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