Equips the reader with the background necessary to understand recent advances in homotopy theory and algebraic geometry
Written by one of the main contributors to the field
Goes beyond the formalism of the theory to explain the development and applications of the main ideas and results
This monograph on the homotopy theory of topologized diagrams of spaces and spectra gives an expert account of a subject at the foundation of motivic homotopy theory and the theory of topological modular forms in stable homotopy theory.
Beginning with an introduction to the homotopy theory of simplicial sets and topos theory, the book covers core topics such as the unstable homotopy theory of simplicial presheaves and sheaves, localized theories, cocycles, descent theory, non-abelian cohomology, stacks, and local stable homotopy theory. A detailed treatment of the formalism of the subject is interwoven with explanations of the motivation, development, and nuances of ideas and results. The coherence of the abstract theory is elucidated through the use of widely applicable tools, such as Barr's theorem on Boolean localization, model structures on the category of simplicial presheaves on a site, and cocycle categories. A wealth of concrete examples convey the vitality and importance of the subject in topology, number theory, algebraic geometry, and algebraic K-theory.
Assuming basic knowledge of algebraic geometry and homotopy theory, Local Homotopy Theory will appeal to researchers and advanced graduate students seeking to understand and advance the applications of homotopy theory in multiple areas of mathematics and the mathematical sciences.
Topics
Category Theory, Homological Algebra
K-Theory
Algebraic Topology
Author(s): John F. Jardine
Series: Springer Monographs in Mathematics
Edition: 2015
Publisher: Springer
Year: 2015
Language: English
Pages: C, ix, 508, B
Tags: Topology Geometry Mathematics Science Math Abstract Algebra Pure Trigonometry New Used Rental Textbooks Specialty Boutique
1 Introduction
Part I Preliminaries
2 Homotopy Theory of Simplicial Sets
2.1 Simplicial Sets
2.2 Model Structure for Simplicial Sets
2.3 Projective Model Structure for Diagrams
3 Some Topos Theory
3.1 Grothendieck Topologies
3.2 Exactness Properties
3.3 Geometric Morphisms
3.4 Points
3.5 Boolean Localization
Part II Simplicial Presheaves and Simplicial Sheaves
4 LocalWeak Equivalences
4.1 LocalWeak Equivalences
4.2 Local Fibrations
4.3 First Applications of Boolean Localization
5 Local Model Structures
5.1 The Injective Model Structure
5.2 Injective Fibrations
5.3 Geometric and Site Morphisms
5.4 Descent Theorems
5.5 Intermediate Model Structures
5.6 Postnikov Sections and n-Types
6 Cocycles
6.1 Cocycle Categories
6.2 The Verdier Hypercovering Theorem
7 Localization Theories
7.1 General Theory
7.2 Localization Theorems for Simplicial Presheaves
7.3 Properness
Part III Sheaf Cohomology Theory
8 Homology Sheaves and Cohomology Groups
8.1 Chain Complexes
8.2 The Derived Category
8.3 Abelian Sheaf Cohomology
8.4 Products and Pairings
8.5 Localized Chain Complexes
8.6 Linear Simplicial Presheaves
9 Non-abelian Cohomology
9.1 Torsors
9.2 Stacks and Homotopy Theory
9.3 Groupoids Enriched in Simplicial Sets
9.4 Presheaves of Groupoids Enriched in Simplicial Sets
9.5 Extensions and Gerbes
Part IV Stable Homotopy Theory
10 Spectra and T-spectra
10.1 Presheaves of Spectra
10.2 T-spectra and Localization
10.3 Stable Model Structures for T-spectra
10.4 Shifts and Suspensions
10.5 Fibre and Cofibre Sequences
10.6 Postnikov Sections and Slice Filtrations
10.7 T-Complexes
11 Symmetric T-spectra
11.1 Symmetric Spaces
11.2 First Model Structures
11.3 Localized Model Structures
11.4 Stable Homotopy Theory of Symmetric Spectra
11.5 Equivalence of Stable Categories
11.6 The Smash Product
11.7 Symmetric T-complexes
References
Index
