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.