Subject: Gaucher's question
From: Tom Goodwillie
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
> 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.