Localization and genus in group theory and homotopy theory

Abstract

To support development in (1) we combine works of Adams [1], Casacuberta-Peschke-Pfenniger [7] and chapter I of present exposition. Thus we arrive at following pivotal point. The objects obtained by applying functors in .c to X E C can always be organized (by intrinsic (!) category theoretical structure) into a unique commutative diagram. Moreover, this process is functorial and is left adjoint to inverse limit functor (if it exists) restricted to such diagrams. Losely speaking this means that production of local is adjoint to the process of assembling local data to global data. The development in (2) employs (1) and constitutes a continuation of earlier work of Sullivan [16], [17], Bousfield-Kan [5] and Hilton-Mislin-Roitberg [10] info in sense that local---tglobal processes familiar from these sources are here info extended in two directions. Firstly, we show that these classicallocal----tglobal processes admit a milch stronger formulation even within traditional environment of simply connected or nilpotent spaces. Secondly, class of spaces for which these processes are valid is enlarged well beyond nilpotent spaces. For this purpose we use P-Iocalization of spaces based on algebrogeometric property that loop space of a P-Iocal space be a P-Iocal group