Subject: holimits and hocolim
Date: Fri, 13 Sep 2002 09:39:36 +0200 (CEST)
From: Marek Golasinski <marek@mat.uni.torun.pl>
To: Don Davis <dmd1@lehigh.edu>

Dear Discussion List,
I would greatly appreciate getting your possible
comments on the following problem.

Let $I$ be a small category and $f : X\to Y$
a map of $I$-diagrams $X$ and $Y$ of simplicial sets.
It is known that the induced maps  $hocolim X\to hocolim Y$
 (resp. $holim X\to holim Y$) are weak equivalences provided
$f_i : X_i\to Y_i$ for $i\in I$ are weak equivalences of cofibrant
(resp. fibrant) objects.

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.
Marek G.