From Categories To Homotopy Theory
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From Categories To Homotopy Theory
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Author : Birgit Richter
language : en
Publisher: Cambridge University Press
Release Date : 2020-04-16
From Categories To Homotopy Theory written by Birgit Richter and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-04-16 with Mathematics categories.
Bridge the gap between category theory and its applications in homotopy theory with this guide for graduate students and researchers.
The Homotopy Theory Of 1 Categories
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Author : Julia E. Bergner
language : en
Publisher: Cambridge University Press
Release Date : 2018-03-15
The Homotopy Theory Of 1 Categories written by Julia E. Bergner and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-03-15 with Mathematics categories.
An introductory treatment to the homotopy theory of homotopical categories, presenting several models and comparisons between them.
Categorical Homotopy Theory
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Author : Emily Riehl
language : en
Publisher: Cambridge University Press
Release Date : 2014-05-26
Categorical Homotopy Theory written by Emily Riehl and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05-26 with Mathematics categories.
This categorical perspective on homotopy theory helps consolidate and simplify one's understanding of derived functors, homotopy limits and colimits, and model categories, among others.
Homotopy Theory Of Higher Categories
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Author : Carlos Simpson
language : en
Publisher: Cambridge University Press
Release Date : 2011-10-20
Homotopy Theory Of Higher Categories written by Carlos Simpson and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-10-20 with Mathematics categories.
The study of higher categories is attracting growing interest for its many applications in topology, algebraic geometry, mathematical physics and category theory. In this highly readable book, Carlos Simpson develops a full set of homotopical algebra techniques and proposes a working theory of higher categories. Starting with a cohesive overview of the many different approaches currently used by researchers, the author proceeds with a detailed exposition of one of the most widely used techniques: the construction of a Cartesian Quillen model structure for higher categories. The fully iterative construction applies to enrichment over any Cartesian model category, and yields model categories for weakly associative n-categories and Segal n-categories. A corollary is the construction of higher functor categories which fit together to form the (n+1)-category of n-categories. The approach uses Tamsamani's definition based on Segal's ideas, iterated as in Pelissier's thesis using modern techniques due to Barwick, Bergner, Lurie and others.
Simplicial Homotopy Theory
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Author : Paul G. Goerss
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-12-05
Simplicial Homotopy Theory written by Paul G. Goerss and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-12-05 with Mathematics categories.
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 book is 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
Categorical Constructions In Stable Homotopy Theory
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Author : Myles Tierney
language : en
Publisher: Springer
Release Date : 2007-01-05
Categorical Constructions In Stable Homotopy Theory written by Myles Tierney and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-01-05 with Mathematics categories.
Homotopy Of Operads And Grothendieck Teichmuller Groups
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Author : Benoit Fresse
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-05-22
Homotopy Of Operads And Grothendieck Teichmuller Groups written by Benoit Fresse and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-05-22 with Grothendieck groups categories.
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.
Homotopy Limit Functors On Model Categories And Homotopical Categories
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Author : William G. Dwyer
language : en
Publisher: American Mathematical Soc.
Release Date : 2004
Homotopy Limit Functors On Model Categories And Homotopical Categories written by William G. Dwyer and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Mathematics categories.
The purpose of this monograph, which is aimed at the graduate level and beyond, is to obtain a deeper understanding of Quillen's model categories. A model category is a category together with three distinguished classes of maps, called weak equivalences, cofibrations, and fibrations. Model categories have become a standard tool in algebraic topology and homological algebra and, increasingly, in other fields where homotopy theoretic ideas are becoming important, such as algebraic $K$-theory and algebraic geometry. The authors' approach is to define the notion of a homotopical category, which is more general than that of a model category, and to consider model categories as special cases of this. A homotopical category is a category with only a single distinguished class of maps, called weak equivalences, subject to an appropriate axiom. This enables one to define ``homotopical'' versions of such basic categorical notions as initial and terminal objects, colimit and limit functors, cocompleteness and completeness, adjunctions, Kan extensions, and universal properties. There are two essentially self-contained parts, and part II logically precedes part I. Part II defines and develops the notion of a homotopical category and can be considered as the beginnings of a kind of ``relative'' category theory. The results of part II are used in part I to obtain a deeper understanding of model categories. The authors show in particular that model categories are homotopically cocomplete and complete in a sense stronger than just the requirement of the existence of small homotopy colimit and limit functors. A reader of part II is assumed to have only some familiarity with the above-mentioned categorical notions. Those who read part I, and especially its introductory chapter, should also know something about model categories.
Model Categories And Their Localizations
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Author : Philip S. Hirschhorn
language : en
Publisher: American Mathematical Soc.
Release Date : 2009-08-24
Model Categories And Their Localizations written by Philip S. Hirschhorn and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-08-24 with Mathematics categories.
From the series that publishes some of the AMS's most distingushed titles, this book stands alone in its class. The authors present a good, detailed introduction to a topic that serves as a standard tool in algebraic topology. It works well as an independent study resource for both students and researchers. A must for bookstores.
Category Theory In Context
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Author : Emily Riehl
language : en
Publisher: Courier Dover Publications
Release Date : 2017-03-09
Category Theory In Context written by Emily Riehl and has been published by Courier Dover Publications this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-09 with Mathematics categories.
Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.