Gaucher's question has a negative answer: In any locally finitely presentable category K, (Mor(K), Iso) is a weak factorization system which is cofibrantly generated by morphisms h having finitely presentable domains and codomains. It follows from the fact that a morphism g has the right lifting property with respect to such morphisms h iff it is an isomorphism. This can be found in the dual form in J.Dydak, F.R.Ruiz del Portal, Isomorphisms in pro-categories, J. Pure Appl. Alg. 190 (2004), 85-120, Proposition 3.1. Also, this follows from the fact that such g is both a pure monomorphism and a pure epimorphism and thus both a regular monomorphism and an epimorphism. Thus it suffices to find a weak factorization system (L,R) in a locally finitely presentable category which is not cofibrantly generated. This was done, for example, in J.Adamek, H.Herrlich, J.Rosicky and W.Tholen, Weak factorization systems and topological functors, Appl. Categ. Struct. 10 (2002), 237-249 - take the category of posets and (L,R) such that L consists of embeddings. Jiri Rosicky
Dear All
Let (L1,R1) and (L2,R2) be two weak factorization systems (by weak, it is usually meant that the diagonal map is not unique) such that L1 \subset L2.
Example of such a thing: in a locally presentable category, take two sets of maps I1 and I2 with I1\subset I2, and let (L1,R1)=(cof(I1),inj(I1)) and (L2,R2)=(cof(I2),inj(I2)). Two such wfs are called cofibrantly generated.
Now come back to (L1,R1) and (L2,R2) with L1 \subset L2.
Question: if (L2,R2) is cofibrantly generated, is (L1,R1) cofibrantly generated ? if we work in a locally presentable category, can one say something under Vopenka's principle ?
The only example I have in mind is the wfs (Serre cofibration, trivial Serre fibration) which is cofibrantly generated by {S^{n-1}->D^n} and (Hurewicz cofibration, Hurewicz fibration which are homotopy equivalences) which is probably not cofibrantly generated. Unfortunately, the inclusion is in the wrong direction : {Serre cofibration}\subset {Hurewicz cofibration} since {Hurewicz fibration which are homotopy equivalences} \subset {trivial Serre fibration} so that does not give a counterexample.
Thanks in advance. pg.