John F. Jardine – författare

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.

This authoritative work provides clinicians, scientists and students with a comprehensive overview of exertional heat illness.  Specifically, it addresses the prevention, recognition, treatment, and care of the various medical conditions that fall within the realm of exertional heat illness.  In doing so, the book also offers a setting-specific (that is, athletics, military, occupational, and road race medicine) discussion of exertional heat illness for the consideration of the varied medical providers working in these settings. Clinicians will benefit from the discussion of the evidence-based best-practice considerations that should be made in the management of exertional heat illness.  Scientists will benefit from this text in that it will provide them with a review of the current scientific evidence related to exertional heat illness and the translation of evidence to clinical practice – while also discussing directions for future research.  Finally, students-- primarily postgraduate students interested in developing a line of research related to exertional heat illness -- will find this title an indispensable text to familiarize themselves with this fascinating field of study.  A major contribution to the literature, Exertional Heat Illness: A Clinical and Evidence-Based Guide will be of significant interest to clinicians and scientists at all levels of training and experience, especially professionals in athletic training, emergency medical services, emergency room care, sports medicine and primary care.

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.

Reviews:

"… a book filling an obvious gap in the literature and the authors have done an excellent job on it. No monograph or expository paper has been published on this topic in the last twenty-eight years." - Analele Universitatii din Timisoara

"… is clearly presented and a brief summary preceding every chapter is useful to the reader. The book should prove enlightening to a broad range of readers including prospective students and researchers who want to apply simplicial techniques for whatever reason." - Zentralblatt MATH

 "… they succeed. The book is an excellent account of simplicial homotopy theory from a modern point of view […] The bookis well written. […] The book can be highly recommended to anybody who wants to learn and to apply simplicial techniques and/or the theory of (simplicial) closed model categories." - Mathematical Reviews

A generalized etale cohomology theory is a theory which is represented by a presheaf of spectra on an etale site for an algebraic variety, in analogy with the way an ordinary spectrum represents a cohomology theory for spaces. Examples include etale cohomology and etale K-theory. This book gives new and complete proofs of both Thomason''s descent theorem for Bott periodic K-theory and the Nisnevich descent theorem. In doing so, it exposes most of the major ideas of the homotopy theory of presheaves of spectra, and generalized etale homology theories in particular. The treatment includes, for the purpose of adequately dealing with cup product structures, a development of stable homotopy theory for n-fold spectra, which is then promoted to the level of presheaves of n-fold spectra.

This book should be of interest to all researchers working in fields related to algebraic K-theory. The techniques presented here are essentially combinatorial, and hence algebraic. An extensive background in traditional stable homotopy theory is not assumed.

------  Reviews

(…) in developing the techniques of the subject, introduces the reader to the stable homotopy category of simplicial presheaves. (…) This book provides the user with the first complete account which is sensitive enough to be compatible with the sort of closed model category necessary in K-theory applications (…). As an application of the techniques the author gives proofs of the descent theorems of R. W. Thomason and Y. A. Nisnevich. (…) The book concludes with a discussion of the Lichtenbaum-Quillen conjecture (an approximation to Thomason’s theorem without Bott periodicity). The recent proof of this conjecture, by V. Voevodsky, (…) makes this volume compulsory reading for all who want to be aufait with current trends in algebraic K-theory!

- Zentralblatt MATH

The presentation of these topics is highly original. The book will be very useful for any researcher interested in subjects related to algebraic K-theory.

- Matematica