Simplicial And Dendroidal Homotopy Theory

Author : Gijs Heuts,Ieke Moerdijk
Publisher : Springer Nature
Page : 622 pages
File Size : 54,9 Mb
Release : 2022-09-03
Category : Mathematics
ISBN : 9783031104473

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Simplicial and Dendroidal Homotopy Theory by Gijs Heuts,Ieke Moerdijk Pdf

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,John F. Jardine
Publisher : Birkhäuser
Page : 520 pages
File Size : 41,6 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 : Ieke Moerdijk,Bertrand Toën
Publisher : Springer Science & Business Media
Page : 186 pages
File Size : 45,6 Mb
Release : 2010-12-01
Category : Mathematics
ISBN : 9783034800525

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Simplicial Methods for Operads and Algebraic Geometry by Ieke Moerdijk,Bertrand Toën Pdf

"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,John F. Jardine
Publisher : Unknown
Page : 528 pages
File Size : 44,9 Mb
Release : 2010-11-20
Category : Electronic
ISBN : 3034601905

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

Author : Paul Gregory Goerss,J. F. Jardine
Publisher : Basel : Birkhäuser Verlag
Page : 510 pages
File Size : 55,9 Mb
Release : 1999-01-01
Category : Mathematics
ISBN : 081766064X

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Simplicial Homotopy Theory by Paul Gregory Goerss,J. 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. 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.

Author : Julia E. Bergner
Publisher : Cambridge University Press
Page : 289 pages
File Size : 53,9 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 : Eric M. Friedlander
Publisher : Princeton University Press
Page : 191 pages
File Size : 42,8 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400881499

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Etale Homotopy of Simplicial Schemes. (AM-104), Volume 104 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 : Benoit Fresse
Publisher : American Mathematical Soc.
Page : 704 pages
File Size : 55,6 Mb
Release : 2017-05-22
Category : Grothendieck groups
ISBN : 9781470434823

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Homotopy of Operads and Grothendieck-Teichmuller Groups by Benoit Fresse Pdf

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.
Page : 220 pages
File Size : 42,8 Mb
Release : 2008
Category : Mathematics
ISBN : 0821844369

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Introduction to Homotopy Theory by Paul Selick Pdf

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
Page : 292 pages
File Size : 41,5 Mb
Release : 1992-01-21
Category : Mathematics
ISBN : 0080872824

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Lectures on Homotopy Theory by R.A. Piccinini Pdf

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 : John F. Jardine
Publisher : Springer
Page : 508 pages
File Size : 40,9 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 : William G. Dwyer,Hans-Werner Henn
Publisher : Birkhäuser
Page : 106 pages
File Size : 46,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034883566

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Homotopy Theoretic Methods in Group Cohomology by William G. Dwyer,Hans-Werner Henn Pdf

This book consists essentially of notes which were written for an Advanced Course on Classifying Spaces and Cohomology of Groups. The course took place at the Centre de Recerca Mathematica (CRM) in Bellaterra from May 27 to June 2, 1998 and was part of an emphasis semester on Algebraic Topology. It consisted of two parallel series of 6 lectures of 90 minutes each and was intended as an introduction to new homotopy theoretic methods in group cohomology. The first part of the book is concerned with methods of decomposing the classifying space of a finite group into pieces made of classifying spaces of appropriate subgroups. Such decompositions have been used with great success in the last 10-15 years in the homotopy theory of classifying spaces of compact Lie groups and p-compact groups in the sense of Dwyer and Wilkerson. For simplicity the emphasis here is on finite groups and on homological properties of various decompositions known as centralizer resp. normalizer resp. subgroup decomposition. A unified treatment of the various decompositions is given and the relations between them are explored. This is preceeded by a detailed discussion of basic notions such as classifying spaces, simplicial complexes and homotopy colimits.

Author : Anonim
Publisher : Academic Press
Page : 367 pages
File Size : 50,7 Mb
Release : 1975-11-12
Category : Mathematics
ISBN : 0080873804

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Homotopy Theory: An Introduction to Algebraic Topology by Anonim Pdf

Homotopy Theory: An Introduction to Algebraic Topology

Author : Fabien Morel
Publisher : American Mathematical Soc.
Page : 116 pages
File Size : 52,8 Mb
Release : 2006
Category : Mathematics
ISBN : 082183164X

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Homotopy Theory of Schemes by Fabien Morel Pdf

In this text, the author presents a general framework for applying the standard methods from homotopy theory to the category of smooth schemes over a reasonable base scheme $k$. He defines the homotopy category $h(\mathcal{E} k)$ of smooth $k$-schemes and shows that it plays the same role for smooth $k$-schemes as the classical homotopy category plays for differentiable varieties. It is shown that certain expected properties are satisfied, for example, concerning the algebraic$K$-theory of those schemes. In this way, advanced methods of algebraic topology become available in modern algebraic geometry.

Author : Anonim
Publisher : Academic Press
Page : 346 pages
File Size : 42,5 Mb
Release : 1959-01-01
Category : Mathematics
ISBN : 0080873162

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Homotopy Theory by Anonim Pdf

Homotopy Theory