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Author | : K Heiner Kamps |
Publisher | : World Scientific |
Total Pages | : 476 |
Release | : 1997-04-11 |
Genre | : Mathematics |
ISBN | : 9814502553 |
Download Abstract Homotopy And Simple Homotopy Theory Book in PDF, ePub and Kindle
The abstract homotopy theory is based on the observation that analogues of much of the topological homotopy theory and simple homotopy theory exist in many other categories (e.g. spaces over a fixed base, groupoids, chain complexes, module categories). Studying categorical versions of homotopy structure, such as cylinders and path space constructions, enables not only a unified development of many examples of known homotopy theories but also reveals the inner working of the classical spatial theory. This demonstrates the logical interdependence of properties (in particular the existence of certain Kan fillers in associated cubical sets) and results (Puppe sequences, Vogt's Iemma, Dold's theorem on fibre homotopy equivalences, and homotopy coherence theory).
Author | : Klaus Heiner Kamps |
Publisher | : World Scientific |
Total Pages | : 474 |
Release | : 1997 |
Genre | : Mathematics |
ISBN | : 9789810216023 |
Download Abstract Homotopy and Simple Homotopy Theory Book in PDF, ePub and Kindle
"This book provides a thorough and well-written guide to abstract homotopy theory. It could well serve as a graduate text in this topic, or could be studied independently by someone with a background in basic algebra, topology, and category theory."
Author | : K. H. Kamps |
Publisher | : |
Total Pages | : 56 |
Release | : 1986 |
Genre | : |
ISBN | : |
Download Abstract Homotopy and Simple Homotopy Theory Book in PDF, ePub and Kindle
Author | : M.M. Cohen |
Publisher | : Springer Science & Business Media |
Total Pages | : 124 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1468493728 |
Download A Course in Simple-Homotopy Theory Book in PDF, ePub and Kindle
This book grew out of courses which I taught at Cornell University and the University of Warwick during 1969 and 1970. I wrote it because of a strong belief that there should be readily available a semi-historical and geo metrically motivated exposition of J. H. C. Whitehead's beautiful theory of simple-homotopy types; that the best way to understand this theory is to know how and why it was built. This belief is buttressed by the fact that the major uses of, and advances in, the theory in recent times-for example, the s-cobordism theorem (discussed in §25), the use of the theory in surgery, its extension to non-compact complexes (discussed at the end of §6) and the proof of topological invariance (given in the Appendix)-have come from just such an understanding. A second reason for writing the book is pedagogical. This is an excellent subject for a topology student to "grow up" on. The interplay between geometry and algebra in topology, each enriching the other, is beautifully illustrated in simple-homotopy theory. The subject is accessible (as in the courses mentioned at the outset) to students who have had a good one semester course in algebraic topology. I have tried to write proofs which meet the needs of such students. (When a proof was omitted and left as an exercise, it was done with the welfare of the student in mind. He should do such exercises zealously.
Author | : |
Publisher | : Univalent Foundations |
Total Pages | : 484 |
Release | : |
Genre | : |
ISBN | : |
Download Homotopy Type Theory: Univalent Foundations of Mathematics Book in PDF, ePub and Kindle
Author | : Emily Riehl |
Publisher | : Cambridge University Press |
Total Pages | : 371 |
Release | : 2014-05-26 |
Genre | : Mathematics |
ISBN | : 1139952633 |
Download Categorical Homotopy Theory Book in PDF, ePub and Kindle
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.
Author | : Sadanand Verma |
Publisher | : |
Total Pages | : 242 |
Release | : 1959 |
Genre | : Homotopy theory |
ISBN | : |
Download Relation Between Abstract Homotopy and Geometric Homotopy Book in PDF, ePub and Kindle
Author | : Brian A. Munson |
Publisher | : Cambridge University Press |
Total Pages | : 649 |
Release | : 2015-10-06 |
Genre | : Mathematics |
ISBN | : 1107030250 |
Download Cubical Homotopy Theory Book in PDF, ePub and Kindle
A modern, example-driven introduction to cubical diagrams and related topics such as homotopy limits and cosimplicial spaces.
Author | : Paul G. Goerss |
Publisher | : Birkhäuser |
Total Pages | : 520 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3034887078 |
Download Simplicial Homotopy Theory Book in PDF, ePub and Kindle
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 | : Hans J. Baues |
Publisher | : Cambridge University Press |
Total Pages | : 490 |
Release | : 1989-02-16 |
Genre | : Mathematics |
ISBN | : 0521333768 |
Download Algebraic Homotopy Book in PDF, ePub and Kindle
This book gives a general outlook on homotopy theory; fundamental concepts, such as homotopy groups and spectral sequences, are developed from a few axioms and are thus available in a broad variety of contexts. Many examples and applications in topology and algebra are discussed, including an introduction to rational homotopy theory in terms of both differential Lie algebras and De Rham algebras. The author describes powerful tools for homotopy classification problems, particularly for the classification of homotopy types and for the computation of the group homotopy equivalences. Applications and examples of such computations are given, including when the fundamental group is non-trivial. Moreover, the deep connection between the homotopy classification problems and the cohomology theory of small categories is demonstrated. The prerequisites of the book are few: elementary topology and algebra. Consequently, this account will be valuable for non-specialists and experts alike. It is an important supplement to the standard presentations of algebraic topology, homotopy theory, category theory and homological algebra.