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Higher Topos Theory (AM-170)

Higher Topos Theory (AM-170)

Jacob Lurie
Copyright Date: 2009
Pages: 960
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  • Book Info
    Higher Topos Theory (AM-170)
    Book Description:

    Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. InHigher 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 infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category 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.

    eISBN: 978-1-4008-3055-8
    Subjects: Mathematics

Table of Contents

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  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Preface
    (pp. vii-xviii)
  4. Chapter One An Overview of Higher Category Theory
    (pp. 1-52)

    This 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 usingtopological 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∞-categories(calledweak Kan complexesin [10] andquasi-categoriesin [43]), which provides a more convenient setting for adaptations of sophisticated category-theoretic ideas. Our goal in §1.1.1 is to introduce both approaches and to explain why they are equivalent to one...

  5. Chapter Two Fibrations of Simplicial Sets
    (pp. 53-144)

    Many classes of morphisms which play an important role in the homotopy theory of simplicial sets can be defined by their lifting properties (we refer the reader to §A.1.2 for a brief discussion and a summary of the terminology employed below).

    Example A morphismp:XSof simplicial sets which has the right lifting property with respect to every horn inclusion$\Lambda _i^n \subseteq \;{\Delta ^n}$is called aKan fibration. A morphismi:ABwhich has the left lifting property with respect to every Kan fibration is said to beanodyne.

    Example A morphismp:XSof simplicial sets which has the right...

  6. Chapter Three The ∞-categories of ∞-categories
    (pp. 145-222)

    The power of category theory lies in its role as a unifying language for mathematics: nearly every class of mathematical structures (groups, manifolds, algebraic varieties, and so on) can be organized into a category. This language is somewhat inadequate in situations where the structures need to be classified up to some notion of equivalence less rigid than isomorphism. For example, in algebraic topology one wishes to study topological spaces up to homotopy equivalence; in homological algebra one wishes to study chain complexes up to quasi-isomorphism. Both of these examples are most naturally described in terms of higher category theory (for...

  7. Chapter Four Limits and Colimits
    (pp. 223-310)

    This chapter is devoted to the study of limits and colimits in the setting of ∞-categories. Our goal is to provide tools for proving the existence of limits and colimits, for analyzing them, and for comparing them to the (perhaps more familiar) notion of homotopy limits and colimits in simplicial categories. We will generally confine our remarks to colimits; analogous results for limits can be obtained by passing to the opposite ∞-categories.

    We begin in §4.1 by introducing the notion of acofinalmap between simplicial sets. Iff:ABis a cofinal map of simplicial sets, then we can identify colimits...

  8. Chapter Five Presentable and Accessible ∞-Categories
    (pp. 311-525)

    Many categories which arise naturally, such as the categoryAof abelian groups, arelarge: they have a proper class of objects even when the objects are considered only up to isomorphism. However, thoughAitself is large, it is in some sense determined by the much smaller categoryA0of finitely generated abelian groups:Ais naturally equivalent to the category of Ind-objects ofA0. This remark carries more than simply philosophical significance. When properly exploited, it can be used to prove statements such as the following:

    Proposition F : A → Set be a contravariant functor from A to the...

  9. Chapter Six ∞-Topoi
    (pp. 526-681)

    In this chapter, we come to the main subject of this book: the theory of ∞-Topoi. Roughly speaking, an ∞-topos is an ∞-category which “looks like” the category of spaces, just as an ordinary topos is a category which “looks like” the category of sets. As in classical topos theory, there are various ways of making this precise. We will begin in §6.1 by reviewing several possible definitions and proving that they are equivalent to one another.

    The main result of §6.1 is Theorem, which asserts that an ∞-categoryXis an ∞-topos if and only ifXarises...

  10. Chapter Seven Higher Topos Theory in Topology
    (pp. 682-780)

    In this chapter, we will sketch three applications of the theory of ∞-topoi to the study of classical topology. We begin in §7.1 by showing that ifXis a paracompact topological space, then the ∞-topos Shv(X) of sheaves onXcan be interpreted as a homotopy theory of topological spacesYequipped with a map toX. We will deduce, as an application, that ifp*: Shv(X) → Shv(*) is the geometric morphism induced by the projectionX→ *, then the composition${p_*}p{\kern 1pt} *$is equivalent to the functor


    from (compactly generated) topological spaces to itself.

    Our second application is to the dimension...

  11. Appendix
    (pp. 781-908)
  12. Bibliography
    (pp. 909-914)
  13. General Index
    (pp. 915-922)
  14. Index of Notation
    (pp. 923-925)