Subject: model categories
Date: Mon, 21 Apr 2003 22:56:54 -0400 (EDT)
From: Tom Goodwillie
To: dmd1@lehigh.edu (Don Davis)
>
> Subject: about cofibrant objects
> Date: Mon, 21 Apr 2003 15:18:20 +0200 (MEST)
> From: Philippe Gaucher
>
> Hello [question for the mailing list]
>
> I would be interested in knowing examples of model categories whose
> terminal object is not cofibrant.
>
> Thanks in advance. pg.
If C is a model category and f:A->B is a morphism of C then consider the
category of "objects between A and B": Objects are factorizations
A->X->B of f, and maps (A->X->B)->(A->Y->B) are commutative diagrams
A -> X -> B
1 | | | 1
V V V
A -> Y -> B
This is a model category, with cofibrations (resp. fibrations
resp. weak equivalences) being maps such that the underlying map in C
is a cofibration (resp. fib resp. wk eq).
If f is not a cofibration in C, then the final object in this new category
is not cofibrant.
- Tom Goodwillie