EMS Tracts in Mathematics 2011; 703 pp; hardcover Volume: 15 ISBN10: 3037190833 ISBN13: 9783037190838 List Price: US$118 Member Price: US$94.40 Order Code: EMSTM/15
 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. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in algebraic topology. Table of Contents Part I. 1 and 2dimensional results  Introduction to Part I
 History
 Homotopy theory and crossed modules
 Basic algebra of crossed modules
 Coproducts of crossed \(P\)modules
 Induced crossed modules
 Double groupoids and the 2dimensional Seifertvan Kampen Theorem
Part II. Crossed complexes  Introduction to Part II
 The basics of crossed complexes
 The higher homotopy Seifertvan Kampen Theorem (HHSvKT) and its applications
 Tensor products and homotopies of crossed complexes
 Resolutions
 The cubical classifying space of a crossed complex
 Nonabelian cohomology: spaces, groupoids
Part III. Cubical \(\omega\)groupoids  Introduction to Part III
 The algebra of crossed complexes and cubical \(\omega\)groupoids
 The cubical homotopy \(\omega\)groupoid of a filtered space
 Tensor products and homotopies
 Future directions?
 Appendices
 Bibliography
 Glossary of symbols
 Index
