Subject: Gaucher's question
From: Tom Goodwillie <tomg@math.brown.edu>
Date: Wed, 17 Jan 2007 11:28:27 -0500

Question: If M is a cofibrantly generated model category with generating 
cofibrations {f:Ai-->Bi}, is it necessarily true that every object of M is 
weakly equivalently to the hocolim of some diagram of objects Bi ?

Answer: No.

Strategy for making a counterexample: Give an example of a cofibrantly 
generated model category in which the codomain of every generating 
cofibration is weakly equivalent to the initial object but not every 
object is weakly equivalent to the initial object. (That's enough, right, 
because hocolim of objects equivalent to initial is equivalent to 
initial?)

Example: The category of all chain complexes of k-vectorspaces, with 
equivalence=quasiisomorphism and fibration=epimorphism. The usual 
generating cofibrations are the inclusions {S^{n-1}-->D^n}, for all 
integers n, where these "algebraic spheres and disks" look like (k<--0) 
and (k<--k).

Comment: These chain complexes were Z-graded. The example would not work 
if we said (Z+)-graded. So maybe "Reedy" has something to do with it.

    Tom Goodwillie
> ____________________________________________________________
>
> Subject: cofibrant replacement as a functorial homotopy colimit ?
> From: Gaucher Philippe <Philippe.Gaucher@pps.jussieu.fr>
> Date: Mon, 15 Jan 2007 13:38:31 +0100
>
> Dear All
>
> On the category of compactly generated topological spaces equipped with 
the
> usual model structure, the cofibrant replacement Q(X) of a space X can 
be
> seen as a colimit of D^n. And moreover, with the degree function 
d(D^n)=n,
> the index category can be made so that it becomes equipped with a 
structure
> of direct Reedy category. So the colimit is also a homotopy colimit.
>
> Is this fact generalizable for other cofibrantly generated model 
categories ? More precisely, if I={f:Ai-->Bi} is the set of generating 
cofibrations and if
> X is an object, and if B is the full small subcategory generated by the 
Bi,
> what is hocolim Bi where the homotopy colimit is calculated over the 
comma
> category (B|X) ?
>
> In the category of simplicial sets B is the fullsubcategory generated by 
the
> simplices Delta[n]. And hocolim Bi is equal to X.
>
> pg.