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Overview
Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinitycategories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics.
The book's first five chapters give an exposition of the theory of infinitycategories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinitycategorical setting, such as limits and colimits, adjoint functors, indobjects and proobjects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinitycategorical version of the theory of Grothendieck topoi, introducing the notion of an infinitytopos, an infinitycategory that resembles the infinitycategory of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.
Editorial Reviews
Mathematical Reviews
This book is a remarkable achievement, and the reviewer thinks it marks the beginning of a major change in algebraic topology.— Mark Hovey
Mathematical Reviews  Mark Hovey
This book is a remarkable achievement, and the reviewer thinks it marks the beginning of a major change in algebraic topology.From the Publisher
"This book is a remarkable achievement, and the reviewer thinks it marks the beginning of a major change in algebraic topology."Mark Hovey, Mathematical Reviews
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Meet the Author
Jacob Lurie is associate professor of mathematics at Massachusetts Institute of Technology.
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Higher Topos Theory
By Jacob Lurie
PRINCETON UNIVERSITY PRESS
Copyright © 2009 Princeton University PressAll right reserved.
ISBN: 9780691140490
Chapter One
An Overview of Higher Category TheoryThis chapter is intended as a general introduction to higher category theory. We begin with what we feel is the most intuitive approach to the subject using topological categories. This approach is easy to understand but difficult to work with when one wishes to perform even simple categorical constructions. As a remedy, we will introduce the more suitable formalism of [infinity]categories (called weak Kan complexes in [10] and quasicategories in [43]), which provides a more convenient setting for adaptations of sophisticated categorytheoretic ideas. Our goal in 1.1.1 is to introduce both approaches and to explain why they are equivalent to one another. The proof of this equivalence will rely on a crucial result (Theorem 1.1.5.13) which we will prove in 2.2.
Our second objective in this chapter is to give the reader an idea of how to work with the formalism of [infinity]categories. In 1.2, we will establish a vocabulary which includes [infinity]categorical analogues (often direct generalizations) of most of the important concepts from ordinary category theory. To keep the exposition brisk, we will postpone the more difficult proofs until later chapters of this book. Our hope is that, after reading this chapter, a reader who does not wish to be burdened with the details will be able to understand (at least in outline) some of the more conceptual ideas described in Chapter 5 and beyond.
1.1 FOUNDATIONS FOR HIGHER CATEGORY THEORY
1.1.1 Goals and Obstacles
Recall that a category C consists of the following data:
(1) A collection {X,Y,Z, ...} whose members are the objects of ITLITL. We typically write X [member of] ITLITL to indicate that X is an object of ITLITL.
(2) For every pair of objects X, Y [member of] ITLITL, a set [Hom.sub.ITLITL](X, Y) of morphisms from X to Y. We will typically write f : X [right arrow] Y to indicate that f [member of] [Hom.sub.ITLITL](X, Y) and say that f is a morphism from X to Y.
(3) For every object X [member of] ITLITL, an identity morphism [id.sub.X] [member of] [Hom.sub.ITLITL](X,X).
(4) For every triple of objects X, Y, Z [member of] ITLITL, a composition map
[Hom.sub.ITLITL](X, Y) x [Hom.sub.ITLITL](Y,Z) [right arrow] [Hom.sub.ITLITL](X,Z).
Given morphisms f : X [right arrow] Y and g : Y [right arrow] Z, we will usually denote the image of the pair (f, g) under the composition map by gf or g [??] f.
These data are furthermore required to satisfy the following conditions, which guarantee that composition is unital and associative:
(5) For every morphism f : X [right arrow] Y, we have [id.sub.Y] [??]f = f = f [??] [id.sub.X] in [Hom.sub.ITLITL](X, Y).
(6) For every triple of composable morphisms
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
we have an equality h [??] (g [??] f) = (h [??] g) [??] f in [Hom.sub.ITLITL](W, Z).
The theory of categories has proven to be a valuable organization tool in many areas of mathematics. Mathematical structures of virtually any type can be viewed as the objects of a suitable category ITLITL, where the morphisms in C are given by structurepreserving maps. There is a veritable legion of examples of categories which fit this paradigm:
The category Set whose objects are sets and whose morphisms are maps of sets.
The category Grp whose objects are groups and whose morphisms are group homomorphisms.
The category Top whose objects are topological spaces and whose morphisms are continuous maps.
The category Cat whose objects are (small) categories and whose morphisms are functors. (Recall that a functor F from ITLITL to D is a map which assigns to each object ITLITL [member of] ITLITL another object FC [member of] D, and to each morphism f : ITLITL [right arrow] ITLITL' in ITLITL a morphism F(f) : FC [right arrow] FC' in D, so that F([id.sub.ITLITL]) = [id.sub.FC] and F(g [??] f) = F(g) [??] F(f).)
???
In general, the existence of a morphism f : X [right arrow] Y in a category ITLITL reflects some relationship that exists between the objects X, Y [member of] ITLITL. In some contexts, these relationships themselves become basic objects of study and can be fruitfully organized into categories:
Example 1.1.1.1. Let Grp be the category whose objects are groups and whose morphisms are group homomorphisms. In the theory of groups, one is often concerned only with group homomorphisms up to conjugacy. The relation of conjugacy can be encoded as follows: for every pair of groups G,H [member of] Grp, there is a category Map(G,H) whose objects are group homomorphisms from G to H (that is, elements of [Hom.sub.Grp](G,H)), where a morphism from f : G [right arrow] H to f' : G [right arrow] H is an element h [member of] H such that hf(g)[h.sup.1] = f'(g) for all g [member of] G. Note that two group homomorphisms f, f' : G [right arrow] H are conjugate if and only if they are isomorphic when viewed as objects of Map(G,H).
Example 1.1.1.2. Let X and Y be topological spaces and let [f.sub.0], [f.sub.1] : X [right arrow] Y be continuous maps. Recall that a homotopy from [f.sub.0] to [f.sub.1] is a continuous map f : X x [0, 1] [right arrow] Y such that fX x {0} coincides with [f.sub.0] and fX x {1} coincides with [f.sub.1]. In algebraic topology, one is often concerned not with the category Top of topological spaces but with its homotopy category: that is, the category obtained by identifying those pairs of morphisms [f.sub.0], [f.sub.1] : X [right arrow] Y which are homotopic to one another. For many purposes, it is better to do something a little bit more sophisticated: namely, one can form a category Map(X, Y) whose objects are continuous maps f : X [right arrow] Y and whose morphisms are given by (homotopy classes of) homotopies.
Example 1.1.1.3. Given a pair of categories ITLITL and D, the collection of all functors from ITLITL to D is itself naturally organized into a category Fun(C,D), where the morphisms are given by natural transformations. (Recall that, given a pair of functors F,G : ITLITL [right arrow] D, a natural transformation [alpha] : F [right arrow] G is a collection of morphisms [{[[alpha].sub.ITLITL] : F(ITLITL) [right arrow] G(ITLITL)}.sub.ITLITL [member of] ITLITL] which satisfy the following condition: for every morphism f : ITLITL [right arrow] C' in ITLITL, the diagram
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
commutes in D.)
In each of these examples, the objects of interest can naturally be organized into what is called a 2category (or bicategory): we have not only a collection of objects and a notion of morphisms between objects but also a notion of morphisms between morphisms, which are called 2morphisms. The vision of higher category theory is that there should exist a good notion of ncategory for all n [greater than or equal to] = 0 in which we have not only objects, morphisms, and 2morphisms but also kmorphisms for all k [less than or equal to] = n. Finally, in some sort of limit we might hope to obtain a theory of [infinity]categories, where there are morphisms of all orders.
Example 1.1.1.4. Let X be a topological space and 0 [less than or equal to] n [less than or equal to] [infinity]. We can extract an ncategory [pi] [less than or equal to] [sub.n]X (roughly) as follows. The objects of [pi] [less than or equal to] [sub.n]X are the points of X. If x, y [member of] X, then the morphisms from x to y in [pi] [less than or equal to] [sub.n]X are given by continuous paths [0, 1] [right arrow] X starting at x and ending at y. The 2morphisms are given by homotopies of paths, the 3morphisms by homotopies between homotopies, and so forth. Finally, if n < [infinity], then two nmorphisms of [pi] [less than or equal to] [sub.n]X are considered to be the same if and only if they are homotopic to one another.
If n = 0, then [pi] [less than or equal to] [sub.n]X can be identified with the set [pi][sub.0]X of path components of X. If n = 1, then our definition of [pi] [less than or equal to] [sub.n]X agrees with the usual definition for the fundamental groupoid of X. For this reason, [pi] [less than or equal to] [sub.n]X is often called the fundamental ngroupoid of X. It is called an ngroupoid (rather than a mere ncategory) because every kmorphism of [[pi].sub.[less than or equal to] [sub.k]]X has an inverse (at least up to homotopy).
There are many approaches to realizing the theory of higher categories. We might begin by defining a 2category to be a "category enriched over ITLITLat." In other words, we consider a collection of objects together with a category of morphisms Hom(A,B) for any two objects A and B and composition functors [[c.sub.ABC] x Hom(A,B) x Hom(B,C) [right arrow] Hom(A,C) (to simplify the discussion, we will ignore identity morphisms for a moment). These functors are required to satisfy an associative law, which asserts that for any quadruple (A,B,C,D) of objects, the diagram
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
commutes; in other words, one has an equality of functors
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
from Hom(A,B) x Hom(B,C) x Hom(C,D) to Hom(A,D). This leads to the definition of a strict 2category.
At this point, we should object that the definition of a strict 2category violates one of the basic philosophical principles of category theory: one should never demand that two functors F and F' be equal to one another. Instead one should postulate the existence of a natural isomorphism between F and F'. This means that the associative law should not take the form of an equation but of additional structure: a collection of isomorphisms [[gamma.sub.[ABCD] : [C.sub.ACD] [??]([c.sub.ABC] x 1) [equivalent] [c.sub.ABD] [??](1 x [c.sub.BCD]). We should further demand that the isomorphisms [[gamma.sub.[ABCD] be functorial in the quadruple (A,B,C,D) and satisfy certain higher associativity conditions, which generalize the "Pentagon axiom" described in A.1.3. After formulating the appropriate conditions, we arrive at the definition of a weak 2category.
Let us contrast the notions of strict 2category and weak 2category. The former is easier to define because we do not have to worry about the higher associativity conditions satisfied by the transformations [[gamma].sub.ABCD]. On the other hand, the latter notion seems more natural if we take the philosophy of category theory seriously. In this case, we happen to be lucky: the notions of strict 2category and weak 2category turn out to be equivalent. More precisely, any weak 2category is equivalent (in the relevant sense) to a strict 2category. The choice of definition can therefore be regarded as a question of aesthetics.
We now plunge onward to 3categories. Following the above program, we might define a strict 3category to consist of a collection of objects together with strict 2categories Hom(A,B) for any pair of objects A and B, together with a strictly associative composition law. Alternatively, we could seek a definition of weak 3category by allowing Hom(A,B) to be a weak 2category, requiring associativity only up to natural 2isomorphisms, which satisfy higher associativity laws up to natural 3isomorphisms, which in turn satisfy still higher associativity laws of their own. Unfortunately, it turns out that these notions are not equivalent.
Both of these approaches have serious drawbacks. The obvious problem with weak 3categories is that an explicit definition is extremely complicated (see [33], where a definition is given along these lines), to the point where it is essentially unusable. On the other hand, strict 3categories have the problem of not being the correct notion: most of the weak 3categories which occur in nature are not equivalent to strict 3categories. For example, the fundamental 3groupoid of the 2sphere [S.sup.2] cannot be described using the language of strict 3categories. The situation only gets worse (from either point of view) as we pass to 4categories and beyond.
Fortunately, it turns out that major simplifications can be introduced if we are willing to restrict our attention to [infinity]categories in which most of the higher morphisms are invertible. From this point forward, we will use the term ([infinity], n)category to refer to [infinity]categories in which all kmorphisms are invertible for k > n. The [infinity]categories described in Example 1.1.1.4 (when n = [infinity]) are all ([infinity], 0)categories. The converse, which asserts that every ([infinity], 0)category has the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some topological space X, is a generally accepted principle of higher category theory. Moreover, the [infinity]groupoid [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] encodes the entire homotopy type of X. In other words, ([infinity], 0)categories (that is, [infinity]categories in which all morphisms are invertible) have been extensively studied from another point of view: they are essentially the same thing as "spaces" in the sense of homotopy theory, and there are many equivalent ways to describe them (for example, we can use CW complexes or simplicial sets).
Convention 1.1.1.5. We will sometimes refer to ([infinity], 0)categories as [infinity]groupoids and ([infinity], 2)categories as [infinity]bicategories. Unless we specify otherwise, the generic term "[infinity]category" will refer to an ([infinity], 1)category.
In this book, we will restrict our attention almost entirely to the theory of [infinity]categories (in which we have only invertible nmorphisms for n [greater than or equal to] 2). Our reasons are threefold:
(1) Allowing noninvertible nmorphisms for n > 1 introduces a number of additional complications to the theory at both technical and conceptual levels. As we will see throughout this book, many ideas from category theory generalize to the [infinity]categorical setting in a natural way. However, these generalizations are not so straightforward if we allow noninvertible 2morphisms. For example, one must distinguish between strict and lax fiber products, even in the setting of "classical" 2categories.
(2) For the applications studied in this book, we will not need to consider ([infinity], n)categories for n > 2. The case n = 2 is of some relevance because the collection of (small) [infinity]categories can naturally be viewed as a (large) [infinity]bicategory. However, we will generally be able to exploit this structure in an ad hoc manner without developing any general theory of [infinity]bicategories.
(Continues...)
Table of Contents
Preface vii
Chapter 1. An Overview of Higher Category Theory 1
1.1 Foundations for Higher Category Theory 1
1.2 The Language of Higher Category Theory 26
Chapter 2. Fibrations of Simplicial Sets 53
2.1 Left Fibrations 55
2.2 Simplicial Categories and 1Categories 72
2.3 Inner Fibrations 95
2.4 Cartesian Fibrations 114
Chapter 3. The 1Category of 1Categories 145
3.1 Marked Simplicial Sets 147
3.2 Straightening and Unstraightening 169
3.3 Applications 204
Chapter 4. Limits and Colimits 223
4.1 Co_nality 223
4.2 Techniques for Computing Colimits 240
4.3 Kan Extensions 261
4.4 Examples of Colimits 292
Chapter 5. Presentable and Accessible 1Categories 311
5.1 1Categories of Presheaves 312
5.2 Adjoint Functors 331
5.3 1Categories of Inductive Limits 377
5.4 Accessible 1Categories 414
5.5 Presentable 1Categories 455
Chapter 6. 1Topoi 526
6.1 1Topoi: De_nitions and Characterizations 527
6.2 Constructions of 1Topoi 569
6.3 The 1Category of 1Topoi 593
6.4 nTopoi 632
6.5 Homotopy Theory in an 1Topos 651
Chapter 7. Higher Topos Theory in Topology 682
7.1 Paracompact Spaces 683
7.2 Dimension Theory 711
7.3 The Proper Base Change Theorem 742
Appendix. Appendix 781
A.1 Category Theory 781
A.2 Model Categories 803
A.3 Simplicial Categories 844
Bibliography 909
General Index 915
Index of Notation 923