Two responses to yesterday's question...........DMD ______________________________________________________ Subject: Re: two postings Date: Thu, 23 Oct 2003 16:45:06 +0200 From: Rainer Vogt > Subject: square of fibrations in a model category > Date: Wed, 22 Oct 2003 17:45:16 +0200 (CEST) > From: Philippe Gaucher > > Question for the mailing list: > > I have a commutative square of fibrations with both horizontal morphisms > > which are weak equivalences. So I have something like that : > > B----->C > ^ ^ > | | > | | > A----->D > > In my case, I can prove that the morphism A --> B \times_{C} D is then a > > fibration. My question is: is it general for any model category ? > > pg. The answer is NO: if it were true, the dual situation would be true too by passage to the opposite category. Now consider the model structure on TOP discovered by Strom: fibrations are Hurewicz fibrations, cofibrations are the closed cofibrations and weak equivalences are the genuine homotopy equivalences. Consider the diagram * ----> B | | | | v v B ----> B from the 1-point space to a ball. All maps of the diagram are cofibrations and weak equivalences, but the map from the pushout to B is not a cofibration, because it is not injective. Rainer Vogt ______________________________________________________________ Date: Thu, 23 Oct 2003 12:09:11 -0400 From: Randall Helmstutler I have a response to the recent query of Gaucher: This is true in any "right proper" model category, i.e. one in which the pullback of a weak equivalence along a fibration is a weak equivalence. In your notation, you get the desired conclusion assuming the horizontal maps are weak equivalences and that the map D ---> C is a fibration (so you don't even need the others to be fibrations). The proof is then automatic by the 2-out-of-3 axiom. --Randall Helmstutler