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.