The Homotopy Theory Of 1 Categories
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The Homotopy Theory of 1 Categories
Author | : Julia E. Bergner |
Publsiher | : Cambridge University Press |
Total Pages | : 289 |
Release | : 2018-03-15 |
Genre | : Mathematics |
ISBN | : 9781107101364 |
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An introductory treatment to the homotopy theory of homotopical categories, presenting several models and comparisons between them.
From Categories to Homotopy Theory
Author | : Birgit Richter |
Publsiher | : Cambridge University Press |
Total Pages | : 401 |
Release | : 2020-04-16 |
Genre | : Mathematics |
ISBN | : 9781108479622 |
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Bridge the gap between category theory and its applications in homotopy theory with this guide for graduate students and researchers.
Categorical Homotopy Theory
Author | : Emily Riehl |
Publsiher | : Cambridge University Press |
Total Pages | : 371 |
Release | : 2014-05-26 |
Genre | : Mathematics |
ISBN | : 9781107048454 |
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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 Type Theory Univalent Foundations of Mathematics
Author | : Anonim |
Publsiher | : Univalent Foundations |
Total Pages | : 484 |
Release | : 2024 |
Genre | : Electronic Book |
ISBN | : 9182736450XXX |
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Simplicial Homotopy Theory
Author | : Paul G. Goerss,John F. Jardine |
Publsiher | : Birkhäuser |
Total Pages | : 520 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 9783034887076 |
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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.
Homotopy Theory of Higher Categories
Author | : Carlos Simpson |
Publsiher | : Cambridge University Press |
Total Pages | : 653 |
Release | : 2011-10-20 |
Genre | : Mathematics |
ISBN | : 9781139502191 |
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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.
Towards Higher Categories
Author | : John C. Baez,J. Peter May |
Publsiher | : Springer Science & Business Media |
Total Pages | : 292 |
Release | : 2009-09-24 |
Genre | : Algebra |
ISBN | : 9781441915368 |
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The purpose of this book is to give background for those who would like to delve into some higher category theory. It is not a primer on higher category theory itself. It begins with a paper by John Baez and Michael Shulman which explores informally, by analogy and direct connection, how cohomology and other tools of algebraic topology are seen through the eyes of n-category theory. The idea is to give some of the motivations behind this subject. There are then two survey articles, by Julie Bergner and Simona Paoli, about (infinity,1) categories and about the algebraic modelling of homotopy n-types. These are areas that are particularly well understood, and where a fully integrated theory exists. The main focus of the book is on the richness to be found in the theory of bicategories, which gives the essential starting point towards the understanding of higher categorical structures. An article by Stephen Lack gives a thorough, but informal, guide to this theory. A paper by Larry Breen on the theory of gerbes shows how such categorical structures appear in differential geometry. This book is dedicated to Max Kelly, the founder of the Australian school of category theory, and an historical paper by Ross Street describes its development.
Modern Classical Homotopy Theory
Author | : Jeffrey Strom |
Publsiher | : American Mathematical Society |
Total Pages | : 862 |
Release | : 2023-01-19 |
Genre | : Mathematics |
ISBN | : 9781470471637 |
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The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.