The proof can be found in the paper J.Adamek, H.Herrlich, J.Rosicky, W.Tholen, On a generalized small-object argument for the injective subcategory problem, Cah. Top. Geom. Diff. Cat. XLIII (2002), 83-106. ----- Forwarded message from Gaucher Philippe <gaucher@pps.jussieu.fr> -----

Dear category theorists

I would be interested in knowing a proof of the following fact (due to J. Smith):

"In a combinatorial model category M (i.e. a locally presentable cofibrantly generated model category), there are functorial factorizations of a map into a trivial cofibration followed by a fibration which preserve lambda-filtered colimits for sufficiently large regular cardinals lambda. The same is true for the factorizations as a cofibration followed by a trivial fibration."

As far as I know about the proof, it suffices to apply the small object argument step-by-step and then to use some property of lambda-filtered colimits. The only property I know close to the problem is that a lambda-filtered colimits of lambda-presentable objects is lambda-presentable. But the underlying diagram of a pushout is not lambda-filtered. So I dont understand...

Thanks in advance. pg.

----- End forwarded message -----