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Homotopy Theory: Tools and Applications

Author: Daniel G. Davis

Publisher: American Mathematical Soc.

Published: 2019-05-30

Total Pages: 268

ISBN-13: 1470442442

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.

Modern Classical Homotopy Theory

Author: Jeffrey Strom

Publisher: American Mathematical Society

Published: 2023-01-19

Total Pages: 862

ISBN-13: 1470471639

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The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.

Simplicial Homotopy Theory

Author: Paul G. Goerss

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 520

ISBN-13: 3034887078

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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.

From Categories to Homotopy Theory

Author: Birgit Richter

Publisher: Cambridge University Press

Published: 2020-04-16

Total Pages: 402

ISBN-13: 1108847625

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Category theory provides structure for the mathematical world and is seen everywhere in modern mathematics. With this book, the author bridges the gap between pure category theory and its numerous applications in homotopy theory, providing the necessary background information to make the subject accessible to graduate students or researchers with a background in algebraic topology and algebra. The reader is first introduced to category theory, starting with basic definitions and concepts before progressing to more advanced themes. Concrete examples and exercises illustrate the topics, ranging from colimits to constructions such as the Day convolution product. Part II covers important applications of category theory, giving a thorough introduction to simplicial objects including an account of quasi-categories and Segal sets. Diagram categories play a central role throughout the book, giving rise to models of iterated loop spaces, and feature prominently in functor homology and homology of small categories.

Categorical Homotopy Theory

Author: Emily Riehl

Publisher: Cambridge University Press

Published: 2014-05-26

Total Pages: 371

ISBN-13: 1139952633

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This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion in certain examples. Emily Riehl unifies these seemingly rival perspectives and demonstrates that model structures on diagram categories are irrelevant. Homotopy (co)limits are explained to be a special case of weighted (co)limits, a foundational topic in enriched category theory. In Part II, Riehl further examines this topic, separating categorical arguments from homotopical ones. Part III treats the most ubiquitous axiomatic framework for homotopy theory - Quillen's model categories. Here, Riehl simplifies familiar model categorical lemmas and definitions by focusing on weak factorization systems. Part IV introduces quasi-categories and homotopy coherence.

Homotopy Theory and Its Applications

Author: Alejandro Adem

Publisher: American Mathematical Soc.

Published: 1995-01-01

Total Pages: 252

ISBN-13: 9780821855256

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This book is the result of a conference held to examine developments in homotopy theory in honor of Samuel Gitler in August 1993 (Cocoyoc, Mexico). It includes several research papers and three expository papers on various topics in homotopy theory. The research papers discuss the following: application of homotopy theory to group theory, fiber bundle theory, and homotopy theory. The expository papers consider the following topics: the Atiyah-Jones conjecture (by C. Boyer), classifying spaces of finite groups (by J. Martino), and instanton moduli spaces (by R. J. Milgram). Homotopy Theory and Its Applications offers a distinctive account of how homotopy-theoretic methods can be applied to a variety of interesting problems.

Cubical Homotopy Theory

Author: Brian A. Munson

Publisher: Cambridge University Press

Published: 2015-10-06

Total Pages: 649

ISBN-13: 1107030250

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A modern, example-driven introduction to cubical diagrams and related topics such as homotopy limits and cosimplicial spaces.

Homotopy Theory: An Introduction to Algebraic Topology

Author:

Publisher: Academic Press

Published: 1975-11-12

Total Pages: 367

ISBN-13: 9780080873800

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Homotopy Theory: An Introduction to Algebraic Topology

Axiomatic Stable Homotopy Theory

Author: Mark Hovey

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 130

ISBN-13: 0821806246

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We define and investigate a class of categories with formal properties similar to those of the homotopy category of spectra. This class includes suitable versions of the derived category of modules over a commutative ring, or of comodules over a commutative Hopf algebra, and is closed under Bousfield localization. We study various notions of smallness, questions about representability of (co)homology functors, and various kinds of localization. We prove theorems analogous to those of Hopkins and Smith about detection of nilpotence and classification of thick subcategories. We define the class of Noetherian stable homotopy categories, and investigate their special properties. Finally, we prove that a number of categories occurring in nature (including those mentioned above) satisfy our axioms.

Nilpotence and Periodicity in Stable Homotopy Theory

Author: Douglas C. Ravenel

Publisher: Princeton University Press

Published: 1992-11-08

Total Pages: 228

ISBN-13: 9780691025728

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Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic. Ravenel's first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.