Local Homotopy Theory

Author : John F. Jardine
Publisher : Unknown
Page : 128 pages
File Size : 50,8 Mb
Release : 2015
Category : Electronic
ISBN : 1493923013

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Local Homotopy Theory by John F. Jardine Pdf

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.

Author : John F. Jardine
Publisher : Springer
Page : 508 pages
File Size : 47,8 Mb
Release : 2015-05-27
Category : Mathematics
ISBN : 9781493923007

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Local Homotopy Theory by John F. Jardine Pdf

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.

Author : Emily Riehl
Publisher : Cambridge University Press
Page : 371 pages
File Size : 50,8 Mb
Release : 2014-05-26
Category : Mathematics
ISBN : 9781107048454

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Categorical Homotopy Theory by Emily Riehl Pdf

This categorical perspective on homotopy theory helps consolidate and simplify one's understanding of derived functors, homotopy limits and colimits, and model categories, among others.

Author : Mamoru Mimura
Publisher : Springer
Page : 246 pages
File Size : 49,7 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540469384

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Homotopy Theory and Related Topics by Mamoru Mimura Pdf

Author : Daniel G. Davis
Publisher : American Mathematical Soc.
Page : 268 pages
File Size : 45,7 Mb
Release : 2019-05-30
Category : Homotopy theory
ISBN : 9781470442446

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Homotopy Theory: Tools and Applications by Daniel G. Davis Pdf

This volume contains the proceedings of the conference Homotopy Theory: Tools and Applications, in honor of Paul Goerss's 60th birthday, held from July 17–21, 2017, at the University of Illinois at Urbana-Champaign, Urbana, IL. The articles cover a variety of topics spanning the current research frontier of homotopy theory. This includes articles concerning both computations and the formal theory of chromatic homotopy, different aspects of equivariant homotopy theory and K-theory, as well as articles concerned with structured ring spectra, cyclotomic spectra associated to perfectoid fields, and the theory of higher homotopy operations.

Author : Paul Gregory Goerss,Stewart Priddy,International Conference on Algebraic Topology
Publisher : American Mathematical Soc.
Page : 520 pages
File Size : 43,6 Mb
Release : 2004
Category : Homotopy theory
ISBN : 9780821832851

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Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic $K$-Theory by Paul Gregory Goerss,Stewart Priddy,International Conference on Algebraic Topology Pdf

As part of its series of Emphasis Years in Mathematics, Northwestern University hosted an International Conference on Algebraic Topology. The purpose of the conference was to develop new connections between homotopy theory and other areas of mathematics. This proceedings volume grew out of that event. Topics discussed include algebraic geometry, cohomology of groups, algebraic $K$-theory, and $\mathbb{A 1$ homotopy theory. Among the contributors to the volume were Alejandro Adem,Ralph L. Cohen, Jean-Louis Loday, and many others. The book is suitable for graduate students and research mathematicians interested in homotopy theory and its relationship to other areas of mathematics.

Author : Paul G. Goerss,John F. Jardine
Publisher : Birkhäuser
Page : 520 pages
File Size : 54,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034887076

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Simplicial Homotopy Theory by Paul G. Goerss,John F. Jardine Pdf

Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic theory. With the development of Quillen's concept of a closed model category and, in particular, a simplicial model category, this collection of methods has become the primary way to describe non-abelian homological algebra and to address homotopy-theoretical issues in a variety of fields, including algebraic K-theory. This book supplies a modern exposition of these ideas, emphasizing model category theoretical techniques. Discussed here are the homotopy theory of simplicial sets, and other basic topics such as simplicial groups, Postnikov towers, and bisimplicial sets. The more advanced material includes homotopy limits and colimits, localization with respect to a map and with respect to a homology theory, cosimplicial spaces, and homotopy coherence. Interspersed throughout are many results and ideas well-known to experts, but uncollected in the literature. Intended for second-year graduate students and beyond, this book introduces many of the basic tools of modern homotopy theory. An extensive background in topology is not assumed.

Author : Haynes Miller
Publisher : CRC Press
Page : 982 pages
File Size : 49,7 Mb
Release : 2020-01-23
Category : Mathematics
ISBN : 9781351251617

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Handbook of Homotopy Theory by Haynes Miller Pdf

The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other mathematical disciplines. The origins of the subject date back to work of Henri Poincaré and Heinz Hopf in the early 20th century, but it has seen enormous progress in the 21st century. A highlight of this volume is an introduction to and diverse applications of the newly established foundational theory of ¥ -categories. The coverage is vast, ranging from axiomatic to applied, from foundational to computational, and includes surveys of applications both geometric and algebraic. The contributors are among the most active and creative researchers in the field. The 22 chapters by 31 contributors are designed to address novices, as well as established mathematicians, interested in learning the state of the art in this field, whose methods are of increasing importance in many other areas.

Author : Bjorn Ian Dundas,Marc Levine,P.A. Østvær,Oliver Röndigs,Vladimir Voevodsky
Publisher : Springer Science & Business Media
Page : 228 pages
File Size : 43,5 Mb
Release : 2007-07-11
Category : Mathematics
ISBN : 9783540458975

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Motivic Homotopy Theory by Bjorn Ian Dundas,Marc Levine,P.A. Østvær,Oliver Röndigs,Vladimir Voevodsky Pdf

This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Aimed at graduate students in algebraic topology and algebraic geometry, it contains background material from both of these fields, as well as the foundations of motivic homotopy theory. It will serve as a good introduction as well as a convenient reference for a broad group of mathematicians to this important and fascinating new subject. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work, and the other authors have all done important work in the subject.

Author : Julia E. Bergner
Publisher : Cambridge University Press
Page : 289 pages
File Size : 45,6 Mb
Release : 2018-03-15
Category : Mathematics
ISBN : 9781107101364

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The Homotopy Theory of (?,1)-Categories by Julia E. Bergner Pdf

An introductory treatment to the homotopy theory of homotopical categories, presenting several models and comparisons between them.

Author : Yves Félix,Gregory Lupton,Samuel B. Smith
Publisher : American Mathematical Soc.
Page : 246 pages
File Size : 51,6 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821849293

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Homotopy Theory of Function Spaces and Related Topics by Yves Félix,Gregory Lupton,Samuel B. Smith Pdf

This volume contains the proceedings of the Workshop on Homotopy Theory of Function Spaces and Related Topics, which was held at the Mathematisches Forschungsinstitut Oberwolfach, in Germany, from April 5-11, 2009. This volume contains fourteen original research articles covering a broad range of topics that include: localization and rational homotopy theory, evaluation subgroups, free loop spaces, Whitehead products, spaces of algebraic maps, gauge groups, loop groups, operads, and string topology. In addition to reporting on various topics in the area, this volume is supposed to facilitate the exchange of ideas within Homotopy Theory of Function Spaces, and promote cross-fertilization between Homotopy Theory of Function Spaces and other areas. With these latter aims in mind, this volume includes a survey article which, with its extensive bibliography, should help bring researchers and graduate students up to speed on activity in this field as well as a problems list, which is an expanded and edited version of problems discussed in sessions held at the conference. The problems list is intended to suggest directions for future work.

Author : Eric M. Friedlander
Publisher : Princeton University Press
Page : 196 pages
File Size : 45,9 Mb
Release : 1982-12-21
Category : Mathematics
ISBN : 0691083177

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Etale Homotopy of Simplical Schemes by Eric M. Friedlander Pdf

This book presents a coherent account of the current status of etale homotopy theory, a topological theory introduced into abstract algebraic geometry by M. Artin and B. Mazur. Eric M. Friedlander presents many of his own applications of this theory to algebraic topology, finite Chevalley groups, and algebraic geometry. Of particular interest are the discussions concerning the Adams Conjecture, K-theories of finite fields, and Poincare duality. Because these applications have required repeated modifications of the original formulation of etale homotopy theory, the author provides a new treatment of the foundations which is more general and more precise than previous versions. One purpose of this book is to offer the basic techniques and results of etale homotopy theory to topologists and algebraic geometers who may then apply the theory in their own work. With a view to such future applications, the author has introduced a number of new constructions (function complexes, relative homology and cohomology, generalized cohomology) which have immediately proved applicable to algebraic K-theory.

Author : Michael Charles Crabb,Ioan Mackenzie James
Publisher : Springer Science & Business Media
Page : 344 pages
File Size : 45,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781447112655

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Fibrewise Homotopy Theory by Michael Charles Crabb,Ioan Mackenzie James Pdf

Topology occupies a central position in modern mathematics, and the concept of the fibre bundle provides an appropriate framework for studying differential geometry. Fibrewise homotopy theory is a very large subject that has attracted a good deal of research in recent years. This book provides an overview of the subject as it stands at present.

Author : Bjørn Ian Dundas
Publisher : Springer Science & Business Media
Page : 228 pages
File Size : 48,8 Mb
Release : 2007
Category : Mathematics
ISBN : 9783540458951

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Motivic Homotopy Theory by Bjørn Ian Dundas Pdf

This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work.

Author : Bjørn Ian Dundas,Thomas G. Goodwillie,Randy McCarthy
Publisher : Springer Science & Business Media
Page : 447 pages
File Size : 49,5 Mb
Release : 2012-09-06
Category : Mathematics
ISBN : 9781447143932

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The Local Structure of Algebraic K-Theory by Bjørn Ian Dundas,Thomas G. Goodwillie,Randy McCarthy Pdf

Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. This book covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology and proves that the difference between the theories are ‘locally constant’. The usefulness of this theorem stems from being more accessible for calculations than K-theory, and hence a single calculation of K-theory can be used with homological calculations to obtain a host of ‘nearby’ calculations in K-theory. For instance, Quillen's calculation of the K-theory of finite fields gives rise to Hesselholt and Madsen's calculations for local fields, and Voevodsky's calculations for the integers give insight into the diffeomorphisms of manifolds. In addition to the proof of the full integral version of the local correspondence between K-theory and topological cyclic homology, the book provides an introduction to the necessary background in algebraic K-theory and highly structured homotopy theory; collecting all necessary tools into one common framework. It relies on simplicial techniques, and contains an appendix summarizing the methods widely used in the field. The book is intended for graduate students and scientists interested in algebraic K-theory, and presupposes a basic knowledge of algebraic topology.