Lax factorization algebras

It is shown that many weak factorization systems appearing in functorialQuillen 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 homomorphisms of such lax algebras is also discussed, with focus on “lax freeness”.