Subject: Re: question and book review Date: Fri, 13 Sep 2002 14:18:44 -0400 From: Tom Goodwillie > >My problem is: let $f,g : X\to Y$ be two maps of $I$-diagrams >such that $f_i$ is homotopic to $g_i$ for $i\in I$. >Is it true that the induced maps by $f$ and $g$ of $hocolim$ >and $holim$ are homotopic? Perhaps under some extra conditions? >Any hints and possible references, please. >I am really looking forward to hearing your >answering. Here's a favorite counterexample: Let A be a nonempty space whose unreduced suspension SA is not contractible. X is the diagram * <- A -> CA Y is the diagram * <- * -> SA (CA is the cone.) Any two maps from X to Y will be "pointwise" homotopic. But one map induces a homotopy equivalence of the hocolims and another induces a map homotopic to a constant. Tom Goodwillie