Cofibrance and Completion
  For a cofibrantly generated Quillen model category, we show that the cofibrant replacement functor constructed using the small object argument admits a cotriple structure. If all acyclic cofibrations are monomorphisms, the fibrant replacement functor constructed using the small object argument admits a triple structure. For a triple in the base category, the associated cosimplicial resolution is not necessarily homotopy invariant. However using a mix of the triple with the cofibrant replacement cotriple we construct a 'homotopically correct' version of the cosimplicial resolution of the triple. This allows us to construct a Bousfield-Kan completion functor with respect to a triple, and for pointed cofibrantly-generated model categories a Bousfield-Kan spectral sequence that computes the relative homotopy groups of the Bousfield-Kan completion of an object.   This is the text of my PhD thesis, worked under the supervision of Prof. Haynes Miller, submitted on Feb. 1999 at MIT. 
Andrei Radulescu-Banu