The ps-file of the following paper is available via my website at www.math.yorku.ca/Who/Faculty/Tholen/research.html Please do not hesitate contacting me in case of difficulty. Walter. -------------------------------------------------------------------------- "Lax factorization algebras" Jiri Rosicky' and Walter Tholen Abstract: It is shown that many weak factorization systems appearing in functorial Quillen model categories, including all those that are cofibrantly generated, come with a rich computational structure, defined by a certain lax algebra with respect to the "squaring monad" on CAT. This structure largely facilitates natural choices for left or right liftings once certain basic natural choices have been made. The use of lax homomorphisms of such lax algebras is also discussed, with focus on "lax freeness".