The main purpose of part I of these notes is to develop for a ring R a functional notion of R-completion of a space X. For R=Zp and X subject to usual finiteness condition, the R-completion coincides up to homotopy, with the p-profinite ...

Author: A. K. Bousfield

Publisher: Springer

ISBN: 9783540381174

Category: Mathematics

Page: 352

View: 903

The main purpose of part I of these notes is to develop for a ring R a functional notion of R-completion of a space X. For R=Zp and X subject to usual finiteness condition, the R-completion coincides up to homotopy, with the p-profinite completion of Quillen and Sullivan; for R a subring of the rationals, the R-completion coincides up to homotopy, with the localizations of Quillen, Sullivan and others. In part II of these notes, the authors have assembled some results on towers of fibrations, cosimplicial spaces and homotopy limits which were needed in the discussions of part I, but which are of some interest in themselves.

The main purpose of part I of these notes is to develop for a ring R a functional notion of R-completion of a space X. For R=Zp and X subject to usual finiteness condition, the R-completion coincides up to homotopy, with the p-profinite ...

The main purpose of part I of these notes is to develop for a ring R a functional notion of R-completion of a space X. For R=Zp and X subject to usual finiteness condition, the R-completion coincides up to homotopy, with the p-profinite ...

( BK72 ] A. K. Bousfield and D. M. Kan , Homotopy limits , completions and localizations , Lect . Notes in Math . , vol . 304 , Springer - Verlag , New York , 1972 . ( DK80a ] W. G. Dwyer and D. M. Kan , Calculating simplicial ...

Author: William G. Dwyer

Publisher: American Mathematical Soc.

ISBN: 0821839756

Category: Mathematics

Page: 181

View: 201

The purpose of this monograph, which is aimed at the graduate level and beyond, is to obtain a deeper understanding of Quillen's model categories. A model category is a category together with three distinguished classes of maps, called weak equivalences, cofibrations, and fibrations. Model categories have become a standard tool in algebraic topology and homological algebra and, increasingly, in other fields where homotopy theoretic ideas are becoming important, such as algebraic $K$-theory and algebraic geometry. The authors' approach is to define the notion of a homotopical category, which is more general than that of a model category, and to consider model categories as special cases of this. A homotopical category is a category with only a single distinguished class of maps, called weak equivalences, subject to an appropriate axiom. This enables one to define ``homotopical'' versions of such basic categorical notions as initial and terminal objects, colimit and limit functors, cocompleteness and completeness, adjunctions, Kan extensions, and universal properties. There are two essentially self-contained parts, and part II logically precedes part I. Part II defines and develops the notion of a homotopical category and can be considered as the beginnings of a kind of ``relative'' category theory. The results of part II are used in part I to obtain a deeper understanding of model categories. The authors show in particular that model categories are homotopically cocomplete and complete in a sense stronger than just the requirement of the existence of small homotopy colimit and limit functors. A reader of part II is assumed to have only some familiarity with the above-mentioned categorical notions. Those who read part I, and especially its introductory chapter, should also know something about model categories.

[Aml A. Amit, Direct limits over categories with contractible nerve, Master Thesis, The Hebrew University of ... [B-K] A. K. Bousfield and D. M. Kan, Homotopy Limit, Completions, and Localizations, Springer Lecture Notes in Math.

Author: Emmanuel D. Farjoun

Publisher: Springer

ISBN: 9783540484493

Category: Mathematics

Page: 206

View: 465

In this monograph we give an exposition of some recent development in homotopy theory. It relates to advances in periodicity in homotopy localization and in cellular spaces. The notion of homotopy localization is treated quite generally and encompasses all the known idempotent homotopy functors. It is applied to K-theory localizations, to Morava-theories, to Hopkins-Smith theory of types. The method of homotopy colimits is used heavily. It is written with an advanced graduate student in topology and research homotopy theorist in mind.

[5] A. K. Bousfield and D. M. Kan. Homotopy limits, completions and localizations. SpringerVerlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304. [6] W. Chachólski and J. Scherer. Homotopy theory of diagrams. Mem. Amer. Math.

Author: Dominique Arlettaz

Publisher: American Mathematical Soc.

ISBN: 9780821836965

Category: Mathematics

Page: 209

View: 890

The second Arolla conference on algebraic topology brought together specialists covering a wide range of homotopy theory and $K$-theory. These proceedings reflect both the variety of talks given at the conference and the diversity of promising research directions in homotopy theory. The articles contained in this volume include significant contributions to classical unstable homotopy theory, model category theory, equivariant homotopy theory, and the homotopy theory of fusion systems, as well as to $K$-theory of both local fields and $C^*$-algebras.

Proceedings of the NATO Advanced Study Institute on Axiomatic, Enriched and Motivic Homotopy Theory Cambridge, ... [13] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Springer-Verlag, Berlin, 1972, ...

Author: John Greenlees

Publisher: Springer Science & Business Media

ISBN: 9789400709485

Category: Mathematics

Page: 392

View: 641

The NATO Advanced Study Institute "Axiomatic, enriched and rna tivic homotopy theory" took place at the Isaac Newton Institute of Mathematical Sciences, Cambridge, England during 9-20 September 2002. The Directors were J.P.C.Greenlees and I.Zhukov; the other or ganizers were P.G.Goerss, F.Morel, J.F.Jardine and V.P.Snaith. The title describes the content well, and both the event and the contents of the present volume reflect recent remarkable successes in model categor ies, structured ring spectra and homotopy theory of algebraic geometry. The ASI took the form of a series of 15 minicourses and a few extra lectures, and was designed to provide background, and to bring the par ticipants up to date with developments. The present volume is based on a number of the lectures given during the workshop. The ASI was the opening workshop of the four month programme "New Contexts for Stable Homotopy Theory" which explored several themes in greater depth. I am grateful to the Isaac Newton Institute for providing such an ideal venue, the NATO Science Committee for their funding, and to all the speakers at the conference, whether or not they were able to contribute to the present volume. All contributions were refereed, and I thank the authors and referees for their efforts to fit in with the tight schedule. Finally, I would like to thank my coorganizers and all the staff at the Institute for making the ASI run so smoothly. J.P.C.GREENLEES.

Author: Philip S. HirschhornPublish On: 2009-08-24

... and bisimplicial sets, Geometric Applications of Homotopy Theory II, Lect. Notes in Math., vol. 658, SpringerVerlag, New York, 1978, pp. 80–130. A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lect.

Author: Philip S. Hirschhorn

Publisher: American Mathematical Soc.

ISBN: 9780821849170

Category: Mathematics

Page: 457

View: 288

From the series that publishes some of the AMS's most distingushed titles, this book stands alone in its class. The authors present a good, detailed introduction to a topic that serves as a standard tool in algebraic topology. It works well as an independent study resource for both students and researchers. A must for bookstores.

A. K. Bousfield and D. M. Kan, Homotopy with respect to a ring, Proc. Symp. Pure Math. Amer. Math. Soc. 22 (1971), 59-64. , Localization and completion in homotopy theory, Bull. Amer. Math. Soc. 77 (1971), 1006–1010. , Homotopy limits, ...

Author: Peter Hilton

Publisher: Elsevier

ISBN: 9781483258744

Category: Mathematics

Page: 166

View: 707

North-Holland Mathematics Studies, 15: Localization of Nilpotent Groups and Spaces focuses on the application of localization methods to nilpotent groups and spaces. The book first discusses the localization of nilpotent groups, including localization theory of nilpotent groups, properties of localization in N, further properties of localization, actions of a nilpotent group on an abelian group, and generalized Serre classes of groups. The book then examines homotopy types, as well as mixing of homotopy types, localizing H-spaces, main (pullback) theorem, quasifinite nilpotent spaces, localization of nilpotent complexes, and nilpotent spaces. The manuscript takes a look at the applications of localization theory, including genus and H-spaces, finite H-spaces, and non-cancellation phenomena. The publication is a vital source of data for mathematicians and researchers interested in the localization of nilpotent groups and spaces.

Proceedings of a Conference on Homotopy Theory, March 23-27, 1997, Northwestern University Mark E. Mahowald, ... A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics 304, ...

Author: Mark E. Mahowald

Publisher: American Mathematical Soc.

ISBN: 9780821808054

Category: Mathematics

Page: 379

View: 592

The academic year 1996-97 was designated as a special year in Algebraic Topology at Northwestern University (Evanston, IL). In addition to guest lecturers and special courses, an international conference was held entitled ""Current trends in algebraic topology with applications to algebraic geometry and physics"". The series of plenary lectures included in this volume indicate the great breadth of the conference and the lively interaction that took place among various areas of mathematics. Original research papers were submitted, and all submissions were refereed to the usual journal standards. It features a paper prepared by C. Rezk on the Hopkins-Miller theorem, and a set of problems presented at a special problem session held at the conference.