Homotopical Algebra
Author: Daniel G. Quillen
Publisher: Springer
Published: 2006-11-14
Total Pages: 165
ISBN-13: 3540355235
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Algebraic Topology from a Homotopical Viewpoint
Author: Marcelo Aguilar
Publisher: Springer Science & Business Media
Published: 2008-02-02
Total Pages: 499
ISBN-13: 0387224890
DOWNLOAD EBOOK →The authors present introductory material in algebraic topology from a novel point of view in using a homotopy-theoretic approach. This carefully written book can be read by any student who knows some topology, providing a useful method to quickly learn this novel homotopy-theoretic point of view of algebraic topology.
Higher Categories and Homotopical Algebra
Author: Denis-Charles Cisinski
Publisher: Cambridge University Press
Published: 2019-05-02
Total Pages: 449
ISBN-13: 1108473202
DOWNLOAD EBOOK →At last, a friendly introduction to modern homotopy theory after Joyal and Lurie, reaching advanced tools and starting from scratch.
Abstract Homotopy And Simple Homotopy Theory
Author: K Heiner Kamps
Publisher: World Scientific
Published: 1997-04-11
Total Pages: 476
ISBN-13: 9814502553
DOWNLOAD EBOOK →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).
Categorical Homotopy Theory
Author: Emily Riehl
Publisher: Cambridge University Press
Published: 2014-05-26
Total Pages: 371
ISBN-13: 1139952633
DOWNLOAD EBOOK →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.
Higher Categories and Homotopical Algebra
Author: Denis-Charles Cisinski
Publisher: Cambridge University Press
Published: 2019-05-02
Total Pages: 450
ISBN-13: 1108643477
DOWNLOAD EBOOK →This book provides an introduction to modern homotopy theory through the lens of higher categories after Joyal and Lurie, giving access to methods used at the forefront of research in algebraic topology and algebraic geometry in the twenty-first century. The text starts from scratch - revisiting results from classical homotopy theory such as Serre's long exact sequence, Quillen's theorems A and B, Grothendieck's smooth/proper base change formulas, and the construction of the Kan–Quillen model structure on simplicial sets - and develops an alternative to a significant part of Lurie's definitive reference Higher Topos Theory, with new constructions and proofs, in particular, the Yoneda Lemma and Kan extensions. The strong emphasis on homotopical algebra provides clear insights into classical constructions such as calculus of fractions, homotopy limits and derived functors. For graduate students and researchers from neighbouring fields, this book is a user-friendly guide to advanced tools that the theory provides for application.
Homotopical Algebraic Geometry II: Geometric Stacks and Applications
Author: Bertrand Toën
Publisher: American Mathematical Soc.
Published: 2008
Total Pages: 242
ISBN-13: 0821840991
DOWNLOAD EBOOK →This is the second part of a series of papers called "HAG", devoted to developing the foundations of homotopical algebraic geometry. The authors start by defining and studying generalizations of standard notions of linear algebra in an abstract monoidal model category, such as derivations, étale and smooth morphisms, flat and projective modules, etc. They then use their theory of stacks over model categories to define a general notion of geometric stack over a base symmetric monoidal model category $C$, and prove that this notion satisfies the expected properties.
Motivic Homotopy Theory
Author: Bjorn Ian Dundas
Publisher: Springer Science & Business Media
Published: 2007-07-11
Total Pages: 228
ISBN-13: 3540458972
DOWNLOAD EBOOK →This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Aimed at graduate students in algebraic topology and algebraic geometry, it contains background material from both of these fields, as well as the foundations of motivic homotopy theory. It will serve as a good introduction as well as a convenient reference for a broad group of mathematicians to this important and fascinating new subject. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work, and the other authors have all done important work in the subject.
Homotopy Type Theory: Univalent Foundations of Mathematics
Modern Classical Homotopy Theory
Author: Jeffrey Strom
Publisher: American Mathematical Society
Published: 2023-01-19
Total Pages: 862
ISBN-13: 1470471639
DOWNLOAD EBOOK →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.
