Cubical Homotopy Theory

Cubical Homotopy Theory
Author: Brian A. Munson
Publisher: Cambridge University Press
Total Pages: 649
Release: 2015-10-06
Genre: Mathematics
ISBN: 1107030250

A modern, example-driven introduction to cubical diagrams and related topics such as homotopy limits and cosimplicial spaces.

Nonabelian Algebraic Topology

Nonabelian Algebraic Topology
Author: Ronald Brown
Publisher: JP Medical Ltd
Total Pages: 714
Release: 2011
Genre: Algebraic topology
ISBN: 9783037190838

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.

From Categories to Homotopy Theory

From Categories to Homotopy Theory
Author: Birgit Richter
Publisher: Cambridge University Press
Total Pages: 402
Release: 2020-04-16
Genre: Mathematics
ISBN: 1108847625

Category theory provides structure for the mathematical world and is seen everywhere in modern mathematics. With this book, the author bridges the gap between pure category theory and its numerous applications in homotopy theory, providing the necessary background information to make the subject accessible to graduate students or researchers with a background in algebraic topology and algebra. The reader is first introduced to category theory, starting with basic definitions and concepts before progressing to more advanced themes. Concrete examples and exercises illustrate the topics, ranging from colimits to constructions such as the Day convolution product. Part II covers important applications of category theory, giving a thorough introduction to simplicial objects including an account of quasi-categories and Segal sets. Diagram categories play a central role throughout the book, giving rise to models of iterated loop spaces, and feature prominently in functor homology and homology of small categories.

Homotopy Theory of C*-Algebras

Homotopy Theory of C*-Algebras
Author: Paul Arne Østvær
Publisher: Springer Science & Business Media
Total Pages: 142
Release: 2010-09-08
Genre: Mathematics
ISBN: 303460565X

Homotopy theory and C* algebras are central topics in contemporary mathematics. This book introduces a modern homotopy theory for C*-algebras. One basic idea of the setup is to merge C*-algebras and spaces studied in algebraic topology into one category comprising C*-spaces. These objects are suitable fodder for standard homotopy theoretic moves, leading to unstable and stable model structures. With the foundations in place one is led to natural definitions of invariants for C*-spaces such as homology and cohomology theories, K-theory and zeta-functions. The text is largely self-contained. It serves a wide audience of graduate students and researchers interested in C*-algebras, homotopy theory and applications.

Cubical Homotopy Theory

Cubical Homotopy Theory
Author: Brian A. Munson
Publisher:
Total Pages:
Release: 2015
Genre: Algebraic topology
ISBN: 9781139343329

Abstract Homotopy And Simple Homotopy Theory

Abstract Homotopy And Simple Homotopy Theory
Author: K Heiner Kamps
Publisher: World Scientific
Total Pages: 476
Release: 1997-04-11
Genre: Mathematics
ISBN: 9814502553

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).

Abstract Homotopy and Simple Homotopy Theory

Abstract Homotopy and Simple Homotopy Theory
Author: Klaus Heiner Kamps
Publisher: World Scientific
Total Pages: 474
Release: 1997
Genre: Mathematics
ISBN: 9789810216023

"This book provides a thorough and well-written guide to abstract homotopy theory. It could well serve as a graduate text in this topic, or could be studied independently by someone with a background in basic algebra, topology, and category theory."