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Higher Topos Theory (AM-170).

Author: Lurie, Jacob.
Publisher: Princeton University Press 2009.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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. 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  Read more...
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Genre/Form: Electronic resource
Electronic books
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Lurie, Jacob.
ISBN: 1282644955 9781282644953 9781400830558 1400830559
OCLC Number: 748357193
Description: 1 online resource (960)
Contents: Cover --
Contents --
Preface --
Chapter 1. An Overview of Higher Category Theory --
1.1 Foundations for Higher Category Theory --
1.2 The Language of Higher Category Theory --
Chapter 2. Fibrations of Simplicial Sets --
2.1 Left Fibrations --
2.2 Simplicial Categories and -Categories --
2.3 Inner Fibrations --
2.4 Cartesian Fibrations --
Chapter 3. The -Category of -Categories --
3.1 Marked Simplicial Sets --
3.2 Straightening and Unstraightening --
3.3 Applications --
Chapter 4. Limits and Colimits --
4.1 Cofinality --
4.2 Techniques for Computing Colimits --
4.3 Kan Extensions --
4.4 Examples of Colimits --
Chapter 5. Presentable and Accessible -Categories --
5.1 -Categories of Presheaves --
5.2 Adjoint Functors --
5.3 -Categories of Inductive Limits --
5.4 Accessible -Categories --
5.5 Presentable -Categories --
Chapter 6.-Topoi --
6.1 -Topoi: Definitions and Characterizations --
6.2 Constructions of -Topoi --
6.3 The -Category of -Topoi --
6.4 n-Topoi --
6.5 Homotopy Theory in an -Topos --
Chapter 7. Higher Topos Theory in Topology --
7.1 Paracompact Spaces --
7.2 Dimension Theory --
7.3 The Proper Base Change Theorem --
Appendix --
A.1 Category Theory --
A.2 Model Categories --
A.3 Simplicial Categories --
Bibliography --
General Index --
A --
B --
C --
D --
E --
F --
G --
H --
I --
J --
K --
L --
M --
N --
O --
P --
Q --
R --
S --
T --
U --
W --
Y --
Index of Notation --
A --
B --
C --
D --
E --
F --
G --
H --
I --
K --
L --
M --
N --
O --
P --
Q --
R --
S --
T --
U --
X.

Abstract:

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. 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 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.

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