A continuous localization and completion

Abstract

The main goal of this paper is to construct a localization and completion of Bousfield-Kan type as a continuous functor for a virtually nilpotent CW-complex. Then the localization and completion of an A n {A_n} -space is given to be an A n {A_n} -homomorphism between A n {A_n} -spaces. For any general compact Lie group, this gives a continuous equivariant localization and completion for a virtually nilpotent G G -CW-complex. More generally, we have a continuous localization with respect to a system of core rings for a virtually nilpotent D \mathbf {D} -CW-complex for a polyhedral category D \mathbf {D} .