re: question about lambda-filtered colimits

re: question about lambda-filtered colimits

[Jiri Rosicky]

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 -----

question about lambda-filtered colimits

[Gaucher Philippe]

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.