## Homotopy Limits, Completions and LocalizationsThe 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. |

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abelian groups acts nilpotently algebra Artin-Mazur base point Bousfield-Kan central series Ch.I Ch.II Ch.IX Ch.V Ch.X Ch.XI chapter classes of maps cofibration cofinal commutative connected fibre convergence cosimplicial space define denote direct limits Ext-p-complete Ext(Z Q fibre lemma fracture lemma function space functor H-space hence holim Hom(Z Q homology homotopy category homotopy direct limits homotopy equivalence homotopy groups homotopy inverse limits homotopy spectral sequence homotopy theory homotopy type implies induced map induces an isomorphism integers let f map f natural isomorphisms natural map nilpotent fibration nilpotent group nilpotent spaces notion obvious map pointed sets principal fibration pro-isomorphism pro-trivial Proof Proposition prove Quillen R C Q R-good R-module R-nilpotent ring short exact sequence simplicial group simplicial replacement simplicial set simply connected spaces small category total space tower lemma tower of fibrations weak equivalence