the maps fα has a fiber and a cofiber in Indκ. It follows immediately that f has a cofiber

(which can be written as a colimit of the cofibers of the maps fα). The existence of fib(f) is slightly

more difficult. Choose a κ-filtered diagram p : I→ Fun(∆1×∆1,C′), where each p(α) is a pullbacksquare

Zα //

��

0

��Xα

fα // Yα.

1.1. FOUNDATIONS 33

Let σ be a colimit of the diagram p; we wish to show that σ is a pullback diagram in Indκ(C). Since

Indκ(C) is stable under κ-small limits in P(C), it will suffice to show that σ is a pullback square in

P(C). Since P(C) is an ∞-topos, filtered colimits in P(C) are left exact (Example HTT.7.3.4.7 ); itwill therefore suffice to show that each p(α) is a pullback diagram in P(C). This is obvious, since

the inclusion C′ ⊆ P(C) preserves all limits which exist in C′ (Proposition HTT.5.1.3.2 ).To complete the proof, we must show that a triangle in Indκ(C) is a fiber sequence if and only

if it is a cofiber sequence. Suppose we are given a fiber sequence

Z //

��

0

��X // Y

in Indκ(C). The above argument shows that we can write this triangle as a filtered colimit of fiber

sequences

Zα //

��

0

��Xα // Yα

in C′. Since C′ is stable, we conclude that these triangles are also cofiber sequences. The original

triangle is therefore a filtered colimit of cofiber sequences in C′, hence a cofiber sequence. The

converse follows by the same argument.

Corollary 1.1.3.7. Let C be a stable ∞-category. Then the idempotent completion of C is alsostable.

Proof. According to Lemma HTT.5.4.2.4 , we can identify the idempotent completion of C with a

full subcategory of Ind(C) which is closed under shifts and finite colimits.

1.1.4 Exact Functors

Let F : C→ C′ be a functor between stable ∞-categories. Suppose that F carries zero objects intozero objects. It follows immediately that F carries triangles into triangles. If, in addition, F carries

fiber sequences to fiber sequences, then we will say that F is exact. The exactness of a functor F

admits the following alternative characterizations:

Proposition 1.1.4.1. Let F : C → C′ be a functor between stable ∞-categories. The followingconditions are equivalent:

(1) The functor F is left exact. That is, F commutes with finite limits.

(2) The functor F is right exact. That is, F commutes with finite colimits.


(3) The functor F is exact.

Proof. We will prove that (2)⇔ (3); the equivalence (1)⇔ (3) will follow by a dual argument. Theimplication (2) ⇒ (3) is obvious. Conversely, suppose that F is exact. The proof of Proposition1.1.3.4 shows that F preserves coequalizers, and the proof of Lemma 1.1.2.9 shows that F pre-

serves finite coproducts. It follows that F preserves all finite colimits (see the proof of Proposition

HTT.4.4.3.2 ).

The identity functor from any stable ∞-category to itself is exact, and a composition of exactfunctors is exact. Consequently, there exists a subcategory CatEx∞ ⊆ Cat∞ in which the objects arestable ∞-categories and the morphisms are the exact functors. Our next few results concern thestability properties of this subcategory.

Proposition 1.1.4.2. Suppose given a homotopy Cartesian diagram of ∞-categories

C′G′ //

F ′��

C

F��

D′G // D .

Suppose further that C, D′, and D are stable, and that the functors F and G are exact. Then:

(1) The ∞-category C′ is stable.

(2) The functors F ′ and G′ are exact.

(3) If E is a stable ∞-category, then a functor H : E → C′ is exact if and only if the functorsF ′ ◦H and G′ ◦H are exact.

Proof. Combine Proposition 1.1.3.4 with Lemma HTT.5.4.5.5 .

Proposition 1.1.4.3. Let {Cα}α∈A be a collection of stable ∞-categories. Then the product

C =∏α∈A

Cα

is stable. Moreover, for any stable ∞-category D, a functor F : D→ C is exact if and only if eachof the compositions

DF→ C→ Cα

is an exact functor.

Proof. This follows immediately from the fact that limits and colimits in C are computed pointwise.

1.1. FOUNDATIONS 35

Theorem 1.1.4.4. The ∞-category CatEx∞ admits small limits, and the inclusion

CatEx∞ ⊆ Cat∞

preserves small limits.

Proof. Using Propositions 1.1.4.2 and 1.1.4.3, one can repeat the argument used to prove Proposi-

tion HTT.5.4.7.3 .

We have the following analogue of Theorem 1.1.4.4.

Proposition 1.1.4.5. Let p : X → S be an inner fibration of simplicial sets. Suppose that:

(i) For each vertex s of S, the fiber Xs = X ×S {s} is a stable ∞-category.

(ii) For every edge s → s′ in S, the restriction X ×S ∆1 → ∆1 is a coCartesian fibration,associated to an exact functor Xs → Xs′.

Then:

(1) The ∞-category MapS(S,X) of sections of p is stable.

(2) If C is an arbitrary stable ∞-category, and f : C → MapS(S,X) induces an exact functorC

f→ MapS(S,X)→ Xs for every vertex s of S, then f is exact.

(3) For every set E of edges of S, let Y (E) ⊆ MapS(S,X) be the full subcategory spanned by thosesections f : S → X of p with the following property:

(∗) For every e ∈ E, f carries e to a pe-coCartesian edge of the fiber product X×S∆1, wherepe : X ×S ∆1 → ∆1 denotes the projection.

Then each Y (E) is a stable subcategory of MapS(S,X).

Proof. Combine Proposition HTT.5.4.7.11 , Theorem 1.1.4.4, and Proposition 1.1.3.1.

Proposition 1.1.4.6. The ∞-category CatEx∞ admits small filtered colimits, and the inclusionCatEx∞ ⊆ Cat∞ preserves small filtered colimits.

Proof. Let I be a filtered ∞-category, p : I→ CatEx∞ a diagram, which we will indicate by {CI}I∈I,and C a colimit of the induced diagram I→ Cat∞. We must prove:

(i) The ∞-category C is stable.

(ii) Each of the canonical functors θI : CI → C is exact.

(iii) Given an arbitrary stable ∞-category D, a functor f : C → D is exact if and only if each ofthe composite functors CI

θI→ C→ D is exact.


In view of Proposition 1.1.4.1, (ii) and (iii) follow immediately from Proposition HTT.5.5.7.11 .

The same result implies that C admits finite limits and colimits, and that each of the functors θIpreserves finite limits and colimits.

To prove that C has a zero object, we select an object I ∈ I. The functor CI → C preservesinitial and final objects. Since CI has a zero object, so does C.

We will complete the proof by showing that every fiber sequence in C is a cofiber sequence

(the converse follows by the same argument). Fix a morphism f : X → Y in C. Without loss ofgenerality, we may suppose that there exists I ∈ I and a morphism f̃ : X̃ → Ỹ in CI such thatf = θI(f̃) (Proposition HTT.5.4.1.2 ). Form a pullback diagram σ̃

W̃ //

��

X̃

��

0 // Ỹ

in CI . Since CI is stable, this diagram is also a pushout. It follows that θI(σ̃) is a triangle

W → X f→ Y which is both a fiber sequence and a cofiber sequence in C.

1.2 Stable ∞-Categories and Homological Algebra

Let A be an abelian category with enough projective objects. In §1.3.2, we will explain how toassociate to A a stable ∞-category D−(A), whose objects are (right-bounded) chain complexesof projective objects of A. The homotopy category D−(A) is a triangulated category, which is

usually called the derived category of A. We can recover A as a full subcategory of the triangulated

category hD−(A) (or even as a full subcategory of the∞-category D−(A)): namely, A is equivalentto the full subcategory spanned by those chain complexes P∗ satisfying Hn(P∗) ' 0 for n 6= 0. Thissubcategory can be described as the intersection

D−(A)≥0 ∩D−(A)≤0,

where D−(A)≤0 is defined to be the full subcategory spanned by those chain complexes P∗ with

Hn(P∗) ' 0 for n > 0, and D−(A)≥0 is spanned by those chain complexes with Hn(P∗) ' 0 forn < 0.

In §1.2.1, we will axiomatize the essence of the situation by reviewing the notion of a t-structureon a stable ∞-category C. A t-structure on C is a pair of full subcategories (C≥0,C≤0) satisfyingsome axioms which reflect the idea that objects of C≥0 (C≤0) are “concentrated in nonnegative

(nonpositive) degrees” (see Definition 1.2.1.1). In this case, one can show that the intersection

C≥0 ∩C≤0 is equivalent to the nerve of an abelian category, which we call the heart of C and denoteby C♥. To any object X ∈ C, we can associate homotopy objects πnX ∈ C♥ (in the special caseC = D−(A), the functor πn associates to each chain complex P∗ its nth homology Hn(P∗)).

1.2. STABLE ∞-CATEGORIES AND HOMOLOGICAL ALGEBRA 37

If C is a stable ∞-category equipped with a t-structure, then it is often possible to relate ques-tions about C to homological algebra in the abelian category C♥. In §1.2.2, we give an illustrationof this principle, by showing that every f

higher algebra - people.math.harvard.edupeople.math.harvard.edu/~lurie/papers/ha.pdf · contents 5...

Documents