Homotopy Limits of Triples

Given a triple J on the category of (pointed) spaces, one uses the cosimplicial. resolution J . X of a space X, to define the functors J(n)X = Tot(n) J.X. When n = infinity this is known as the completion functor.

We show that when J is a module triple, then the Bousfield-Kan functors J(n) are triples on the homotopy category of spaces. In particular, when E is the spectrum of an S-algebra (or a symmetric spectrum), then the E-completion functor is up to homotopy a triple. (C) 2002 Elsevier Science B.V. All rights reserved.