Subject: Re: three postings
From: Brian Munson
Date: Wed, 11 Jul 2007 15:07:56 -0400
> Subject: A question about homotopy limits
> From: Johannes Ebert
> Date: Tue, 10 Jul 2007 18:58:51 +0200 (CEST)
>
> I have the following question for the list.
>
> I have a small category C and a functor F: C \to Top
> (topological spaces). I want to know whether the natural map for c \in
> Ob(C)
>
> f_c:holim F \to F(c)
>
> is a homotopy equivalence. In the example I am interested in, the
> classifying space BC is contractible and F maps all arrows in C to weak
> homotopy equivalences. Apart from that, I can assume that all F(c) are
> of the homotopy type of a CW, but nothing else. It sounds reasonable
that
> f_c is a (weak) homotopy equivalence, but is it true?
here is a reply to ebert's question:
if F takes all morphisms to weak equivalences, then hocolim F quasifibers
over BC with fiber F(c) over c, and the space of sections of the
associated fibration is holim F. i seem to remember reading this in a
paper of dwyer's which appeared in some conference proceedings somewhere
(i am pretty sure the paper is titled "the centralizer decomposition of
BG"), but i am unable to track this down at the moment. since BC is
contractible, F(c) is equivalent to hocolim F and the associated section
space is equivalent to F(c). maybe i was sloppy and misapplied the fact,
but it seems like a useful fact nonetheless.
brian