Two responses to yesterday's question...........DMD
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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
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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