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Higher Algebra

September 18, 2017

Contents

• 1 Stable ∞-Categories
  • 1.1 Foundations
    • 1.1.1 Stability
    • 1.1.2 The Homotopy Category of a Stable ∞-Category
    • 1.1.3 Closure Properties of Stable ∞-Categories
    • 1.1.4 Exact Functors
  • 1.2 Stable ∞-Categories and Homological Algebra
    • 1.2.1 t-Structures on Stable ∞-Categories
    • 1.2.2 Filtered Objects and Spectral Sequences
    • 1.2.3 The Dold-Kan Correspondence
    • 1.2.4 The ∞-Categorical Dold-Kan Correspondence
  • 1.3 Homological Algebra and Derived Categories
    • 1.3.1 Nerves of Differential Graded Categories
    • 1.3.2 Derived ∞-Categories
    • 1.3.3 The Universal Property of D−(A)
    • 1.3.4 Inverting Quasi-Isomorphisms
    • 1.3.5 Grothendieck Abelian Categories
  • 1.4 Spectra and Stabilization
    • 1.4.1 The Brown Representability Theorem
    • 1.4.2 Spectrum Objects
    • 1.4.3 The ∞-Category of Spectra
    • 1.4.4 Presentable Stable ∞-Categories
• 2 ∞-Operads
  • 2.1 Foundations
    • 2.1.1 From Colored Operads to ∞-Operads
    • 2.1.2 Maps of ∞-Operads
    • 2.1.3 Algebra Objects
    • 2.1.4 ∞-Preoperads
• CONTENTS
  • 2.2 Constructions of ∞-Operads
    • 2.2.1 Subcategories of O-Monoidal ∞-Categories
    • 2.2.2 Slicing ∞-Operads
    • 2.2.3 Coproducts of ∞-Operads
    • 2.2.4 Monoidal Envelopes
    • 2.2.5 Tensor Products of ∞-Operads
    • 2.2.6 Day Convolution
  • 2.3 Disintegration and Assembly
    • 2.3.1 Unital ∞-Operads
    • 2.3.2 Generalized ∞-Operads
    • 2.3.3 Approximations to ∞-Operads
    • 2.3.4 Disintegration of ∞-Operads
  • 2.4 Products and Coproducts
    • 2.4.1 Cartesian Symmetric Monoidal Structures
    • 2.4.2 Monoid Objects
    • 2.4.3 CoCartesian Symmetric Monoidal Structures
    • 2.4.4 Wreath Products
• 3 Algebras and Modules over ∞-Operads
  • 3.1 Free Algebras
    • 3.1.1 Operadic Colimit Diagrams
    • 3.1.2 Operadic Left Kan Extensions
    • 3.1.3 Construction of Free Algebras
    • 3.1.4 Transitivity of Operadic Left Kan Extensions
  • 3.2 Limits and Colimits of Algebras
    • 3.2.1 Unit Objects and Trivial Algebras
    • 3.2.2 Limits of Algebras
    • 3.2.3 Colimits of Algebras
    • 3.2.4 Tensor Products of Commutative Algebras
  • 3.3 Modules over ∞-Operads
    • 3.3.1 Coherent ∞-Operads
    • 3.3.2 A Coherence Criterion
    • 3.3.3 Module Objects
  • 3.4 General Features of Module ∞-Categories
    • 3.4.1 Algebra Objects of ∞-Categories of Modules
    • 3.4.2 Modules over Trivial Algebras
    • 3.4.3 Limits of Modules
    • 3.4.4 Colimits of Modules
• 4 Associative Algebras and Their Modules 4 CONTENTS
  • 4.1 Associative Algebras
    • 4.1.1 The Associative ∞-Operad
    • 4.1.2 Monoid Objects of ∞-Categories
    • 4.1.3 Planar ∞-Operads and A∞-Algebras
    • 4.1.4 Nonunital An-Algebras and Nonunital An-Monoids
    • 4.1.5 From An-Algebras to An+1-Algebras
    • 4.1.6 The Associahedron
    • 4.1.7 Monoidal Model Categories
    • 4.1.8 Rectification of Associative Algebras
  • 4.2 Left and Right Modules
    • 4.2.1 The ∞-Operad LM⊗
    • 4.2.2 Simplicial Models for Algebras and Modules
    • 4.2.3 Limits and Colimits of Modules
    • 4.2.4 Free Modules
  • 4.3 Bimodules
    • 4.3.1 The ∞-Operad BM⊗
    • 4.3.2 Bimodules, Left Modules, and Right Modules
    • 4.3.3 Limits, Colimits, and Free Bimodules
  • 4.4 The Relative Tensor Product
    • 4.4.1 Multilinear Maps
    • 4.4.2 Tensor Products and the Bar Construction
    • 4.4.3 Associativity of the Tensor Product
  • 4.5 Modules over Commutative Algebras
    • 4.5.1 Left and Right Modules over Commutative Algebras
    • 4.5.2 Tensor Products over Commutative Algebras
    • 4.5.3 Change of Algebra
    • 4.5.4 Rectification of Commutative Algebras
  • 4.6 Duality
    • 4.6.1 Duality in Monoidal ∞-Categories
    • 4.6.2 Duality of Bimodules
    • 4.6.3 Exchanging Right and Left Actions
    • 4.6.4 Smooth and Proper Algebras
    • 4.6.5 Frobenius Algebras
  • 4.7 Monads and the Barr-Beck Theorem
    • 4.7.1 Endomorphism ∞-Categories
    • 4.7.2 Split Simplicial Objects
    • 4.7.3 The Barr-Beck Theorem
• CONTENTS - 4.7.4 BiCartesian Fibrations - 4.7.5 Descent and the Beck-Chevalley Condition
  • 4.8 Tensor Products of ∞-Categories
    • 4.8.1 Tensor Products of ∞-Categories
    • 4.8.2 Smash Products of Spectra
    • 4.8.3 Algebras and their Module Categories
    • 4.8.4 Properties of RModA(C)
    • 4.8.5 Behavior of the Functor Θ
• 5 Little Cubes and Factorizable Sheaves
  • 5.1 Definitions and Basic Properties
    • 5.1.1 Little Cubes and Configuration Spaces
    • 5.1.2 The Additivity Theorem
    • 5.1.3 Tensor Products of Ek-Modules
    • 5.1.4 Comparison of Tensor Products
  • 5.2 Bar Constructions and Koszul Duality
    • 5.2.1 Twisted Arrow ∞-Categories
    • 5.2.2 The Bar Construction for Associative Algebras
    • 5.2.3 Iterated Bar Constructions
    • 5.2.4 Reduced Pairings
    • 5.2.5 Koszul Duality for Ek-Algebras
    • 5.2.6 Iterated Loop Spaces
  • 5.3 Centers and Centralizers
    • 5.3.1 Centers and Centralizers
    • 5.3.2 The Adjoint Representation
    • 5.3.3 Tensor Products of Free Algebras
  • 5.4 Little Cubes and Manifold Topology
    • 5.4.1 Embeddings of Topological Manifolds
    • 5.4.2 Variations on the Little Cubes Operads
    • 5.4.3 Digression: Nonunital Associative Algebras and their Modules
    • 5.4.4 Nonunital Ek-Algebras
    • 5.4.5 Little Cubes in a Manifold
  • 5.5 Topological Chiral Homology
    • 5.5.1 The Ran Space
    • 5.5.2 Topological Chiral Homology
    • 5.5.3 Properties of Topological Chiral Homology
    • 5.5.4 Factorizable Cosheaves and Ran Integration
    • 5.5.5 Verdier Duality
    • 5.5.6 Nonabelian Poincare Duality
• 6 The Calculus of Functors 6 CONTENTS
  • 6.1 The Calculus of Functors
    • 6.1.1 n-Excisive Functors
    • 6.1.2 The Taylor Tower
    • 6.1.3 Functors of Many Variables
    • 6.1.4 Symmetric Functors
    • 6.1.5 Functors from Spaces to Spectra
    • 6.1.6 Norm Maps
  • 6.2 Differentiation
    • 6.2.1 Derivatives of Functors
    • 6.2.2 Stabilization of Differentiable Fibrations
    • 6.2.3 Differentials of Functors
    • 6.2.4 Generalized Smash Products
    • 6.2.5 Stabilization of ∞-Operads
    • 6.2.6 Uniqueness of Stabilizations
  • 6.3 The Chain Rule
    • 6.3.1 Cartesian Structures
    • 6.3.2 Composition of Correspondences
    • 6.3.3 Derivatives of the Identity Functor
    • 6.3.4 Differentiation and Reduction
    • 6.3.5 Consequences of Theorem 6.3.3.14
    • 6.3.6 The Dual Chain Rule
• 7 Algebra in the Stable Homotopy Category
  • 7.1 Structured Ring Spectra
    • 7.1.1 E 1 -Rings and Their Modules
    • 7.1.2 Recognition Principles
    • 7.1.3 Change of Ring
    • 7.1.4 Algebras over Commutative Rings
  • 7.2 Properties of Rings and Modules
    • 7.2.1 Free Resolutions and Spectral Sequences
    • 7.2.2 Flat and Projective Modules
    • 7.2.3 Localizations and Ore Conditions
    • 7.2.4 Finiteness Properties of Rings and Modules
  • 7.3 The Cotangent Complex Formalism
    • 7.3.1 Stable Envelopes and Tangent Bundles
    • 7.3.2 Relative Adjunctions
    • 7.3.3 The Relative Cotangent Complex
    • 7.3.4 Tangent Bundles to ∞-Categories of Algebras
• CONTENTS - 7.3.5 The Cotangent Complex of an Ek-Algebra - 7.3.6 The Tangent Correspondence
  • 7.4 Deformation Theory
    • 7.4.1 Square-Zero Extensions
    • 7.4.2 Deformation Theory of E∞-Algebras
    • 7.4.3 Connectivity and Finiteness of the Cotangent Complex
  • 7.5 EtaleMorphisms´
    • 7.5.1 EtaleMorphisms of´ E 1 -Rings
    • 7.5.2 The Nonconnective Case
    • 7.5.3 Cocentric Morphisms
    • 7.5.4 EtaleMorphisms of´ Ek-Rings
• A Constructible Sheaves and Exit Paths
  • A.1 Locally Constant Sheaves
  • A.2 Homotopy Invariance
  • A.3 The Seifert-van Kampen Theorem
  • A.4 Singular Shape
  • A.5 Constructible Sheaves
  • A.6 ∞-Categories of Exit Paths
  • A.7 A Seifert-van Kampen Theorem for Exit Paths
  • A.8 Digression: Recollement
  • A.9 Exit Paths and Constructible Sheaves
• B Categorical Patterns
  • B.1 P-Anodyne Morphisms
  • B.2 The Model Structure on (Set+∆)/ P
  • B.3 Flat Inner Fibrations
  • B.4 Functoriality
  • General Index
  • Notation Index

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 generally a sequence of abelian groups {Kn(X, Y )}n∈Z for every inclusion of topological spaces Y ⊆ X, in such 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. According to 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 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 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 cohomology theory 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 × 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′) ' (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

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 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 MapC(X, Y ) for every pair of objects X, Y ∈ C (which are equipped with coherently 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 collection of objects together with a space of operations MulO({Xi} 1 ≤i≤n, Y )} for every finite collection of objects X 1 ,... , 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 AlgO(C), and refer to as the ∞-category of O-algebra objects of C. An important special case is when O is the commutative ∞-operad and C is a symmetric monoidal ∞-category: in this case, we will refer to AlgO(C) as the ∞-category of commutative algebra 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 ∞-category of associative algebra objects of C. The ∞-categorical theory of associative algebra objects is an excellent 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 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 Ek-algebra for each 0 ≤ k ≤ ∞. This definition reduces to 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 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

CONTENTS 11

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 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 ordinary noncommutative 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 extension condition:

(∗) Every map of simplicial sets f 0 : 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 the name 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 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 [98].

Remark 0.0.0.3. For a different treatment of the theory of ∞-categories, we refer the reader to Joyal’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.

CONTENTS 13

• In Chapter 1, we will construct an ∞-category Sp, whose homotopy category hSp can be identified 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 the classical smash product of spectra, which is usually denoted by (X, Y ) 7 → X ∧ Y. We will adopt 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 the undercategory and overcategory defined in §HTT.1.2.9. We will generally abuse notation by identifying objects of these ∞-categories with their images in C. If we are given a morphism f : X → Y , we can identify X with an object of C/Y and Y with an object of CX/, so that the ∞-categories (CX/)/Y (C/Y )X/

are defined (and canonically isomorphic as simplicial sets). We will denote these ∞-categories by 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 to LFun(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 fibration X → 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 a homotopy equivalence FunT (T, X) → FunT (S, X) for every left fibration X → T. If S and T are ∞-categories, then f is left cofinal if and only if for every object t ∈ T , the fiber product S ×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 Br¨ugmann, 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 Sch¨urg, 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 To¨en, 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.

16 CHAPTER 1. STABLE ∞-CATEGORIES

We will return to the setting of homological algebra in §1.3. To any abelian category A with enough 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 for organizing 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 of spectra. 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 of the 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 C is 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 which are not of this form (see [114]). Our next goal is to study the properties of stable ∞-categories in greater depth. In §1.1.3, we will show that a stable ∞-category C admits all finite limits and colimits, and that pullback squares and 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 C is 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 (Proposition 1.1.3.1). In §1.1.4, we shift our focus somewhat. Rather than concerning ourselves with the properties of 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 show that the collection of all (small) stable ∞-categories and exact functors between them can itself be organized 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 presented here 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 show that 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-module spectra. 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 complexes of 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 some definitions from [98].

Definition 1.1.1.1. Let C be an ∞-category. A zero object of C is an object which is both initial and 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 both contractible for every object X ∈ C. Note that if C contains a zero object, then that object is 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 conditions are 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.

1.1. FOUNDATIONS 19

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

 

X

g   0 //Y.

Dually, a cofiber of g is a cofiber sequence

X g^ //

 

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 of f , 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 forgetful functor, which associates to a diagram

X g^ //

 

Y

  0 //Z

the morphism g : X → Y. Applying Proposition HTT.4.3.2.15 twice, we deduce that θ is a Kan fibration, 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 is a trivial Kan fibration, and therefore admits a section cofib : Fun(∆^1 , C) → Fun(∆^1 × ∆^1 , C), which is 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 the left Kan extension functor C ' Fun({ 1 }, C) → Fun(∆^1 , C), which associates to each object X ∈ C a 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.

20 CHAPTER 1. STABLE ∞-CATEGORIES

(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 space functor. The collection of spectra can be organized into a stable ∞-category Sp. Moreover, Sp is in some sense the universal example of a stable ∞-category. This motivates the terminology of Definition 1.1.1.9: an ∞-category C is stable if it resembles the ∞-category Sp, whose homotopy category 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 derived category 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 is a property of ∞-categories, rather than additional data. The situation for additive categories is similar. 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 is similar: we will see later that every stable ∞-category is canonically enriched over the ∞-category of 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

· · · → P 2 → P 1 → P 0 → 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. This phenomenon 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