Suppose C is a category and T is a monad on C. One knows that one can factor T into a composite GF, where F,G are an adjoint pair of functors, and in fact one knows that there are two universal ways to do this, a Kleisli/initial construction and an Eilenberg-Moore/terminal construction. Now suppose C is a model category and T is a monad on C which preserves weak equivalences. One would like to know that T factors as GF, where F,G are a Quillen pair. Is this always possible and does one have Kleisli-like and Eilenberg-Moore-like constructions with appropriate universal properties? I am sure people have worked on these questions before; where can I read about this? Thanks, Andrew S. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]