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Author : Brandon Doherty
Publisher : American Mathematical Society
ISBN 13 : 1470468948
Total Pages : 122 pages
Book Rating : 4.41/5 ( download)
Download or read book Cubical Models of $(infty ,1)$-Categories written by Brandon Doherty and published by American Mathematical Society. This book was released on 2024-06-07 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
Author : Tom Leinster
Publisher : Cambridge University Press
ISBN 13 : 0521532159
Total Pages : 451 pages
Book Rating : 4.50/5 ( download)
Download or read book Higher Operads, Higher Categories written by Tom Leinster and published by Cambridge University Press. This book was released on 2004-07-22 with total page 451 pages. Available in PDF, EPUB and Kindle. Book excerpt: Foundations of higher dimensional category theory for graduate students and researchers in mathematics and mathematical physics.
Author : Ronald Brown
Publisher : JP Medical Ltd
ISBN 13 : 9783037190838
Total Pages : 714 pages
Book Rating : 4.33/5 ( download)
Download or read book Nonabelian Algebraic Topology written by Ronald Brown and published by JP Medical Ltd. This book was released on 2011 with total page 714 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main theme of this book is that the use of filtered spaces rather than just topological spaces allows the development of basic algebraic topology in terms of higher homotopy groupoids; these algebraic structures better reflect the geometry of subdivision and composition than those commonly in use. Exploration of these uses of higher dimensional versions of groupoids has been largely the work of the first two authors since the mid 1960s. The structure of the book is intended to make it useful to a wide class of students and researchers for learning and evaluating these methods, primarily in algebraic topology but also in higher category theory and its applications in analogous areas of mathematics, physics, and computer science. Part I explains the intuitions and theory in dimensions 1 and 2, with many figures and diagrams, and a detailed account of the theory of crossed modules. Part II develops the applications of crossed complexes. The engine driving these applications is the work of Part III on cubical $\omega$-groupoids, their relations to crossed complexes, and their homotopically defined examples for filtered spaces. Part III also includes a chapter suggesting further directions and problems, and three appendices give accounts of some relevant aspects of category theory. Endnotes for each chapter give further history and references.
Author : Emily Riehl
Publisher : Cambridge University Press
ISBN 13 : 1108952194
Total Pages : 782 pages
Book Rating : 4.94/5 ( download)
Download or read book Elements of ∞-Category Theory written by Emily Riehl and published by Cambridge University Press. This book was released on 2022-02-10 with total page 782 pages. Available in PDF, EPUB and Kindle. Book excerpt: The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.
Author : Denis-Charles Cisinski
Publisher : Cambridge University Press
ISBN 13 : 1108473202
Total Pages : 449 pages
Book Rating : 4.00/5 ( download)
Download or read book Higher Categories and Homotopical Algebra written by Denis-Charles Cisinski and published by Cambridge University Press. This book was released on 2019-05-02 with total page 449 pages. Available in PDF, EPUB and Kindle. Book excerpt: At last, a friendly introduction to modern homotopy theory after Joyal and Lurie, reaching advanced tools and starting from scratch.
Author : Philip S. Hirschhorn
Publisher : American Mathematical Soc.
ISBN 13 : 0821849174
Total Pages : 482 pages
Book Rating : 4.70/5 ( download)
Download or read book Model Categories and Their Localizations written by Philip S. Hirschhorn and published by American Mathematical Soc.. This book was released on 2003 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to explain modern homotopy theory in a manner accessible to graduate students yet structured so that experts can skip over numerous linear developments to quickly reach the topics of their interest. Homotopy theory arises from choosing a class of maps, called weak equivalences, and then passing to the homotopy category by localizing with respect to the weak equivalences, i.e., by creating a new category in which the weak equivalences are isomorphisms. Quillen defined a model category to be a category together with a class of weak equivalences and additional structure useful for describing the homotopy category in terms of the original category. This allows you to make constructions analogous to those used to study the homotopy theory of topological spaces. A model category has a class of maps called weak equivalences plus two other classes of maps, called cofibrations and fibrations. Quillen's axioms ensure that the homotopy category exists and that the cofibrations and fibrations have extension and lifting properties similar to those of cofibration and fibration maps of topological spaces. During the past several decades the language of model categories has become standard in many areas of algebraic topology, and it is increasingly being used in other fields where homotopy theoretic ideas are becoming important, including modern algebraic $K$-theory and algebraic geometry. All these subjects and more are discussed in the book, beginning with the basic definitions and giving complete arguments in order to make the motivations and proofs accessible to the novice. The book is intended for graduate students and research mathematicians working in homotopy theory and related areas.
Author : J. P. May
Publisher : University of Chicago Press
ISBN 13 : 9780226511832
Total Pages : 262 pages
Book Rating : 4.39/5 ( download)
Download or read book A Concise Course in Algebraic Topology written by J. P. May and published by University of Chicago Press. This book was released on 1999-09 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field.
Author : Bertrand Toën
Publisher : American Mathematical Soc.
ISBN 13 : 0821840991
Total Pages : 242 pages
Book Rating : 4.93/5 ( download)
Download or read book Homotopical Algebraic Geometry II: Geometric Stacks and Applications written by Bertrand Toën and published by American Mathematical Soc.. This book was released on 2008 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : Julia E. Bergner
Publisher : Cambridge University Press
ISBN 13 : 1108565042
Total Pages : 290 pages
Book Rating : 4.42/5 ( download)
Download or read book The Homotopy Theory of (∞,1)-Categories written by Julia E. Bergner and published by Cambridge University Press. This book was released on 2018-03-15 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: The notion of an (∞,1)-category has become widely used in homotopy theory, category theory, and in a number of applications. There are many different approaches to this structure, all of them equivalent, and each with its corresponding homotopy theory. This book provides a relatively self-contained source of the definitions of the different models, the model structure (homotopy theory) of each, and the equivalences between the models. While most of the current literature focusses on how to extend category theory in this context, and centers in particular on the quasi-category model, this book offers a balanced treatment of the appropriate model structures for simplicial categories, Segal categories, complete Segal spaces, quasi-categories, and relative categories, all from a homotopy-theoretic perspective. Introductory chapters provide background in both homotopy and category theory and contain many references to the literature, thus making the book accessible to graduates and to researchers in related areas.
Author : Emily Riehl
Publisher : Courier Dover Publications
ISBN 13 : 0486820807
Total Pages : 273 pages
Book Rating : 4.04/5 ( download)
Download or read book Category Theory in Context written by Emily Riehl and published by Courier Dover Publications. This book was released on 2017-03-09 with total page 273 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
