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Final print in Autumn 2010, 670 pages, ISBN 978-3-03719-083-8 (v01 in 2009, 4 Mb Free download PDF file, beta version-Preprint)

Abstract:

The book will be published in the Autumn of 2010 by the European Mathematical Society Publishing House,ISBN 978-3-03719-083-8; see announcement and the authors' Book Abstract in the EMS Tracts in Mathematics, Vol. 15, attached PDF file--" RonnieBrownbuk_vol15.pdf " (uploaded with permission from Professor Ronald Brown). Ronald Brown, Bangor University, UK,Philip J. Higgins, Durham University, UK Rafael Sivera, University of Valencia, Spain. Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids 2010. EMS Tracts in Mathematics, Vol. 15, 2010, 17 x 24 cm. Approx. 670 pages. =================Authors' Abstract, as a quote:================ ``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 omegagroupoids, 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." The attached 4 Mb file was the first 516-page version of the two-volume book, in its beta version to present in detail non-Abelian Algebraic Topology with emphasis on higher dimensional groupoids. The first part in Volume I consisted of 159 pages divided in six chapters. The first chapter presents the relevant background, including the van Kampen Theorem and its proof for the groupoid case, as well as the definition of higher dimensional groupoids. The second chapter presents Homotopy theory and crossed modules, whereas the following chapter focuses on the basic algebra of crossed modules including the van Kampen diagrams. Chapter six presents Double groupoid, double categories and the 2-dimensional van Kampen Theorem together with its proof published by the first author. Part II in the second volume of this book is entitled "Crossed Complexes" and begins by introducing the basics of crossed complexes: "This first Chapter of Part II gives the background on crossed complexes which is required for the statement and applications of the Higher Homotopy van Kampen Theorem (HHvKT) for the functor \Pi from the category of filtered spaces to the category of crossed complexes." Ch. 8 presents the Higher Homotopy van Kampen Theorem (HHvKT) and its applications, including HHvKT for crossed complexes. The following chapter introduces Tensor products and homotopies of crossed complexes, and Ch.10 continues with the classifying space of a crossed complex. The concluding Ch.12 of Part II presents the Nonabelian cohomology of spaces and of groups. Part III focuses on omega-groupoids and begins with the algebra of crossed complexes and cubical omega-groupoids. Ch.14: The cubical homotopy omega-groupoid of a filtered space; Ch. 15: Tensor products and homotopies ; Ch.16 contains the Conclusion including Problems. Two extensive appendices present respectively, fundamentals of category theory and groupoids, and closed categories, concluding with Crossed modules and quotients of groups in the last section B7 of Appendix B. http://www.bangor.ac.uk/~mas010/pdffiles/rbrsbookb-e040609.pdf http://www.bangor.ac.uk/~mas010/nonab-a-t.html http://www.bangor.ac.uk/~mas010/pdffiles/EMSTracts_vol15.pdf

Rights:

Free downloads of the 516 -page beta version only were available in 2009 at: http://www.bangor.ac.uk/~mas010/pdffiles/rbrsbookb-e040609.pdf Copyright@2007--2010 by Ronald Brown, Philip J. Higgins and Rafael Sivera, ISBN 978-3-03719-083-8 COPYRIGHT NOTICE "This book is copyright. Subject to statutory exception, and the provision of relevant collective licensing arrangements, no reproduction may take place of any part without the written permission of the authors or their agents."

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