Simplicial And Dendroidal Homotopy Theory
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Author |
: Gijs Heuts |
Publisher |
: Springer Nature |
Total Pages |
: 622 |
Release |
: 2022-09-03 |
ISBN-10 |
: 9783031104473 |
ISBN-13 |
: 3031104471 |
Rating |
: 4/5 (73 Downloads) |
Synopsis Simplicial and Dendroidal Homotopy Theory by : Gijs Heuts
This open access book offers a self-contained introduction to the homotopy theory of simplicial and dendroidal sets and spaces. These are essential for the study of categories, operads, and algebraic structure up to coherent homotopy. The dendroidal theory combines the combinatorics of trees with the theory of Quillen model categories. Dendroidal sets are a natural generalization of simplicial sets from the point of view of operads. In this book, the simplicial approach to higher category theory is generalized to a dendroidal approach to higher operad theory. This dendroidal theory of higher operads is carefully developed in this book. The book also provides an original account of the more established simplicial approach to infinity-categories, which is developed in parallel to the dendroidal theory to emphasize the similarities and differences. Simplicial and Dendroidal Homotopy Theory is a complete introduction, carefully written with the beginning researcher in mind and ideally suited for seminars and courses. It can also be used as a standalone introduction to simplicial homotopy theory and to the theory of infinity-categories, or a standalone introduction to the theory of Quillen model categories and Bousfield localization.
Author |
: Paul G. Goerss |
Publisher |
: Birkhäuser |
Total Pages |
: 520 |
Release |
: 2012-12-06 |
ISBN-10 |
: 9783034887076 |
ISBN-13 |
: 3034887078 |
Rating |
: 4/5 (76 Downloads) |
Synopsis Simplicial Homotopy Theory by : Paul G. Goerss
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 |
: Ieke Moerdijk |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 186 |
Release |
: 2010-12-01 |
ISBN-10 |
: 9783034800525 |
ISBN-13 |
: 3034800525 |
Rating |
: 4/5 (25 Downloads) |
Synopsis Simplicial Methods for Operads and Algebraic Geometry by : Ieke Moerdijk
"This book is an introduction to two higher-categorical topics in algebraic topology and algebraic geometry relying on simplicial methods. It is based on lectures delivered at the Centre de Recerca Matemàtica in February 2008, as part of a special year on Homotopy Theory and Higher Categories"--Foreword
Author |
: Paul G. Goerss |
Publisher |
: |
Total Pages |
: 528 |
Release |
: 2010-11-20 |
ISBN-10 |
: 3034601905 |
ISBN-13 |
: 9783034601900 |
Rating |
: 4/5 (05 Downloads) |
Synopsis Simplicial Homotopy Theory by : Paul G. Goerss
Author |
: Eric M. Friedlander |
Publisher |
: Princeton University Press |
Total Pages |
: 196 |
Release |
: 1982-12-21 |
ISBN-10 |
: 0691083177 |
ISBN-13 |
: 9780691083179 |
Rating |
: 4/5 (77 Downloads) |
Synopsis Etale Homotopy of Simplical Schemes by : Eric M. Friedlander
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 |
: Julia E. Bergner |
Publisher |
: Cambridge University Press |
Total Pages |
: 289 |
Release |
: 2018-03-15 |
ISBN-10 |
: 9781107101364 |
ISBN-13 |
: 1107101360 |
Rating |
: 4/5 (64 Downloads) |
Synopsis The Homotopy Theory of (?,1)-Categories by : Julia E. Bergner
An introductory treatment to the homotopy theory of homotopical categories, presenting several models and comparisons between them.
Author |
: Benoit Fresse |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 743 |
Release |
: 2017-05-22 |
ISBN-10 |
: 9781470434823 |
ISBN-13 |
: 1470434822 |
Rating |
: 4/5 (23 Downloads) |
Synopsis Homotopy of Operads and Grothendieck-Teichmuller Groups by : Benoit Fresse
The ultimate goal of this book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2-disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads. The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2-disc operad. This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory.
Author |
: Paul Selick |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 220 |
Release |
: 2008 |
ISBN-10 |
: 0821844369 |
ISBN-13 |
: 9780821844366 |
Rating |
: 4/5 (69 Downloads) |
Synopsis Introduction to Homotopy Theory by : Paul Selick
Offers a summary for students and non-specialists who are interested in learning the basics of algebraic topology. This book covers fibrations and cofibrations, Hurewicz and cellular approximation theorems, topics in classical homotopy theory, simplicial sets, fiber bundles, Hopf algebras, and generalized homology and cohomology operations.
Author |
: R.A. Piccinini |
Publisher |
: Elsevier |
Total Pages |
: 307 |
Release |
: 1992-01-21 |
ISBN-10 |
: 9780080872827 |
ISBN-13 |
: 0080872824 |
Rating |
: 4/5 (27 Downloads) |
Synopsis Lectures on Homotopy Theory by : R.A. Piccinini
The central idea of the lecture course which gave birth to this book was to define the homotopy groups of a space and then give all the machinery needed to prove in detail that the nth homotopy group of the sphere Sn, for n greater than or equal to 1 is isomorphic to the group of the integers, that the lower homotopy groups of Sn are trivial and that the third homotopy group of S2 is also isomorphic to the group of the integers. All this was achieved by discussing H-spaces and CoH-spaces, fibrations and cofibrations (rather thoroughly), simplicial structures and the homotopy groups of maps.Later, the book was expanded to introduce CW-complexes and their homotopy groups, to construct a special class of CW-complexes (the Eilenberg-Mac Lane spaces) and to include a chapter devoted to the study of the action of the fundamental group on the higher homotopy groups and the study of fibrations in the context of a category in which the fibres are forced to live; the final material of that chapter is a comparison of various kinds of universal fibrations. Completing the book are two appendices on compactly generated spaces and the theory of colimits. The book does not require any prior knowledge of Algebraic Topology and only rudimentary concepts of Category Theory are necessary; however, the student is supposed to be well at ease with the main general theorems of Topology and have a reasonable mathematical maturity.
Author |
: Paul Gregory Goerss |
Publisher |
: Basel : Birkhäuser Verlag |
Total Pages |
: 510 |
Release |
: 1999-01-01 |
ISBN-10 |
: 081766064X |
ISBN-13 |
: 9780817660642 |
Rating |
: 4/5 (4X Downloads) |
Synopsis Simplicial Homotopy Theory by : Paul Gregory Goerss
Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic theory. This book supplies a modern and detailed exposition of simplicial methods and introduces many of the halle tools of modern holotopy theory. The basic topics as well as more advanced material are discussed, and many results and ideas that are known to experts, but uncollected in the literature, are interspersed throughout the presentation.