Subject: about homotopy colimit and Reedy structures From: Philippe Gaucher Date: Mon, 7 Mar 2005 10:10:46 +0100 To: Don Davis Dear Don Here is a question for the mailing-list. Sincerely yours. pg. --------------------------------------------------------------------------------------------------------- Where is it possible to find a proof of the fact that a homotopy colimit of contractible cofibrant topological spaces is contractible ? I ask you the question because I have encountered the following situation : I is a connected Reedy category such that the colimit functor is a left Quillen adjoint. I have a Reedy cofibrant diagram D of contractible cofibrant topological spaces, and I need to prove that colim D is contractible. If the terminal object of Top^I is cofibrant, it's obvious. But the terminal object is not cofibrant in my situation. If I is small, I have a method for "fixing" the diagram in order to make the terminal diagram cofibrant but I am not able to generalize it (by now). It is already known that the fundamendal groupoid of the homotopy colimit is the homotopy colimit of the fundamental groupoids. In Farjoun's paper "Fundamental groups of homotopy colimits", a "folklore" theorem about the preservation of homotopy colimits by pi_0 and C_* (the singular complex ?) is mentioned. Any reference is welcome. Thanks in advance. pg.