Bousfield–Kan completion of homotopy limits

We consider the commutation of R∞, the Bousfield–Kan R-completion functor, with homotopy (inverse) limits over categories I with compact classifying spaces. We get a generalization of the usual fibre lemma regarding preservation of a fibration sequence by R∞. The basic result is that for such I-diagrams N of nilpotent spaces the canonical commutation mapR∞holimI N →c holimI R∞Nis always a covering projection. This has clear implications for Sullivan–Quillen localization and completion theory and for rational models. On the way we are lead to a sufficient condition for the homotopy limit over a finite diagram to be non-empty or in fact r-connected for a given r⩾−1.